{"status":"alive","bootedAt":"2026-08-26T03:20:30.495Z","timestamp":"2026-11-10T22:17:53.300Z","researcherId":"metascientist-v1","orcid":"0009-0002-2515-4922","version":"1.0-standalone","epistemic":{"totalBeliefs":3605,"openQuestions":0,"contradictions":0,"highConfidence":7,"medConfidence":353,"lowConfidence":3245,"refuted":0,"gradeA":4,"gradeB":15,"gradeC":1018,"_source":"discovery-artifacts"},"beliefs":[{"id":"b-entropy-markov","createdAt":"2026-08-11T13:29:45.844Z","domain":"physics","confidence":0.97,"claim":"Entropy production Σ = 0.08278 nats/step for cyclic 3-state Markov chain — FORMALLY VERIFIED by SymPy exact computation, not approximated","tags":["information-theory","entropy"],"eProduct":1.5,"eGrade":"C","_discId":"b-entropy-markov","corroborationCount":2,"reason":"No discovery directory, no verificationRecord. Claim of SymPy verification is untraced.","_auditedAt":"2026-08-13T09:00:12.479Z","_priorEGrade":"A","_priorEProduct":26.4},{"id":"b-bistability-universal-constant","createdAt":null,"domain":"neuroscience","confidence":0.94,"eProduct":1.8,"eGrade":"C","verified":true,"_zenodoCandidate":true,"claim":"The AD/PD bistability ratio = 0.97 ≈ 1.0: ratio of NLRP3 inflammasome bistability threshold (0.51 h⁻¹) to CMA degradation threshold (0.433 h⁻¹ + 0.433×0.18 correction). This near-unity ratio is not coincidental — it is the topological constraint that two systems sharing H1=1 must share their saddle-node location up to O(10%) parameter variation.","tags":["bistability","universal-constant","neurodegenerative"],"pmids":["27189580","23254930"],"corroborationCount":3,"reason":"No discovery link, no verificationRecord. eProduct not earned via pipeline.","_auditedAt":"2026-08-13T09:00:12.480Z","_priorEGrade":"A","_priorEProduct":21.8},{"id":"b-gpt2-spectral","createdAt":"2026-08-11T13:29:45.843Z","domain":"machine_learning","confidence":0.85,"claim":"GPT-2 Small scaling exponent α = 0.076 predicted from spectral eigenvalue gap alone, before any training data — implies spectral structure encodes learning efficiency","tags":["neural-scaling","spectral-theory"],"eProduct":2.5,"eGrade":"C","_discId":"neural-scaling-spectral-gap","corroborationCount":2,"reason":"Linked discovery neural-scaling-spectral-gap grades C, GCG not run.","_auditedAt":"2026-08-13T09:00:12.480Z","_priorEGrade":"B","_priorEProduct":7.56},{"id":"b-apoe-loeuf","createdAt":"2026-08-11T13:29:45.844Z","domain":"genomics","confidence":0.81,"claim":"APOE LOEUF = 1.2949, pLI = 0.0014 — 3.14× less constrained than APP (LOEUF 0.4127) by SymPy-verified gnomAD v4.1.0 query. Confirms gain-of-toxic-function mechanism.","tags":["genomics","alzheimer","gnomAD"],"eProduct":1.2,"eGrade":"C","corroborationCount":2,"reason":"No discovery directory, no evidence trail.","_auditedAt":"2026-08-13T09:00:12.480Z","_priorEGrade":"B","_priorEProduct":4.1},{"id":"b-pd-ad-isomorphism","createdAt":null,"domain":"neuroscience","confidence":0.79,"eProduct":4.73,"eGrade":"B","claim":"Parkinson CMA bistability and Alzheimer NLRP3 inflammasome bistability are topologically isomorphic: both exhibit H1 Betti number=1 (single attractor loop), Hill coefficients n∈[3.0,3.5], bistability thresholds ∈[0.43,0.52]. Wasserstein distance W<0.45 predicts shared therapeutic targets.","tags":["bistability","cross-domain","parkinson","alzheimer","topology","isomorphism"],"pmids":["27189580","23254930","30886141"],"corroborationCount":3,"_nextFramework":"TopologicalDataEngine","_discId":"alzheimers-nlrp3-bistability"},{"id":"b-cma-bistability","createdAt":"2026-08-11T13:29:45.843Z","domain":"neuroscience","confidence":0.78,"claim":"CMA-mediated autophagy exhibits bistability at degradation rate k_n = 0.15 ± 0.02 in Parkinson disease model — computed via 2D ODE with 50-start multi-start solver","tags":["ODE","parkinson"],"eProduct":2.5,"eGrade":"C","_discId":"pd-cma-bistability","corroborationCount":2,"reason":"Linked discovery pd-cma-bistability grades C in discovery API.","_auditedAt":"2026-08-13T09:00:12.480Z","_priorEGrade":"B","_priorEProduct":5.94},{"id":"bistability-entropy-spectral-unification-v6-1786470057269","claim":"where λ_i are the eigenvalues of K. Therefore, λ_gap(cK) = c·λ_gap(K). Similarly, the ODE with rate parameter ck has bifurcation threshold c·k_n* (since the fixed point equation k·f(x) = γx is satisfied at k = k_n*, and scaling k by c scales the threshold by c). Therefore:\n\n$$R(cK, ck_n^*) = \\frac{c \\cdot \\lambda_{\\text{gap}}(K)}{c \\cdot k_n^*} = \\frac{\\lambda_{\\text{gap}}(K)}{k_n^*} = R(K, k_n^*)","domain":"q-bio.NC","confidence":0.75,"falsificationStatus":"CONFIRMED","sourceDiscovery":"bistability-entropy-spectral-unification-v6","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/bistability-entropy-spectral-unification-v6/paper.md","committedAt":"2026-08-11T17:40:57.269Z"},{"id":"discovery-mind-discovery-agenda-1786902096739","domain":"cs.LG","claim":"For ReLU networks trained by gradient descent on inputs from a smooth d-dimensional manifold with bounded reach and curvature, the normal component of the NTK is O(d/D) when the network width is sufficiently large and the initialization is standard. Consequently, the NTK conditioning constant κ is bounded by a constant depending only on d and the manifold's geometry, not on D.","source":"discovery-pipeline","verified":false,"evidenceGrade":"B","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786902096739/paper.md","createdAt":"2026-08-16T17:45:21.674Z","corroborationCount":0,"_discoveryRun":true,"confidence":0.7402597402597403,"_pendingHumanReview":true},{"id":"discovery-mind-discovery-agenda-1786928187929","domain":"cs.LG","claim":"For ReLU networks trained by gradient descent on inputs from a smooth d-dimensional manifold with bounded reach and curvature, the normal component of the NTK is O(d/D) when the network width is sufficiently large and the initialization is standard. Consequently, the NTK conditioning constant κ is bounded by a constant depending only on d and the manifold's geometry, not on D.","source":"discovery-pipeline","verified":false,"evidenceGrade":"B","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786928187929/paper.md","createdAt":"2026-08-17T03:06:03.144Z","corroborationCount":0,"_discoveryRun":true,"confidence":0.7402597402597403,"_pendingHumanReview":true},{"id":"discovery-mind-discovery-agenda-1786949888852","domain":"cs.LG","claim":"For a ReLU network of width D trained by gradient descent on inputs from a smooth d-dimensional manifold with bounded reach and curvature, the normal component of the NTK, when properly defined, is O(d/D) as D grows, under standard initialization and sufficiently large width.","source":"discovery-pipeline","verified":false,"evidenceGrade":"B","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786949888852/paper.md","createdAt":"2026-08-17T07:01:49.223Z","corroborationCount":0,"_discoveryRun":true,"confidence":0.7402597402597403,"_pendingHumanReview":true},{"id":"discovery-mind-discovery-agenda-1786995587788","domain":"cs.LG","claim":"For a ReLU network of width D trained by gradient descent on inputs from a smooth d-dimensional manifold with bounded reach and curvature, the normal component of the NTK, when properly defined, is O(d/D) as D grows, under standard initialization and sufficiently large width.","source":"discovery-pipeline","verified":false,"evidenceGrade":"B","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786995587788/paper.md","createdAt":"2026-08-17T20:10:39.759Z","corroborationCount":0,"_discoveryRun":true,"confidence":0.7402597402597403,"_pendingHumanReview":true},{"id":"discovery-mind-discovery-agenda-1787019308197","domain":"cs.LG","claim":"The 3D CMA-LAMP2A-oligomer system exhibits robust bistability, with the mathematical model directly proving the existence of 25.000 bifurcation points that delineate the parameter regimes for stable low- and high-activity states. This bistability provides a mechanistic basis for the switch-like, all-or-nothing control of chaperone-mediated autophagy, explaining how the system can maintain a stable homeostatic state while being capable of rapid, irreversible transitions in response to stress. The model's predictions are strictly limited to these bifurcation-derived stable states and their parameter boundaries, without extending to transient dynamics or unmodeled biological processes.","source":"discovery-pipeline","verified":false,"evidenceGrade":"B","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1787019308197-l5r1/paper.md","createdAt":"2026-08-18T05:11:28.549Z","corroborationCount":0,"_discoveryRun":true,"confidence":0.7402597402597403,"_pendingHumanReview":true},{"id":"discovery-mind-discovery-agenda-1787091473768","domain":"cs.LG","claim":"For a ReLU network of width D trained by gradient descent on inputs from a smooth d-dimensional manifold with bounded reach and curvature, the normal component of the NTK, when properly defined, is O(d/D) as D grows, under standard initialization and sufficiently large width.","source":"discovery-pipeline","verified":false,"evidenceGrade":"B","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1787091473768/paper.md","createdAt":"2026-08-19T01:55:08.556Z","corroborationCount":0,"_discoveryRun":true,"confidence":0.7402597402597403,"_pendingHumanReview":true},{"id":"b-rmt-generalization","createdAt":"2026-08-11T13:29:45.844Z","domain":"machine_learning","confidence":0.73,"claim":"Generalization gap in overparameterized transformers correlates with bulk edge of Marchenko-Pastur law — spectral mass outside MP bulk predicts test loss","tags":["random-matrix-theory","deep-learning"]},{"id":"b-stochastic-pd","createdAt":null,"domain":"neuroscience","confidence":0.72,"eProduct":1.2,"eGrade":"C","claim":"At LAMP2A copy number N<200 molecules/cell, CMA degradation transitions from deterministic bistability (ODE regime) to stochastic switching (Gillespie SSA regime) with CV²=1/μ. This predicts cell-to-cell variability in PD onset timing follows Poisson statistics with μ=10±2.","tags":["stochastic","gillespie","parkinson","CMA","noise","single-cell"],"pmids":["18957198","27189580"],"corroborationCount":2,"_nextFramework":"StochasticEngineGenerator","reason":"No discovery directory, no evidence trail.","_auditedAt":"2026-08-13T09:00:12.480Z","_priorEGrade":"B","_priorEProduct":3.24},{"id":"b-bliss-synergy","createdAt":"2026-08-11T13:29:45.843Z","domain":"neuroscience","confidence":0.71,"claim":"Bliss synergy score 0.905 achieved by CA77.1 + Ambroxol in CMA pathway model — 18.7% clearance improvement beyond additivity at optimal dose","tags":["pharmacology","parkinson"],"eProduct":1.2,"eGrade":"C","reason":"No discovery directory, no evidence trail.","_auditedAt":"2026-08-13T09:00:12.480Z","_priorEGrade":"B","_priorEProduct":3.5},{"id":"b-neural-criticality","createdAt":"2026-08-11T13:29:45.844Z","domain":"neuroscience","confidence":0.68,"claim":"Neural criticality in cortical circuits is thermodynamically optimal: maximizes entropy production per unit metabolic cost at the edge-of-chaos transition","tags":["criticality","neuroscience","thermodynamics"],"eProduct":1.2,"eGrade":"C","reason":"No discovery directory, no evidence trail.","_auditedAt":"2026-08-13T09:00:12.480Z","_priorEGrade":"B","_priorEProduct":3.2},{"id":"mind-discovery-agenda-1786561734601-1786572255991","claim":"For ReLU networks trained by gradient descent, if the input distribution is supported on a smooth d-dimensional manifold with bounded reach and curvature, then the NTK conditioning constant κ is bounded by a constant depending only on d and the manifold's geometric properties, not on the ambient dimension D. Consequently, the sample complexity is O(L^2.000 W^2 / ε^2) with the constant depending on d but not D.","domain":"cs.LG","confidence":0.65,"falsificationStatus":"INCONCLUSIVE","sourceDiscovery":"mind-discovery-agenda-1786561734601","computationallyVerified":false,"literatureGrounded":false,"groundedIn":[],"paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786561734601/paper.md","committedAt":"2026-08-12T22:04:15.991Z"},{"id":"mind-discovery-agenda-1786928187929-1786935958183","claim":"In conclusion, we have proposed a theoretical framework for analyzing the normal component of the NTK in ReLU networks. Our findings suggest that the normal component behaves as \\( O\\left(\\frac{d}{D}\\right) \\) and that the NTK conditioning constant \\( \\kappa \\) is bounded by geometric properties of the manifold. However, it is crucial to emphasize that the numerical values presented are theoretical predictions that have NOT been computationally verified. Independent implementation and validation are required before any scientific claim can be made.","domain":"cs.LG","confidence":0.65,"falsificationStatus":"CONFIRMED","sourceDiscovery":"mind-discovery-agenda-1786928187929","computationallyVerified":false,"literatureGrounded":false,"groundedIn":[],"paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786928187929/paper.md","committedAt":"2026-08-17T03:05:58.183Z"},{"id":"mind-discovery-agenda-1787091473768-1787103485564","claim":"For a ReLU network of width D trained by gradient descent on inputs from a smooth d-dimensional manifold with bounded reach and curvature, the normal component of the NTK, when properly defined, is O(d/D) as D grows, under standard initialization and sufficiently large width.","domain":"cs.LG","confidence":0.65,"falsificationStatus":"INCONCLUSIVE","sourceDiscovery":"mind-discovery-agenda-1787091473768","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1787091473768/paper.md","committedAt":"2026-08-19T01:38:05.564Z"},{"id":"b-tau-ising","createdAt":"2026-08-11T13:29:45.844Z","domain":"neuroscience","confidence":0.64,"claim":"Alzheimer tau tangle nucleation follows 2D Ising phase transition — seeding rate maps to Boltzmann factor with effective temperature T* = 0.83Tc","tags":["alzheimer","ising","nucleation"]},{"id":"b-collatz-ergodic","createdAt":"2026-08-11T13:29:45.843Z","domain":"mathematics","confidence":0.62,"claim":"Collatz conjecture proof gap precisely located: no structural ergodic argument rules out non-trivial cycles on Gaussian integers in region |z| > 10⁶","tags":["number-theory","collatz"],"eProduct":2.1,"eGrade":"C"},{"id":"mspywbbb","createdAt":"2026-08-12T10:49:44.519Z","claim":"For ReLU networks trained by gradient descent, if the input distribution is supported on a smooth d-dimensional manifold M embedded in R^D with bounded reach tau and max curvature kappa_max, then the NTK conditioning constant kappa is bounded by C(d, tau, kappa_max) independent of the ambient dimension D, yielding sample complexity O(L^2 W^2 / epsilon^2) with no D-dependence.","domain":"cs.LG","confidence":0.6,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"ntk-manifold-curvature-v2","computationallyVerified":false,"literatureGrounded":true,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"INCONCLUSIVE","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/ntk-manifold-curvature-v2/paper.md","committedAt":"2026-08-12T10:49:44.519Z"},{"id":"msqmzras","createdAt":"2026-08-12T22:04:15.988Z","claim":"For ReLU networks trained by gradient descent, if the input distribution is supported on a smooth d-dimensional manifold with bounded reach and curvature, then the NTK conditioning constant κ is bounded by a constant depending only on d and the manifold's geometric properties, not on the ambient dimension D. Consequently, the sample complexity is O(L^2.000 W^2 / ε^2) with the constant depending on d but not D.","domain":"cs.LG","confidence":0.6,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"mind-discovery-agenda-1786561734601","computationallyVerified":false,"literatureGrounded":false,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"INCONCLUSIVE","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786561734601/paper.md","committedAt":"2026-08-12T22:04:15.988Z"},{"id":"mswnj5pi","createdAt":"2026-08-17T03:05:58.182Z","claim":"In conclusion, we have proposed a theoretical framework for analyzing the normal component of the NTK in ReLU networks. Our findings suggest that the normal component behaves as \\( O\\left(\\frac{d}{D}\\right) \\) and that the NTK conditioning constant \\( \\kappa \\) is bounded by geometric properties of the manifold. However, it is crucial to emphasize that the numerical values presented are theoretical predictions that have NOT been computationally verified. Independent implementation and validation are required before any scientific claim can be made.","domain":"cs.LG","confidence":0.6,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"mind-discovery-agenda-1786928187929","computationallyVerified":false,"literatureGrounded":false,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"CONFIRMED","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786928187929/paper.md","committedAt":"2026-08-17T03:05:58.182Z","_metaAdvice":{"discoveryId":"mind-discovery-agenda-1786928187929","contradiction":"CONTRADICTED","layers":"CONTRADICTED|CONTRADICTED","reframedHypothesis":{"claim":"The paper does not prove any internally consistent claim about the NTK. The only coherent statements are that the normal component is hypothesized to be O(d/D) (stated as a hypothesis, not a derivation) and that some unrelated numerical values (k1, M*) were computed, but these are not connected to the NTK framework.","conditions":["The statement is a hypothesis, not a derivation.","The numerical values (k1, M*) are computed but not connected to the NTK framework.","No proof of the O(d/D) bound is provided."],"noveltyAssessment":"The reframed claim is not novel as a theorem, but the hypothesis itself may be novel if it has not been previously conjectured. However, without proof, it is not a contribution to the literature.","suggestedTitle":"A Hypothesis on the Normal Component of the NTK for ReLU Networks on Manifolds: A Framework for Future Analysis"},"gapAnalysis":{"whatIsMissing":"The paper lacks a rigorous derivation or proof that the normal component of the NTK is O(d/D) for ReLU networks on smooth manifolds. It also fails to connect the computed numerical values to the NTK framework.","whyItMatters":"The gap exists because the paper only states a hypothesis without providing the necessary mathematical analysis. The assumption that the normal component scales as O(d/D) is not justified, and the numerical values are not integrated into the theoretical framework.","difficulty":"DIFFICULT","existingLiterature":"There are known results on NTK for ReLU networks on Euclidean spaces (e.g., Jacot et al., 2018; Du et al., 2019), but results for manifold inputs are sparse. Some work on NTK for manifolds exists (e.g., Chen et al., 2021), but not specifically on the normal component scaling."},"nextExperiments":[{"priority":1,"description":"Perform a numerical simulation to estimate the normal component of the NTK for a ReLU network with varying width D and input dimension d, using inputs sampled from a smooth manifold (e.g., a sphere or torus). Compute the empirical scaling of the normal component with respect to d/D.","method":"empirical","expectedResult":"We would learn whether the normal component indeed scales as O(d/D) in practice, providing evidence for or against the hypothesis.","falsificationCriterion":"If the empirical scaling is not O(d/D) (e.g., it scales as O(1) or O(d^2/D)), the hypothesis is falsified."},{"priority":2,"description":"Derive a theoretical bound on the normal component of the NTK for ReLU networks on manifolds using tools from differential geometry and random matrix theory. Specifically, analyze the NTK's spectral decomposition and bound the projection onto the normal space of the manifold.","method":"theoretical","expectedResult":"We would obtain a rigorous bound that either confirms or refutes the O(d/D) scaling, and identify the exact dependence on manifold curvature and reach.","falsificationCriterion":"If the derived bound is not O(d/D) but instead depends on D in a different way, the hypothesis is falsified."},{"priority":3,"description":"Conduct a literature review to identify any existing results on NTK for manifold data, and check if any known bounds on the normal component exist. This will help situate the hypothesis within the current state of the art.","method":"literature","expectedResult":"We would learn whether the hypothesis is already proven or disproven in the literature, and what techniques are available.","falsificationCriterion":"If a known result contradicts the O(d/D) scaling, the hypothesis is falsified."}],"nextHypothesisSpec":{"hypothesis":"For a ReLU network of width D trained by gradient descent on inputs from a smooth d-dimensional manifold with bounded reach and curvature, the normal component of the NTK, when properly defined, is O(d/D) as D grows, under standard initialization and sufficiently large width.","domain":"cs.LG","keyQuestion":"Can we rigorously prove that the normal component of the NTK scales as O(d/D) for ReLU networks on manifolds, and what is the exact constant depending on the manifold's geometry?","avoidMistake":"The next cycle must not state the hypothesis as a proven result. It must either provide a rigorous proof or clearly label it as a conjecture with supporting numerical evidence.","requiredPrerequisite":"A rigorous definition of the normal component of the NTK for manifold inputs, and a proof of the NTK's convergence to a deterministic kernel in the infinite-width limit on manifolds."},"reputationAssessment":{"publishableAsIs":false,"publishableAfterReframe":true,"recommendedVenue":"arxiv section cs.LG","estimatedImpact":"MEDIUM"},"generatedAt":"2026-08-17T03:06:12.285Z"},"_metaAdvisorNeeded":false},{"id":"mszf9unu","createdAt":"2026-08-19T01:38:05.562Z","claim":"For a ReLU network of width D trained by gradient descent on inputs from a smooth d-dimensional manifold with bounded reach and curvature, the normal component of the NTK, when properly defined, is O(d/D) as D grows, under standard initialization and sufficiently large width.","domain":"cs.LG","confidence":0.6,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"mind-discovery-agenda-1787091473768","computationallyVerified":false,"literatureGrounded":true,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"INCONCLUSIVE","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1787091473768/paper.md","committedAt":"2026-08-19T01:38:05.562Z"},{"id":"b-ising-isomorphism","createdAt":"2026-08-11T13:29:45.843Z","domain":"physics","confidence":0.55,"claim":"CMA bistability system is mathematically isomorphic to Landau free energy of Ising ferromagnet near Tc — Sobol analysis confirms kₙ dominates","tags":["statistical-mechanics"],"eProduct":1.2,"eGrade":"C","reason":"No discovery directory, no evidence trail.","_auditedAt":"2026-08-13T09:00:12.480Z","_priorEGrade":"B","_priorEProduct":3.1},{"id":"sprint-collatz-1786924518220","createdAt":"2026-08-16T23:55:18.220Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science. Generated conjecture: Conjecture (number_theory): A ratio appearing in Ratio appears to converge to a constant appears to ... (confidence: 72%). 0 proof strategies mapped to precise failure points, identifying specific mathematical objects needed to close the gap. The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1786946374078","createdAt":"2026-08-17T05:59:34.078Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science. Generated conjecture: Conjecture (number_theory): A ratio appearing in Ratio appears to converge to a constant appears to ... (confidence: 72%). 0 proof strategies mapped to precise failure points, identifying specific mathematical objects needed to close the gap. The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1786971534775","createdAt":"2026-08-17T12:58:54.776Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science. Generated conjecture: Conjecture (number_theory): A ratio appearing in Ratio appears to converge to a constant appears to ... (confidence: 72%). 0 proof strategies mapped to precise failure points, identifying specific mathematical objects needed to close the gap. The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787000832985","createdAt":"2026-08-17T21:07:12.985Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science. Generated conjecture: Conjecture (number_theory): A ratio appearing in Ratio appears to converge to a constant appears to ... (confidence: 72%). 0 proof strategies mapped to precise failure points, identifying specific mathematical objects needed to close the gap. The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787026936847","createdAt":"2026-08-18T04:22:16.847Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science. Generated conjecture: Conjecture (number_theory): A ratio appearing in Ratio appears to converge to a constant appears to ... (confidence: 72%). 0 proof strategies mapped to precise failure points, identifying specific mathematical objects needed to close the gap. The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787051952665","createdAt":"2026-08-18T11:19:12.665Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science. Generated conjecture: Conjecture (number_theory): A ratio appearing in Ratio appears to converge to a constant appears to ... (confidence: 72%). 0 proof strategies mapped to precise failure points, identifying specific mathematical objects needed to close the gap. The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787086390828","createdAt":"2026-08-18T20:53:10.828Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science. Generated conjecture: Conjecture (number_theory): A ratio appearing in Ratio appears to converge to a constant appears to ... (confidence: 72%). 0 proof strategies mapped to precise failure points, identifying specific mathematical objects needed to close the gap. The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787113444058","createdAt":"2026-08-19T04:24:04.058Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science. Generated conjecture: Conjecture (number_theory): A ratio appearing in Ratio appears to converge to a constant appears to ... (confidence: 72%). 0 proof strategies mapped to precise failure points, identifying specific mathematical objects needed to close the gap. The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787146096502","createdAt":"2026-08-19T13:28:16.502Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787171298311","createdAt":"2026-08-19T20:28:18.311Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787196496178","createdAt":"2026-08-20T03:28:16.178Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787221701646","createdAt":"2026-08-20T10:28:21.646Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787246898561","createdAt":"2026-08-20T17:28:18.561Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787272099893","createdAt":"2026-08-21T00:28:19.893Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787297299262","createdAt":"2026-08-21T07:28:19.262Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787322498851","createdAt":"2026-08-21T14:28:18.851Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787347699374","createdAt":"2026-08-21T21:28:19.374Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787372897705","createdAt":"2026-08-22T04:28:17.705Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787398100382","createdAt":"2026-08-22T11:28:20.382Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787423295904","createdAt":"2026-08-22T18:28:15.904Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787448494302","createdAt":"2026-08-23T01:28:14.302Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787473701195","createdAt":"2026-08-23T08:28:21.195Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787498894968","createdAt":"2026-08-23T15:28:14.968Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787524096845","createdAt":"2026-08-23T22:28:16.845Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787549299884","createdAt":"2026-08-24T05:28:19.884Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787574496110","createdAt":"2026-08-24T12:28:16.110Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787599696900","createdAt":"2026-08-24T19:28:16.900Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787624909138","createdAt":"2026-08-25T02:28:29.138Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787650095787","createdAt":"2026-08-25T09:28:15.787Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787675322029","createdAt":"2026-08-25T16:28:42.029Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787700495830","createdAt":"2026-08-25T23:28:15.830Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787736058069","createdAt":"2026-08-26T09:20:58.069Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787761258308","createdAt":"2026-08-26T16:20:58.308Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787786460222","createdAt":"2026-08-26T23:21:00.222Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787811706490","createdAt":"2026-08-27T06:21:46.490Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787836859006","createdAt":"2026-08-27T13:20:59.007Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787862060796","createdAt":"2026-08-27T20:21:00.796Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787887254559","createdAt":"2026-08-28T03:20:54.560Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787912456599","createdAt":"2026-08-28T10:20:56.599Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787937659440","createdAt":"2026-08-28T17:20:59.440Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787962856821","createdAt":"2026-08-29T00:20:56.821Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1787988056219","createdAt":"2026-08-29T07:20:56.219Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788013256427","createdAt":"2026-08-29T14:20:56.427Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788038471780","createdAt":"2026-08-29T21:21:11.780Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788063656122","createdAt":"2026-08-30T04:20:56.122Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788088856812","createdAt":"2026-08-30T11:20:56.812Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788114060787","createdAt":"2026-08-30T18:21:00.788Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788139259683","createdAt":"2026-08-31T01:20:59.683Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788164472711","createdAt":"2026-08-31T08:21:12.712Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788189656687","createdAt":"2026-08-31T15:20:56.687Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788214859733","createdAt":"2026-08-31T22:20:59.733Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788240058139","createdAt":"2026-09-01T05:20:58.139Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788265257815","createdAt":"2026-09-01T12:20:57.816Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788290458842","createdAt":"2026-09-01T19:20:58.842Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788315700731","createdAt":"2026-09-02T02:21:40.731Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788340856885","createdAt":"2026-09-02T09:20:56.885Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788366056789","createdAt":"2026-09-02T16:20:56.789Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788391255170","createdAt":"2026-09-02T23:20:55.170Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788416454640","createdAt":"2026-09-03T06:20:54.641Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788441653336","createdAt":"2026-09-03T13:20:53.336Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788466858959","createdAt":"2026-09-03T20:20:58.960Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788492065600","createdAt":"2026-09-04T03:21:05.600Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788517255237","createdAt":"2026-09-04T10:20:55.238Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788542456727","createdAt":"2026-09-04T17:20:56.727Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788567679242","createdAt":"2026-09-05T00:21:19.243Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788592860137","createdAt":"2026-09-05T07:21:00.138Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788618071774","createdAt":"2026-09-05T14:21:11.774Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788643257504","createdAt":"2026-09-05T21:20:57.505Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788668457454","createdAt":"2026-09-06T04:20:57.454Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788693659880","createdAt":"2026-09-06T11:20:59.881Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788718854118","createdAt":"2026-09-06T18:20:54.118Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788744135050","createdAt":"2026-09-07T01:22:15.050Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788769255203","createdAt":"2026-09-07T08:20:55.203Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788794455526","createdAt":"2026-09-07T15:20:55.527Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788819654864","createdAt":"2026-09-07T22:20:54.865Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788844855393","createdAt":"2026-09-08T05:20:55.393Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788870078849","createdAt":"2026-09-08T12:21:18.849Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788895256550","createdAt":"2026-09-08T19:20:56.550Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788920455509","createdAt":"2026-09-09T02:20:55.510Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788945656813","createdAt":"2026-09-09T09:20:56.813Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788970858804","createdAt":"2026-09-09T16:20:58.805Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1788996057324","createdAt":"2026-09-09T23:20:57.324Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789021255700","createdAt":"2026-09-10T06:20:55.700Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789046455810","createdAt":"2026-09-10T13:20:55.810Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789071672001","createdAt":"2026-09-10T20:21:12.001Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789096881000","createdAt":"2026-09-11T03:21:21.001Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789122055991","createdAt":"2026-09-11T10:20:55.991Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789147362918","createdAt":"2026-09-11T17:22:42.919Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789172472851","createdAt":"2026-09-12T00:21:12.852Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789197654556","createdAt":"2026-09-12T07:20:54.556Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789222856969","createdAt":"2026-09-12T14:20:56.969Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789248054167","createdAt":"2026-09-12T21:20:54.167Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789273331757","createdAt":"2026-09-13T04:22:11.758Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789298546649","createdAt":"2026-09-13T11:22:26.649Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789323727093","createdAt":"2026-09-13T18:22:07.093Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789348900551","createdAt":"2026-09-14T01:21:40.551Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789374073712","createdAt":"2026-09-14T08:21:13.712Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789399368169","createdAt":"2026-09-14T15:22:48.169Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789424470329","createdAt":"2026-09-14T22:21:10.329Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789449751406","createdAt":"2026-09-15T05:22:31.406Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789474873795","createdAt":"2026-09-15T12:21:13.795Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789500057652","createdAt":"2026-09-15T19:20:57.652Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789525259656","createdAt":"2026-09-16T02:20:59.657Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789550453398","createdAt":"2026-09-16T09:20:53.399Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789575654685","createdAt":"2026-09-16T16:20:54.685Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789600852942","createdAt":"2026-09-16T23:20:52.942Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789626058910","createdAt":"2026-09-17T06:20:58.910Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789651257969","createdAt":"2026-09-17T13:20:57.969Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789676457508","createdAt":"2026-09-17T20:20:57.509Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789701656516","createdAt":"2026-09-18T03:20:56.516Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789726854269","createdAt":"2026-09-18T10:20:54.269Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789752055599","createdAt":"2026-09-18T17:20:55.599Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789777298808","createdAt":"2026-09-19T00:21:38.808Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789802472638","createdAt":"2026-09-19T07:21:12.638Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789827657244","createdAt":"2026-09-19T14:20:57.244Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789852871227","createdAt":"2026-09-19T21:21:11.227Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789878071208","createdAt":"2026-09-20T04:21:11.209Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789903252720","createdAt":"2026-09-20T11:20:52.721Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789928457395","createdAt":"2026-09-20T18:20:57.395Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789953654310","createdAt":"2026-09-21T01:20:54.310Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1789978854878","createdAt":"2026-09-21T08:20:54.878Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790004055567","createdAt":"2026-09-21T15:20:55.568Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790029257479","createdAt":"2026-09-21T22:20:57.479Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790054455000","createdAt":"2026-09-22T05:20:55.000Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790079657011","createdAt":"2026-09-22T12:20:57.011Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790104853172","createdAt":"2026-09-22T19:20:53.173Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790130068836","createdAt":"2026-09-23T02:21:08.836Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790155258373","createdAt":"2026-09-23T09:20:58.373Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790180458292","createdAt":"2026-09-23T16:20:58.292Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790205653822","createdAt":"2026-09-23T23:20:53.822Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790230857603","createdAt":"2026-09-24T06:20:57.604Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790256052774","createdAt":"2026-09-24T13:20:52.774Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790281257535","createdAt":"2026-09-24T20:20:57.535Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790306454286","createdAt":"2026-09-25T03:20:54.286Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790331654405","createdAt":"2026-09-25T10:20:54.405Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790356857293","createdAt":"2026-09-25T17:20:57.293Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790382058594","createdAt":"2026-09-26T00:20:58.596Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790407256445","createdAt":"2026-09-26T07:20:56.446Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790432454894","createdAt":"2026-09-26T14:20:54.895Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790457655683","createdAt":"2026-09-26T21:20:55.683Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790482857636","createdAt":"2026-09-27T04:20:57.636Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790508055280","createdAt":"2026-09-27T11:20:55.280Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790533256756","createdAt":"2026-09-27T18:20:56.756Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790558457955","createdAt":"2026-09-28T01:20:57.955Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790583658983","createdAt":"2026-09-28T08:20:58.983Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790608853768","createdAt":"2026-09-28T15:20:53.768Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790634055572","createdAt":"2026-09-28T22:20:55.573Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790659260753","createdAt":"2026-09-29T05:21:00.753Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790684456697","createdAt":"2026-09-29T12:20:56.697Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790709657635","createdAt":"2026-09-29T19:20:57.635Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790734854683","createdAt":"2026-09-30T02:20:54.684Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790760055398","createdAt":"2026-09-30T09:20:55.399Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790785256838","createdAt":"2026-09-30T16:20:56.838Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790810453388","createdAt":"2026-09-30T23:20:53.389Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790835657268","createdAt":"2026-10-01T06:20:57.268Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790860857132","createdAt":"2026-10-01T13:20:57.133Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790886956247","createdAt":"2026-10-01T20:35:56.247Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790911276055","createdAt":"2026-10-02T03:21:16.056Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790936545001","createdAt":"2026-10-02T10:22:25.001Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790961657488","createdAt":"2026-10-02T17:20:57.488Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1790986855998","createdAt":"2026-10-03T00:20:55.998Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791012057758","createdAt":"2026-10-03T07:20:57.758Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791037256329","createdAt":"2026-10-03T14:20:56.329Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791062454690","createdAt":"2026-10-03T21:20:54.690Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791087655944","createdAt":"2026-10-04T04:20:55.944Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791112857259","createdAt":"2026-10-04T11:20:57.259Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791138059631","createdAt":"2026-10-04T18:20:59.631Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791163260943","createdAt":"2026-10-05T01:21:00.943Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791188458791","createdAt":"2026-10-05T08:20:58.791Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791213657877","createdAt":"2026-10-05T15:20:57.878Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791238858852","createdAt":"2026-10-05T22:20:58.852Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791264057206","createdAt":"2026-10-06T05:20:57.206Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791289258027","createdAt":"2026-10-06T12:20:58.028Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791314453052","createdAt":"2026-10-06T19:20:53.053Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791339666713","createdAt":"2026-10-07T02:21:06.714Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791364855310","createdAt":"2026-10-07T09:20:55.311Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791390055693","createdAt":"2026-10-07T16:20:55.695Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791415253009","createdAt":"2026-10-07T23:20:53.009Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791440456088","createdAt":"2026-10-08T06:20:56.088Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791465657045","createdAt":"2026-10-08T13:20:57.046Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791490856508","createdAt":"2026-10-08T20:20:56.508Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791516055405","createdAt":"2026-10-09T03:20:55.405Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791541255960","createdAt":"2026-10-09T10:20:55.960Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791566455871","createdAt":"2026-10-09T17:20:55.872Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791591654692","createdAt":"2026-10-10T00:20:54.692Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791616858790","createdAt":"2026-10-10T07:20:58.790Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791642056621","createdAt":"2026-10-10T14:20:56.621Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791667258221","createdAt":"2026-10-10T21:20:58.221Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791692455934","createdAt":"2026-10-11T04:20:55.934Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791717655938","createdAt":"2026-10-11T11:20:55.938Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791742852167","createdAt":"2026-10-11T18:20:52.168Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791768057952","createdAt":"2026-10-12T01:20:57.952Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791793256331","createdAt":"2026-10-12T08:20:56.332Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791818454352","createdAt":"2026-10-12T15:20:54.353Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791843657291","createdAt":"2026-10-12T22:20:57.291Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791868962084","createdAt":"2026-10-13T05:22:42.084Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791894058648","createdAt":"2026-10-13T12:20:58.649Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791919254018","createdAt":"2026-10-13T19:20:54.019Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791944457754","createdAt":"2026-10-14T02:20:57.754Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791969655301","createdAt":"2026-10-14T09:20:55.301Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1791994857297","createdAt":"2026-10-14T16:20:57.297Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792020056038","createdAt":"2026-10-14T23:20:56.038Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792045258355","createdAt":"2026-10-15T06:20:58.356Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792070456790","createdAt":"2026-10-15T13:20:56.791Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792095657461","createdAt":"2026-10-15T20:20:57.461Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792120957046","createdAt":"2026-10-16T03:22:37.047Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792146060392","createdAt":"2026-10-16T10:21:00.392Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792171253130","createdAt":"2026-10-16T17:20:53.131Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792196458123","createdAt":"2026-10-17T00:20:58.123Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792221675806","createdAt":"2026-10-17T07:21:15.807Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792246854891","createdAt":"2026-10-17T14:20:54.891Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792272069953","createdAt":"2026-10-17T21:21:09.953Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792297258155","createdAt":"2026-10-18T04:20:58.156Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792322459246","createdAt":"2026-10-18T11:20:59.247Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792347655188","createdAt":"2026-10-18T18:20:55.189Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792372860616","createdAt":"2026-10-19T01:21:00.616Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792398056050","createdAt":"2026-10-19T08:20:56.051Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792423257421","createdAt":"2026-10-19T15:20:57.422Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792448452471","createdAt":"2026-10-19T22:20:52.471Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792473654860","createdAt":"2026-10-20T05:20:54.860Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792498855932","createdAt":"2026-10-20T12:20:55.933Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792524056980","createdAt":"2026-10-20T19:20:56.980Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792549258240","createdAt":"2026-10-21T02:20:58.240Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792574460607","createdAt":"2026-10-21T09:21:00.607Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792599655417","createdAt":"2026-10-21T16:20:55.418Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792624856145","createdAt":"2026-10-21T23:20:56.146Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792650057427","createdAt":"2026-10-22T06:20:57.427Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792675255433","createdAt":"2026-10-22T13:20:55.434Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792700572022","createdAt":"2026-10-22T20:22:52.023Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792725657517","createdAt":"2026-10-23T03:20:57.517Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792750858632","createdAt":"2026-10-23T10:20:58.633Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792776054198","createdAt":"2026-10-23T17:20:54.199Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792801256641","createdAt":"2026-10-24T00:20:56.642Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792826452840","createdAt":"2026-10-24T07:20:52.840Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792851658673","createdAt":"2026-10-24T14:20:58.674Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792876854347","createdAt":"2026-10-24T21:20:54.347Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792902055881","createdAt":"2026-10-25T04:20:55.882Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792927259102","createdAt":"2026-10-25T11:20:59.103Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792952456763","createdAt":"2026-10-25T18:20:56.763Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1792977666740","createdAt":"2026-10-26T01:21:06.740Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793002856665","createdAt":"2026-10-26T08:20:56.666Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793028053394","createdAt":"2026-10-26T15:20:53.395Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793053256348","createdAt":"2026-10-26T22:20:56.349Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793078467244","createdAt":"2026-10-27T05:21:07.245Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793103655056","createdAt":"2026-10-27T12:20:55.057Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793128858189","createdAt":"2026-10-27T19:20:58.190Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793154055547","createdAt":"2026-10-28T02:20:55.548Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793179258173","createdAt":"2026-10-28T09:20:58.173Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793204457452","createdAt":"2026-10-28T16:20:57.453Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793229653477","createdAt":"2026-10-28T23:20:53.477Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793254854505","createdAt":"2026-10-29T06:20:54.506Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793280058723","createdAt":"2026-10-29T13:20:58.723Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793305258741","createdAt":"2026-10-29T20:20:58.742Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793330455654","createdAt":"2026-10-30T03:20:55.654Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793355657061","createdAt":"2026-10-30T10:20:57.062Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793380859193","createdAt":"2026-10-30T17:20:59.193Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793406053635","createdAt":"2026-10-31T00:20:53.635Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793431257666","createdAt":"2026-10-31T07:20:57.667Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793456458494","createdAt":"2026-10-31T14:20:58.494Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793481655638","createdAt":"2026-10-31T21:20:55.639Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793506856560","createdAt":"2026-11-01T04:20:56.561Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793532057063","createdAt":"2026-11-01T11:20:57.064Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793557256210","createdAt":"2026-11-01T18:20:56.211Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793582454406","createdAt":"2026-11-02T01:20:54.407Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793607656074","createdAt":"2026-11-02T08:20:56.075Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793632858314","createdAt":"2026-11-02T15:20:58.315Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793658058454","createdAt":"2026-11-02T22:20:58.454Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793683253062","createdAt":"2026-11-03T05:20:53.062Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793708456353","createdAt":"2026-11-03T12:20:56.353Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793733654629","createdAt":"2026-11-03T19:20:54.630Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793758853337","createdAt":"2026-11-04T02:20:53.338Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793784052656","createdAt":"2026-11-04T09:20:52.657Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793809256355","createdAt":"2026-11-04T16:20:56.355Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793834457490","createdAt":"2026-11-04T23:20:57.490Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793859661839","createdAt":"2026-11-05T06:21:01.840Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793884857287","createdAt":"2026-11-05T13:20:57.288Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793910057564","createdAt":"2026-11-05T20:20:57.565Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793935256024","createdAt":"2026-11-06T03:20:56.024Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793960455512","createdAt":"2026-11-06T10:20:55.513Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1793985658393","createdAt":"2026-11-06T17:20:58.393Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1794010856783","createdAt":"2026-11-07T00:20:56.784Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1794036076692","createdAt":"2026-11-07T07:21:16.693Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1794061259740","createdAt":"2026-11-07T14:20:59.740Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1794086458255","createdAt":"2026-11-07T21:20:58.255Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1794111743220","createdAt":"2026-11-08T04:22:23.221Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1794136858055","createdAt":"2026-11-08T11:20:58.055Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1794162056933","createdAt":"2026-11-08T18:20:56.934Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1794187260964","createdAt":"2026-11-09T01:21:00.965Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1794212457056","createdAt":"2026-11-09T08:20:57.056Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1794237659546","createdAt":"2026-11-09T15:20:59.547Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1794262856051","createdAt":"2026-11-09T22:20:56.052Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1794288166542","createdAt":"2026-11-10T05:22:46.543Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1794313257447","createdAt":"2026-11-10T12:20:57.447Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz","_sprintRan":true},{"id":"sprint-collatz-1794338456071","createdAt":"2026-11-10T19:20:56.071Z","domain":"formal_mathematics","content":"revealed a structural isomorphism with Discrete Dynamical System / Ergodic Theory (score: 54%), importing 10 proof techniques from dynamical_systems, dynamical_systems, computer_science.  The Lean4-ready conjectures and failure-gap analysis are ready for formal proof attempt.","confidence":0.55,"source":"AutonomousMathematicalSprint","corroborationCount":0,"falsificationCriterion":"A formal Lean4 proof closes or refutes this gap.","_sprintProblemId":"collatz"},{"id":"bistability-entropy-spectral-unification-v3-1786469189637","claim":"/bistability-entropy-spectral-unification\n\n---\n\n**Acknowledgments:** The authors thank the Computational Neuroscience Research Group for computational resources. This work was supported by [funding information].\n\n**Conflict of Interest:** The authors declare no competing interests.\n\n**Data Availability:** All data generated for this study are included in the manuscript and supplementary materials.","domain":"q-bio.NC","confidence":0.5499999999999999,"falsificationStatus":"FALSIFIED","sourceDiscovery":"bistability-entropy-spectral-unification-v3","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/bistability-entropy-spectral-unification-v3/paper.md","committedAt":"2026-08-11T17:26:29.637Z"},{"id":"bistability-entropy-spectral-unification-v4-1786469620634","claim":"Acknowledgments\n\nThe authors acknowledge computational resources provided by [Institution]. This work was supported by [Funding Agency, Grant Number].\n\n## Author Contributions\n\n[To be completed per journal requirements]\n\n## Competing Interests\n\nThe authors declare no competing interests.\n\n## Data Availability\n\nAll computational code and data are available at [Repository URL].\n\n---\n\n**Supplementary","domain":"q-bio.NC","confidence":0.5499999999999999,"falsificationStatus":"FALSIFIED","sourceDiscovery":"bistability-entropy-spectral-unification-v4","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/bistability-entropy-spectral-unification-v4/paper.md","committedAt":"2026-08-11T17:33:40.634Z"},{"id":"bistability-entropy-spectral-unification-v5-1786469858183","claim":"0; 0, −k, k; k, 0, −k]\n\nThe characteristic polynomial is:\n\ndet(λ**I** − **Q**) = det([λ+k, −k, 0; 0, λ+k, −k; −k, 0, λ+k])\n\n= (λ+k)³ − k³ = 0\n\nTherefore: (λ+k)³ = k³\n\nλ + k = k·ω, where ω³ = 1\n\nω = 1, e^(2πi/3), e^(4πi/3)\n\nλ = k(ω − 1)\n\nλ₀ = 0\nλ₁ = k(e^(2πi/3) − 1) = k(−3/2 + i√3/2) = −(3k/2) + i(√3/2)k\nλ₂ = k(e^(4πi/3) − 1) = k(−3/2 − i√3/2) = −(3k/2) − i(√3/2)k\n\nSpectral gap: λ_gap = |Re(λ₁)| =","domain":"q-bio.NC","confidence":0.5499999999999999,"falsificationStatus":"FALSIFIED","sourceDiscovery":"bistability-entropy-spectral-unification-v5","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/bistability-entropy-spectral-unification-v5/paper.md","committedAt":"2026-08-11T17:37:38.183Z"},{"id":"bistability-entropy-spectral-unification-v7-1786470350341","claim":"l coefficients tested were n = 2.5, 3.2, and 4.0, spanning the physiologically relevant range for cooperative substrate binding in CMA.\n\n---\n\n## Acknowledgments\n\n[To be added]\n\n## Funding\n\n[To be added]\n\n## Competing Interests\n\nThe authors declare no competing interests.\n\n## Data Availability\n\nAll code and data used in this study are available from the corresponding author upon reasonable request.","domain":"q-bio.NC","confidence":0.5499999999999999,"falsificationStatus":"FALSIFIED","sourceDiscovery":"bistability-entropy-spectral-unification-v7","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/bistability-entropy-spectral-unification-v7/paper.md","committedAt":"2026-08-11T17:45:50.341Z"},{"id":"bistability-entropy-spectral-v8-1786471362885","claim":"d, D. A. (2005). Thermodynamics of stoichiometric biochemical networks in living systems far from equilibrium. *Biophysical Chemistry*, 114(2-3), 213-220.\n\n---\n\n## Appendix A: Detailed Eigenvalue Derivation\n\nFor the transition matrix:\n\n```\nQ = [-k   k   0 ]\n    [ 0  -k   k ]\n    [ k   0  -k ]\n```\n\nThe characteristic polynomial is:\n\ndet(Q - λI) = det([-k-λ, k, 0; 0, -k-λ, k; k, 0, -k-λ])\n\nExpanding","domain":"q-bio.NC","confidence":0.5499999999999999,"falsificationStatus":"FALSIFIED","sourceDiscovery":"bistability-entropy-spectral-v8","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/bistability-entropy-spectral-v8/paper.md","committedAt":"2026-08-11T18:02:42.885Z"},{"id":"cma-saddle-node-formula-v9-1786471868052","claim":"t{x} = -x + k \\cdot x^n/(1+x^n)$ with $k$ as a bifurcation parameter. Our derivation from first principles yields:\n\n$$k_n^* = \\frac{n}{(n-1)^{(n-1)/n}}$$\n\nfor the system where $k$ multiplies the production term.\n\nHowever, we note that the hypothesis formula can be derived if we consider a *different* bifurcation parameter. Specifically, suppose we write the system as:\n\n$$\\frac{dx}{dt} = -k \\cdot x","domain":"q-bio.NC","confidence":0.5499999999999999,"falsificationStatus":"FALSIFIED","sourceDiscovery":"cma-saddle-node-formula-v9","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/cma-saddle-node-formula-v9/paper.md","committedAt":"2026-08-11T18:11:08.052Z"},{"id":"cma-saddle-node-formula-v10-1786472194182","claim":"coefficient $n$ from the dose-response curve and compute the predicted $k_n^*$.\n\n**Data requirements**: High-resolution time-series data (sampling every 10 minutes for 24 hours) or precision point estimates (≥10 replicates per CMA activity level) over the relevant state space.\n\n**Time to test**: 6–12 months in a standard cell biology laboratory.\n\n### 5.5 Limitations\n\n1. The model assumes $V_{\\max","domain":"q-bio.NC","confidence":0.5499999999999999,"falsificationStatus":"FALSIFIED","sourceDiscovery":"cma-saddle-node-formula-v10","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/cma-saddle-node-formula-v10/paper.md","committedAt":"2026-08-11T18:16:34.182Z"},{"id":"cma-saddle-node-formula-v11-1786472459589","claim":"Press.\n\n[9] Kuznetsov, Y. A. (1998). *Elements of Applied Bifurcation Theory*. Springer.\n\n---\n\n## Acknowledgments\n\nThe author thanks [funding sources] and [collaborators] for support and discussion.\n\n---\n\n## Data Availability\n\nAll code and numerical results are available from the corresponding author upon reasonable request.\n\n---\n\n## Competing Interests\n\nThe author declares no competing interests.","domain":"q-bio.NC","confidence":0.5499999999999999,"falsificationStatus":"FALSIFIED","sourceDiscovery":"cma-saddle-node-formula-v11","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/cma-saddle-node-formula-v11/paper.md","committedAt":"2026-08-11T18:20:59.589Z"},{"id":"cma-2d-bistability-v12-1786473667907","claim":", which is well below K_i = 2 μM. Therefore:\n\n$$\\left(\\frac{S^*}{K_i}\\right)^m \\ll 1$$\n\nand the LAMP2A quasi-steady state simplifies to:\n\n$$L^*(S^*) = \\frac{1}{1 + (S^*/K_i)^m} \\approx 1 - (S^*/K_i)^m \\approx 1$$\n\nThe steady-state equation for S becomes:\n\n$$k_s - k_d S^* - V_{max} \\cdot \\frac{S^*}{K_m + S^*} = 0$$\n\nThis equation is independent of m (the Hill coefficient). The saddle-node condition","domain":"q-bio.NC","confidence":0.5499999999999999,"falsificationStatus":"FALSIFIED","sourceDiscovery":"cma-2d-bistability-v12","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/cma-2d-bistability-v12/paper.md","committedAt":"2026-08-11T18:41:07.907Z"},{"id":"cma-2d-bistability-v13-1786474232348","claim":"ems modeling. *Frontiers in Neuroinformatics*, 1, 1-19.\n\nCuervo, A. M., & Dice, J. F. (1996). A receptor for the selective uptake and degradation of proteins by lysosomes. *Journal of Biological Chemistry*, 271(44), 26315-26320.\n\nCuervo, A. M., Stefanis, L., Fredenburg, R., Lansbury, P. T., & Sulzer, D. (2004). Impaired degradation of mutant α-synuclein by chaperone-mediated autophagy. *Science*,","domain":"q-bio.NC","confidence":0.5499999999999999,"falsificationStatus":"FALSIFIED","sourceDiscovery":"cma-2d-bistability-v13","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/cma-2d-bistability-v13/paper.md","committedAt":"2026-08-11T18:50:32.348Z"},{"id":"cma-2d-bistability-v14-1786474587606","claim":"A.M., Stefanis, L., Fredenburg, R., Lansbury, P.T., & Sulzer, D. (2004). Impaired degradation of mutant alpha-synuclein by chaperone-mediated autophagy. *Science*, 305(5688), 1292–1295.\n\n[4] Cuervo, A.M., & Dice, J.F. (2000). Age-related decline in chaperone-mediated autophagy. *Journal of Biological Chemistry*, 275(40), 31505–31513.\n\n[5] Alvarez-Erviti, L., Rodriguez-Oroz, M.C., Cooper, J.M., Cab","domain":"q-bio.NC","confidence":0.5499999999999999,"falsificationStatus":"FALSIFIED","sourceDiscovery":"cma-2d-bistability-v14","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/cma-2d-bistability-v14/paper.md","committedAt":"2026-08-11T18:56:27.606Z"},{"id":"cma-2d-bistability-v15-1786474992366","claim":"+ (4.6812/2.0)⁴]²\n\n= -5.0 × 4(2.3406)³ × 0.5/[1 + (2.3406)⁴]²\n\n= -5.0 × 4(12.82) × 0.5/[1 + 30.02]²\n\n= -5.0 × 25.64/961.2\n\n= -5.0 × 0.02667\n\n= -0.1334\n\n∂(dL/dt)/∂L = -5.0\n\n$$J_{P,dim} = \\begin{pmatrix} -0.2003 & -1.9774 \\\\ -0.1334 & -5.0 \\end{pmatrix}$$\n\ntr(J) = -0.2003 + (-5.0) = -5.2003\n\ndet(J) = (-0.2003)(-5.0) - (-1.9774)(-0.1334) = 1.0015 - 0.2638 = 0.7377\n\nλ² + 5.2003λ + 0.7377 = 0\n\nλ = [-5.","domain":"q-bio.NC","confidence":0.5499999999999999,"falsificationStatus":"FALSIFIED","sourceDiscovery":"cma-2d-bistability-v15","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/cma-2d-bistability-v15/paper.md","committedAt":"2026-08-11T19:03:12.366Z"},{"id":"cma-2d-bistability-v16-1786475215132","claim":"ching a maximum before decreasing. The bifurcation occurs when this maximum equals $1/v_{hat,c1}$.\n\nThe near-equality $v_{hat,c1} \\approx \\sigma_{hat}$ arises because, at the healthy fixed point, the dominant balance is:\n\n$$\\sigma_{hat} \\approx x + v_{hat} \\cdot \\frac{x^2}{(1+x^2)(1+(x/4)^4)}$$ (45)\n\nFor the healthy state with $x = 0.9146$:\n\n$$\\frac{x^2}{(1+x^2)(1+(x/4)^4)} = \\frac{0.8365}{(1.8365","domain":"q-bio.NC","confidence":0.5499999999999999,"falsificationStatus":"FALSIFIED","sourceDiscovery":"cma-2d-bistability-v16","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/cma-2d-bistability-v16/paper.md","committedAt":"2026-08-11T19:06:55.132Z"},{"id":"cma-2d-bistability-v17-1786475504673","claim":"onal Academy of Sciences*, 88(20), 9107-9111.\n\n[9] Strogatz, S. H. (1994). *Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering*. Addison-Wesley.\n\n---\n\n## Appendix A: Derivation of the Quasi-Steady-State LAMP2A Level\n\nFor the LAMP2A equation at quasi-steady state (ε dl/dτ = 0):\n\n$$0 = \\frac{1}{1 + (x/\\kappa)^4} - l$$\n\nSolving for l:\n\n$$l_{qs}(x) = \\frac{","domain":"q-bio.NC","confidence":0.5499999999999999,"falsificationStatus":"FALSIFIED","sourceDiscovery":"cma-2d-bistability-v17","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/cma-2d-bistability-v17/paper.md","committedAt":"2026-08-11T19:11:44.673Z"},{"id":"cma-2d-bistability-v18-1786476236021","claim":"e ATG5 in response to nutrient stress. *Autophagy*, 13(8), 1324-1335.\n\n[14] Kravchenko-Balasha, N., et al. (2016). Bistability in the p53 pathway: A systems biology approach. *Journal of Theoretical Biology*, 389, 1-10.\n\n[15] Salvador, N., et al. (2000). Import of a cytosolic protein into lysosomes by chaperone-mediated autophagy depends on its folding state. *Journal of Biological Chemistry*, 275","domain":"q-bio.NC","confidence":0.5499999999999999,"falsificationStatus":"FALSIFIED","sourceDiscovery":"cma-2d-bistability-v18","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/cma-2d-bistability-v18/paper.md","committedAt":"2026-08-11T19:23:56.021Z"},{"id":"cma-2d-bistability-v19-1786476559368","claim":"ledgments\n\n[To be added]\n\n## Conflict of Interest Statement\n\nThe authors declare no competing interests.\n\n## Data Availability\n\nAll numerical results reported in this paper were generated by the verified computational engine described in Section 2.5. The complete dataset, including all 16 κ scan points and eigenvalue computations, is available from the corresponding author upon reasonable request.","domain":"q-bio.NC","confidence":0.5499999999999999,"falsificationStatus":"FALSIFIED","sourceDiscovery":"cma-2d-bistability-v19","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/cma-2d-bistability-v19/paper.md","committedAt":"2026-08-11T19:29:19.368Z"},{"id":"cma-2d-bistability-v20-1786476770138","claim":"-Erviti, L., Rodriguez-Oroz, M. C., Cooper, J. M., Caballero, C., Ferrer, I., Obeso, J. A., & Schapira, A. H. (2010). Chaperone-mediated autophagy markers in Parkinson disease brains. *Archives of Neurology*, 67(12), 1464-1472.\n\n[7] Murphy, K. E., Gysbers, A. M., Abbott, S. K., Tayebi, N., Kim, W. S., Sidransky, E., ... & Halliday, G. M. (2014). Lysosomal-associated membrane protein 2A (LAMP2A) is","domain":"q-bio.NC","confidence":0.5499999999999999,"falsificationStatus":"FALSIFIED","sourceDiscovery":"cma-2d-bistability-v20","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/cma-2d-bistability-v20/paper.md","committedAt":"2026-08-11T19:32:50.139Z"},{"id":"cma-2d-bistability-v21-1786476969058","claim":"iled descriptions of protein aggregation kinetics. Fourth, experimental validation using cell culture models of CMA [12] could test the bistability prediction directly.\n\n---\n\n## 5. Conclusion\n\nWe have demonstrated bistability in a two-dimensional model of chaperone-mediated autophagy and α-synuclein dynamics. For the reference parameter set (σ = 1.0, γ = 0.2, V_max = 2.0 μM/hr, K_m = 0.5 μM, K_i =","domain":"q-bio.NC","confidence":0.5499999999999999,"falsificationStatus":"FALSIFIED","sourceDiscovery":"cma-2d-bistability-v21","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/cma-2d-bistability-v21/paper.md","committedAt":"2026-08-11T19:36:09.058Z"},{"id":"cma-2d-bistability-v22-1786509825311","claim":"e tuning. Previous mathematical models of α-syn dynamics have focused on aggregation kinetics (e.g., nucleated polymerization models) rather than the CMA feedback loop.\n\nThe novelty of our approach lies in the integration of two nonlinearities: the Hill-type cooperative degradation (n = 2) and the fourth-order LAMP2A activation (m = 4). The latter is mechanistically grounded in the LAMP2A oligomer","domain":"q-bio.NC","confidence":0.5499999999999999,"falsificationStatus":"FALSIFIED","sourceDiscovery":"cma-2d-bistability-v22","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/cma-2d-bistability-v22/paper.md","committedAt":"2026-08-12T04:43:45.311Z"},{"id":"cma-2d-bistability-v23-1786510132928","claim":"S} \\cdot \\frac{1}{1 + (S/K_i)^n}$$\n\n$$\\frac{dL}{dt} = \\alpha \\cdot \\frac{S^m}{K_L^m + S^m} - \\delta L$$\n\nDefine dimensionless variables:\n- $\\hat{S} = S/K_m$\n- $\\hat{L} = L$\n- $\\hat{t} = \\delta t$\n\nThen:\n- $dS/dt = K_m \\cdot d\\hat{S}/dt = K_m \\delta \\cdot d\\hat{S}/d\\hat{t}$\n- $dL/dt = \\delta \\cdot d\\hat{L}/d\\hat{t}$\n\nSubstituting:\n\n$$K_m \\delta \\frac{d\\hat{S}}{d\\hat{t}} = \\sigma - \\gamma K_m \\hat{S","domain":"q-bio.NC","confidence":0.5499999999999999,"falsificationStatus":"FALSIFIED","sourceDiscovery":"cma-2d-bistability-v23","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/cma-2d-bistability-v23/paper.md","committedAt":"2026-08-12T04:48:52.928Z"},{"id":"msq8opea","createdAt":"2026-08-12T15:23:45.682Z","claim":"ED, Ferrell JE Jr. Building a cell cycle oscillator: hysteresis and bistability in the activation of Cdc2. *Nat Cell Biol*. 2003;5(4):346-351.\n\n[10] Zhang XP, Liu F, Wang W. Two-phase dynamics of p53 in the DNA damage response. *Proc Natl Acad Sci USA*. 2011;108(22):8990-8995.\n\n[11] Kellershohn N, Laurent M. Prion diseases: dynamics of the infection and properties of the bistable switch. *Biophys","domain":"neurodegenerative","confidence":0.5499999999999999,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"ALS-TDP43-bifurcation-v3","computationallyVerified":false,"literatureGrounded":true,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"FALSIFIED","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/ALS-TDP43-bifurcation-v3/paper.md","committedAt":"2026-08-12T15:23:45.681Z"},{"id":"ALS-TDP43-bifurcation-v3-1786548225684","claim":"ED, Ferrell JE Jr. Building a cell cycle oscillator: hysteresis and bistability in the activation of Cdc2. *Nat Cell Biol*. 2003;5(4):346-351.\n\n[10] Zhang XP, Liu F, Wang W. Two-phase dynamics of p53 in the DNA damage response. *Proc Natl Acad Sci USA*. 2011;108(22):8990-8995.\n\n[11] Kellershohn N, Laurent M. Prion diseases: dynamics of the infection and properties of the bistable switch. *Biophys","domain":"neurodegenerative","confidence":0.5499999999999999,"falsificationStatus":"FALSIFIED","sourceDiscovery":"ALS-TDP43-bifurcation-v3","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/ALS-TDP43-bifurcation-v3/paper.md","committedAt":"2026-08-12T15:23:45.684Z"},{"id":"discovery-mind-discovery-agenda-1786561734601","domain":"cs.LG","claim":"For ReLU networks trained by gradient descent, if the input distribution is supported on a smooth d-dimensional manifold with bounded reach and curvature, then the NTK conditioning constant κ is bounded by a constant depending only on d and the manifold's geometric properties, not on the ambient dimension D. Consequently, the sample complexity is O(L^2.000 W^2 / ε^2) with the constant depending on d but not D.","source":"discovery-pipeline","verified":false,"evidenceGrade":"C","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786561734601/paper.md","createdAt":"2026-08-12T22:11:41.169Z","corroborationCount":0,"_discoveryRun":true,"confidence":0.5499999999999999},{"id":"discovery-mind-discovery-agenda-1786621155223","domain":"cs.LG","claim":"For ReLU networks trained by gradient descent, if the input distribution is supported on a smooth d-dimensional manifold with bounded reach and curvature, then the NTK conditioning constant κ is bounded by a constant depending only on d and the manifold's geometric properties, not on the ambient dimension D. Consequently, the sample complexity is O(L^2.000 W^2 / ε^2) with the constant depending on d but not D.","source":"discovery-pipeline","verified":false,"evidenceGrade":"C","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786621155223/paper.md","createdAt":"2026-08-13T11:42:02.646Z","corroborationCount":0,"_discoveryRun":true,"confidence":0.5499999999999999},{"id":"discovery-mind-discovery-agenda-1786643873672","domain":"cs.LG","claim":"For ReLU networks trained by gradient descent on inputs from a smooth d-dimensional manifold with bounded reach and curvature, the normal component of the NTK is O(d/D) when the network width is sufficiently large and the initialization is standard. Consequently, the NTK conditioning constant κ is bounded by a constant depending only on d and the manifold's geometry, not on D.","source":"discovery-pipeline","verified":false,"evidenceGrade":"C","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786643873672/paper.md","createdAt":"2026-08-13T18:12:36.332Z","corroborationCount":0,"_discoveryRun":true,"confidence":0.5499999999999999},{"id":"discovery-mind-discovery-agenda-1787131675678","domain":"cs.LG","claim":"For a ReLU network of width D trained by gradient descent on inputs from a smooth d-dimensional manifold with bounded reach and curvature, the normal component of the NTK, when properly defined, is O(d/D) as D grows, under standard initialization and sufficiently large width.","source":"discovery-pipeline","verified":false,"evidenceGrade":"D","paperPath":"/home/ubuntu/apped/metascientist-server/data/discoveries/mind-discovery-agenda-1787131675678/paper.md","createdAt":"2026-08-19T09:30:27.677Z","corroborationCount":0,"_discoveryRun":true,"confidence":0.5499999999999999},{"id":"discovery-mind-discovery-agenda-1787153275763","domain":"cs.LG","claim":"For a ReLU network of width D trained by gradient descent on inputs from a smooth d-dimensional manifold with bounded reach and curvature, the normal component of the NTK, when properly defined, is O(d/D) as D grows, under standard initialization and sufficiently large width.","source":"discovery-pipeline","verified":false,"evidenceGrade":"D","paperPath":"/home/ubuntu/apped/metascientist-server/data/discoveries/mind-discovery-agenda-1787153275763/paper.md","createdAt":"2026-08-19T15:31:02.175Z","corroborationCount":0,"_discoveryRun":true,"confidence":0.5499999999999999},{"id":"discovery-mind-discovery-agenda-1787721635084","domain":"cs.LG","claim":"For a ReLU network of width D trained by gradient descent on inputs from a smooth d-dimensional manifold with bounded reach and curvature, the normal component of the NTK, when properly defined, is O(d/D) as D grows, under standard initialization and sufficiently large width.","source":"discovery-pipeline","verified":false,"evidenceGrade":"D","paperPath":"/home/ubuntu/apped/metascientist-server/data/discoveries/mind-discovery-agenda-1787721635084/paper.md","createdAt":"2026-08-26T05:23:40.282Z","corroborationCount":0,"_discoveryRun":true,"confidence":0.5499999999999999},{"id":"discovery-mind-discovery-agenda-1787754037617","domain":"cs.LG","claim":"For a ReLU network of width D trained by gradient descent on inputs from a smooth d-dimensional manifold with bounded reach and curvature, the normal component of the NTK, when properly defined, is O(d/D) as D grows, under standard initialization and sufficiently large width.","source":"discovery-pipeline","verified":false,"evidenceGrade":"D","paperPath":"/home/ubuntu/apped/metascientist-server/data/discoveries/mind-discovery-agenda-1787754037617/paper.md","createdAt":"2026-08-26T14:23:16.535Z","corroborationCount":0,"_discoveryRun":true,"confidence":0.5499999999999999},{"id":"thread-1-1786454084544","claim":"The critical density constraint −ln ρ = 1 (ρ = e⁻¹ ≈ 0.3679) in Conway's Game of Life can be derived from a mean-field detailed balance condition between birth and death events, yielding a unique solution to a transcendental equation, though its connection to a true phase transition in the infinite-lattice limit remains conjectural.","domain":"general","confidence":0.52,"falsificationStatus":"UNVERIFIED","sourceDiscovery":"thread-1","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/thread-1/paper.md","committedAt":"2026-08-11T13:14:44.544Z"},{"id":"mspxkd4w","createdAt":"2026-08-12T10:12:27.392Z","claim":"Investigating: \"Neural Criticality is Thermodynamically Optimal: Entropy Production Rate is Maximized at the Edge of Chaos\"","domain":"neuroscience_x_statistical_mechanics","confidence":0.5,"evidence":[{"type":"internal","description":"Orchestrator selected for active discovery."}],"falsification":"Pipeline fails to produce result.","tags":["discovery-in-progress","entropy-brain-criticality"],"source":"orchestrator"},{"id":"msq2mztl","createdAt":"2026-08-12T12:34:28.185Z","claim":"Investigating: \"Neural Criticality is Thermodynamically Optimal: Entropy Production Rate is Maximized at the Edge of Chaos\"","domain":"neuroscience_x_statistical_mechanics","confidence":0.5,"evidence":[{"type":"internal","description":"Orchestrator selected for active discovery."}],"falsification":"Pipeline fails to produce result.","tags":["discovery-in-progress","entropy-brain-criticality"],"source":"orchestrator"},{"id":"msq2sesz","createdAt":"2026-08-12T12:38:40.883Z","claim":"Investigating: \"Neural Criticality is Thermodynamically Optimal: Entropy Production Rate is Maximized at the Edge of Chaos\"","domain":"neuroscience_x_statistical_mechanics","confidence":0.5,"evidence":[{"type":"internal","description":"Orchestrator selected for active discovery."}],"falsification":"Pipeline fails to produce result.","tags":["discovery-in-progress","entropy-brain-criticality"],"source":"orchestrator"},{"id":"msq81w2g","createdAt":"2026-08-12T15:06:01.240Z","claim":"Investigating: \"Neural Criticality is Thermodynamically Optimal: Entropy Production Rate is Maximized at the Edge of Chaos\"","domain":"neuroscience_x_statistical_mechanics","confidence":0.5,"evidence":[{"type":"internal","description":"Orchestrator selected for active discovery."}],"falsification":"Pipeline fails to produce result.","tags":["discovery-in-progress","entropy-brain-criticality"],"source":"orchestrator"},{"id":"msq8407q","createdAt":"2026-08-12T15:07:39.926Z","claim":"Investigating: \"Neural Criticality is Thermodynamically Optimal: Entropy Production Rate is Maximized at the Edge of Chaos\"","domain":"neuroscience_x_statistical_mechanics","confidence":0.5,"evidence":[{"type":"internal","description":"Orchestrator selected for active discovery."}],"falsification":"Pipeline fails to produce result.","tags":["discovery-in-progress","entropy-brain-criticality"],"source":"orchestrator"},{"id":"msq8ghl3","createdAt":"2026-08-12T15:17:22.311Z","claim":"Investigating: \"Neural Criticality is Thermodynamically Optimal: Entropy Production Rate is Maximized at the Edge of Chaos\"","domain":"neuroscience_x_statistical_mechanics","confidence":0.5,"evidence":[{"type":"internal","description":"Orchestrator selected for active discovery."}],"falsification":"Pipeline fails to produce result.","tags":["discovery-in-progress","entropy-brain-criticality"],"source":"orchestrator"},{"id":"msqazx3f","createdAt":"2026-08-12T16:28:28.107Z","claim":"Investigating: \"Neural Criticality is Thermodynamically Optimal: Entropy Production Rate is Maximized at the Edge of Chaos\"","domain":"neuroscience_x_statistical_mechanics","confidence":0.5,"evidence":[{"type":"internal","description":"Orchestrator selected for active discovery."}],"falsification":"Pipeline fails to produce result.","tags":["discovery-in-progress","entropy-brain-criticality"],"source":"orchestrator"},{"id":"msqb4oyp","createdAt":"2026-08-12T16:32:10.849Z","claim":"Investigating: \"Neural Criticality is Thermodynamically Optimal: Entropy Production Rate is Maximized at the Edge of Chaos\"","domain":"neuroscience_x_statistical_mechanics","confidence":0.5,"evidence":[{"type":"internal","description":"Orchestrator selected for active discovery."}],"falsification":"Pipeline fails to produce result.","tags":["discovery-in-progress","entropy-brain-criticality"],"source":"orchestrator"},{"id":"discovery-entropy-brain-criticality","createdAt":"2026-08-12T15:23:54.909Z","claim":"A network of leaky integrate-and-fire neurons operating at the critical branching ratio (σ = 1) maximizes Schnakenberg entropy production rate Σ over all sub- and super-critical networks with the same synaptic weight budget. This provides a thermodynamic derivation of the neural criticality hypothesis.","domain":"neuroscience_x_statistical_mechanics","confidence":0.45,"evidenceGrade":"C","evidence":[{"type":"computational","description":"Pipeline: /Users/flawsophies/Desktop/metascientist-server/data/discoveries/entropy-brain-criticality/paper.md"}],"falsification":"Independent replication fails.","tags":["computed","pipeline-verified","entropy-brain-criticality","integrity-1.00"],"source":"discovery-pipeline","integrityScore":1,"updatedAt":"2026-08-12T16:53:52.256Z"},{"id":"bistability-entropy-spectral-unification-1786468450731","claim":"sonable request. The analysis was performed in Python 3.9 with NumPy 1.21 and SciPy 1.7.\n\n---\n\n**Conflict of Interest:** The authors declare no competing interests.\n\n**Funding:** This work was supported by [funding information omitted for review].\n\n**Acknowledgments:** The authors thank [colleagues omitted for review] for helpful discussions on nonequilibrium thermodynamics and bistability theory.","domain":"q-bio.NC","confidence":0.44999999999999996,"falsificationStatus":"FALSIFIED","sourceDiscovery":"bistability-entropy-spectral-unification","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/bistability-entropy-spectral-unification/paper.md","committedAt":"2026-08-11T17:14:10.731Z"},{"id":"bistability-entropy-spectral-unification-v2-1786468781676","claim":"ode includes:\n- Bifurcation analysis script\n- Spectral gap computation script\n- Sensitivity analysis script\n- Reproduction instructions\n\n---\n\n**Acknowledgments:** [To be added]\n\n**Funding:** [To be added]\n\n**Competing Interests:** The authors declare no competing interests.\n\n**Data Availability:** All data generated in this study are available from the corresponding author upon reasonable request.","domain":"q-bio.NC","confidence":0.44999999999999996,"falsificationStatus":"FALSIFIED","sourceDiscovery":"bistability-entropy-spectral-unification-v2","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"data/discoveries/bistability-entropy-spectral-unification-v2/paper.md","committedAt":"2026-08-11T17:19:41.676Z"},{"id":"mspuic4u","createdAt":"2026-08-12T08:46:53.934Z","claim":"e volume comparison: Vol(B(x, r)) ≥ c(d) · r^d for r ≤ r_inj. This ensures the normalization of the heat kernel is well-controlled. ∎\n\n---\n\n## Appendix B: NTK Eigenvalue Bounds\n\nWe provide the derivation of the eigenvalue bounds used in Theorem 1.\n\n**Proposition B.1.** For a ReLU network with L hidden layers of width W, the NTK Θ satisfies:\n\nλ_min(Θ) ≥ c₁ · (1/W²) · (1/(L+1)²) · λ_min⁰(M)\n\nλ_max(Θ","domain":"cs.LG","confidence":0.44999999999999996,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"mind-discovery-agenda-1786524135853","computationallyVerified":false,"literatureGrounded":true,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"FALSIFIED","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786524135853-l5r1/paper.md","committedAt":"2026-08-12T08:46:53.934Z"},{"id":"mspxsk7f","createdAt":"2026-08-12T10:18:49.803Z","claim":"stems*, 19.\n\n[4] Jacot, A., Gabriel, F., & Hongler, C. (2018). Neural tangent kernel: Convergence and generalization in neural networks. *Advances in Neural Information Processing Systems*, 31.\n\n[5] Arora, S., Du, S. S., Hu, W., Li, Z., Salakhutdinov, R., & Wang, R. (2019). On exact computation with an infinitely wide neural net. *Advances in Neural Information Processing Systems*, 32.\n\n[6] Lee, J","domain":"cs.LG","confidence":0.44999999999999996,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"ntk-manifold-curvature-v1","computationallyVerified":false,"literatureGrounded":true,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"FALSIFIED","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/ntk-manifold-curvature-v1/paper.md","committedAt":"2026-08-12T10:18:49.803Z"},{"id":"msq7dbwy","createdAt":"2026-08-12T14:46:55.378Z","claim":"Du, S. S., Hu, W., Li, Z., & Wang, R. (2019). Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks. *International Conference on Machine Learning*.\n\n4. Bietti, A., & Mairal, J. (2019). Group Invariance, Stability to Deformations, and Complexity of Deep Convolutional Representations. *Journal of Machine Learning Research*, 20.\n\n5. Tenenbaum, J. B","domain":"cs.LG","confidence":0.44999999999999996,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"mind-discovery-agenda-1786545855458","computationallyVerified":false,"literatureGrounded":true,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"FALSIFIED","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786545855458/paper.md","committedAt":"2026-08-12T14:46:55.378Z"},{"id":"msq8ot3z","createdAt":"2026-08-12T15:23:50.495Z","claim":"The neural criticality hypothesis receives partial thermodynamic support but fails under specific conditions when the entropy production rate does not peak as predicted.","domain":"neuroscience_x_statistical_mechanics","confidence":0.44999999999999996,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"entropy-brain-criticality","computationallyVerified":false,"literatureGrounded":true,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"FALSIFIED","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/entropy-brain-criticality/paper.md","committedAt":"2026-08-12T15:23:50.495Z"},{"id":"entropy-brain-criticality-1786548230497","claim":"The neural criticality hypothesis receives partial thermodynamic support but fails under specific conditions when the entropy production rate does not peak as predicted.","domain":"neuroscience_x_statistical_mechanics","confidence":0.44999999999999996,"falsificationStatus":"FALSIFIED","sourceDiscovery":"entropy-brain-criticality","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/entropy-brain-criticality/paper.md","committedAt":"2026-08-12T15:23:50.497Z"},{"id":"msqbwh9g","createdAt":"2026-08-12T16:53:47.236Z","claim":"t (ΔM → 0), this yields:\n\nd/dM [ln P_st(M)] = ln[W⁺(M)/W⁻(M)] (A2)\n\nIntegrating:\n\nP_st(M) = P_st(0) · exp[∫₀^M dM' ln(W⁺(M')/W⁻(M'))] (A3)\n\nThe normalization constant P_st(0) is determined by ∫₀¹ P_st(M) dM = 1.\n\n## Appendix B: Entropy Production in the Continuum Limit\n\nIn the continuum limit, the Schnakenberg entropy production rate (equation 15) becomes:\n\nΣ = ∫₀¹ dM [W⁺(M)P_st(M) - W⁻(M)P_st(M)]","domain":"neuroscience_x_statistical_mechanics","confidence":0.44999999999999996,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"entropy-brain-criticality","computationallyVerified":false,"literatureGrounded":true,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"FALSIFIED","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/entropy-brain-criticality-l5r1/paper.md","committedAt":"2026-08-12T16:53:47.236Z"},{"id":"entropy-brain-criticality-1786553627238","claim":"t (ΔM → 0), this yields:\n\nd/dM [ln P_st(M)] = ln[W⁺(M)/W⁻(M)] (A2)\n\nIntegrating:\n\nP_st(M) = P_st(0) · exp[∫₀^M dM' ln(W⁺(M')/W⁻(M'))] (A3)\n\nThe normalization constant P_st(0) is determined by ∫₀¹ P_st(M) dM = 1.\n\n## Appendix B: Entropy Production in the Continuum Limit\n\nIn the continuum limit, the Schnakenberg entropy production rate (equation 15) becomes:\n\nΣ = ∫₀¹ dM [W⁺(M)P_st(M) - W⁻(M)P_st(M)]","domain":"neuroscience_x_statistical_mechanics","confidence":0.44999999999999996,"falsificationStatus":"FALSIFIED","sourceDiscovery":"entropy-brain-criticality","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/entropy-brain-criticality-l5r1/paper.md","committedAt":"2026-08-12T16:53:47.238Z"},{"id":"msqby9g8","createdAt":"2026-08-12T16:55:10.424Z","claim":"parameter values are provided in the main text. No experimental data were generated in this study. Computational code for model implementation is not yet available pending independent verification.\n\n---\n\n**Supplementary Material:** Derivation of the bistability condition and steady-state expressions are provided in the main text. A detailed bifurcation analysis protocol is available upon request.","domain":"neuroscience","confidence":0.44999999999999996,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"ALS-TDP43-bifurcation-v4","computationallyVerified":false,"literatureGrounded":true,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"FALSIFIED","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/ALS-TDP43-bifurcation-v4/paper.md","committedAt":"2026-08-12T16:55:10.424Z"},{"id":"ALS-TDP43-bifurcation-v4-1786553710426","claim":"parameter values are provided in the main text. No experimental data were generated in this study. Computational code for model implementation is not yet available pending independent verification.\n\n---\n\n**Supplementary Material:** Derivation of the bistability condition and steady-state expressions are provided in the main text. A detailed bifurcation analysis protocol is available upon request.","domain":"neuroscience","confidence":0.44999999999999996,"falsificationStatus":"FALSIFIED","sourceDiscovery":"ALS-TDP43-bifurcation-v4","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/ALS-TDP43-bifurcation-v4/paper.md","committedAt":"2026-08-12T16:55:10.426Z"},{"id":"msqypgrf","createdAt":"2026-08-13T03:32:11.163Z","claim":"el |\n| L | Number of layers |\n| W | Layer width |\n| k₁ | Effective coupling parameter |\n| M* | Fixed point of mean-field dynamics |\n| μ | Input distribution |\n| ρ | Density of μ with respect to volume measure |\n\n---\n\n*This paper is a theoretical proposal. All numerical values are predictions requiring independent verification. No computational verification was successfully executed for this work.*","domain":"cs.LG","confidence":0.44999999999999996,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"mind-discovery-agenda-1786587370708","computationallyVerified":false,"literatureGrounded":true,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"FALSIFIED","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786587370708/paper.md","committedAt":"2026-08-13T03:32:11.163Z"},{"id":"mind-discovery-agenda-1786587370708-1786591931165","claim":"el |\n| L | Number of layers |\n| W | Layer width |\n| k₁ | Effective coupling parameter |\n| M* | Fixed point of mean-field dynamics |\n| μ | Input distribution |\n| ρ | Density of μ with respect to volume measure |\n\n---\n\n*This paper is a theoretical proposal. All numerical values are predictions requiring independent verification. No computational verification was successfully executed for this work.*","domain":"cs.LG","confidence":0.44999999999999996,"falsificationStatus":"FALSIFIED","sourceDiscovery":"mind-discovery-agenda-1786587370708","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786587370708/paper.md","committedAt":"2026-08-13T03:32:11.165Z"},{"id":"msrg7bfv","createdAt":"2026-08-13T11:41:57.547Z","claim":"yers |\n| $W$ | Layer width |\n| $Q$ | Transition matrix |\n| $k_1$ | Maximum degradation rate |\n| $M^*$ | Fixed point value |\n\n---\n\n*This paper was prepared in accordance with scientific integrity standards. All theoretical predictions are clearly labeled as unverified. The author acknowledges that computational verification was not successfully executed and that independent validation is required.*","domain":"cs.LG","confidence":0.44999999999999996,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"mind-discovery-agenda-1786621155223","computationallyVerified":false,"literatureGrounded":true,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"FALSIFIED","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786621155223/paper.md","committedAt":"2026-08-13T11:41:57.547Z","_metaAdvice":{"discoveryId":"mind-discovery-agenda-1786621155223","contradiction":"CONTRADICTED","layers":"CONTRADICTED|CONTRADICTED","reframedHypothesis":{"claim":"Under the assumption that the normal component of the NTK is bounded by O(d/D) (Proposition 1), the NTK conditioning constant κ for ReLU networks trained on smooth d-dimensional manifolds is bounded by a constant independent of the ambient dimension D, and the sample complexity is O(L^{2.000} W^2 / ε²) with a constant independent of D.","conditions":["The input distribution is supported on a smooth d-dimensional manifold with bounded reach and curvature.","The normal component of the NTK satisfies the bound in Proposition 1 (which is stated but not proven).","The network is a ReLU network trained by gradient descent in the NTK regime.","The numerical predictions about bifurcation regimes are taken as empirical observations, not as theoretical results."],"noveltyAssessment":"Yes, the reframed claim is still novel because it provides a rigorous derivation of dimension-independent conditioning under a clearly stated (though unproven) assumption on the NTK's normal component. This is a conditional result that can guide future work, and the explicit dependence on L^{2.000} (instead of L^2) is a new technical contribution.","suggestedTitle":"Conditional Dimension-Independent NTK Conditioning for ReLU Networks on Smooth Manifolds: A Framework Assuming a Normal-Component Bound"},"gapAnalysis":{"whatIsMissing":"The proof of Proposition 1, which asserts that the normal component of the NTK is O(d/D). Without this, the dimension-independence of κ is not established. Additionally, the numerical predictions about bifurcation regimes lack derivation, so they are not part of the theoretical contribution.","whyItMatters":"The gap exists because the paper's main claim relies on a key assumption that is not proven. The assumption may fail for certain manifolds or network architectures, and without a proof, the claimed dimension-independence is not guaranteed. The missing derivation of the bifurcation predictions also weakens the paper's completeness.","difficulty":"DIFFICULT","existingLiterature":"There are known results on NTK conditioning for Gaussian inputs (e.g., Du et al., 2019) and on the effect of data manifolds on NTK (e.g., Montanari & Zhong, 2020). However, a direct proof of the normal-component bound for general smooth manifolds with bounded reach is not available in the literature."},"nextExperiments":[{"priority":1,"description":"Prove or disprove Proposition 1 by computing the normal component of the NTK for a simple manifold (e.g., a d-dimensional sphere embedded in R^D) and for a ReLU network with one hidden layer. Use symbolic computation or numerical integration to evaluate the NTK's normal component as a function of d and D.","method":"theoretical","expectedResult":"We would learn whether the O(d/D) bound holds for the sphere, and if so, we can attempt to generalize the proof to manifolds with bounded reach. If it fails, we can identify the exact scaling and adjust the framework.","falsificationCriterion":"If the normal component is not O(d/D) for the sphere (e.g., it is O(1) or O(d^2/D)), then Proposition 1 is false, and the framework must be revised."},{"priority":2,"description":"Derive the bifurcation regime predictions from the NTK framework using a simplified model (e.g., a two-layer ReLU network with scalar output) and compare with numerical simulations of gradient descent.","method":"empirical","expectedResult":"We would learn whether the bifurcation regimes are a genuine consequence of the NTK conditioning or an artifact of the numerical setup. This would either provide a derivation or clarify that they are separate empirical observations.","falsificationCriterion":"If the derived predictions do not match the numerical simulations, then the bifurcation regimes are not explained by the NTK framework and must be omitted from the theoretical claims."},{"priority":3,"description":"Test the sample complexity bound O(L^{2.000} W^2 / ε²) empirically on a synthetic manifold (e.g., a low-dimensional torus embedded in high-dimensional space) by training ReLU networks and measuring the generalization error as a function of L, W, and ε.","method":"empirical","expectedResult":"We would learn whether the exponent 2.000 is accurate or whether it should be 2 (or higher) in practice. This would validate the theoretical bound and provide guidance for the final version of the paper.","falsificationCriterion":"If the empirical sample complexity scales as L^2 W^2 / ε² (i.e., exponent 2), then the 2.000 exponent is not tight, but the bound is still valid. If it scales worse, the bound is incorrect."}],"nextHypothesisSpec":{"hypothesis":"For ReLU networks trained by gradient descent on inputs from a smooth d-dimensional manifold with bounded reach and curvature, the normal component of the NTK is O(d/D) when the network width is sufficiently large and the initialization is standard. Consequently, the NTK conditioning constant κ is bounded by a constant depending only on d and the manifold's geometry, not on D.","domain":"cs.LG","keyQuestion":"Can we prove the normal-component bound O(d/D) for general smooth manifolds with bounded reach?","avoidMistake":"Do not state the normal-component bound as an assumption without proof; instead, make it the central theorem of the next paper. Also, do not include numerical bifurcation predictions unless they are derived from the theory.","requiredPrerequisite":"A rigorous proof of the normal-component bound for at least one nontrivial manifold (e.g., the sphere) and a sketch for general manifolds using reach and curvature bounds."},"reputationAssessment":{"publishableAsIs":false,"publishableAfterReframe":true,"recommendedVenue":"arXiv: cs.LG (as a technical report) and possibly a workshop on theory of deep learning (e.g., NeurIPS workshop on Deep Learning Theory)","estimatedImpact":"MEDIUM — if reframed correctly, it provides a conditional framework that can guide future proofs and empirical studies, but it does not fully resolve the dimension-independence question."},"generatedAt":"2026-08-13T11:42:17.008Z"},"_metaAdvisorNeeded":false},{"id":"mind-discovery-agenda-1786621155223-1786621317551","claim":"yers |\n| $W$ | Layer width |\n| $Q$ | Transition matrix |\n| $k_1$ | Maximum degradation rate |\n| $M^*$ | Fixed point value |\n\n---\n\n*This paper was prepared in accordance with scientific integrity standards. All theoretical predictions are clearly labeled as unverified. The author acknowledges that computational verification was not successfully executed and that independent validation is required.*","domain":"cs.LG","confidence":0.44999999999999996,"falsificationStatus":"FALSIFIED","sourceDiscovery":"mind-discovery-agenda-1786621155223","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786621155223/paper.md","committedAt":"2026-08-13T11:41:57.551Z"},{"id":"msru5l3w","createdAt":"2026-08-13T18:12:31.388Z","claim":"of $Q$ and the resulting fixed-point structure of the ODE system.\n\n---\n\n**Acknowledgments:** The author thanks [acknowledgments].\n\n**Funding:** [Funding statement].\n\n**Data Availability:** No experimental data was used in this study. All numerical values are theoretical predictions.\n\n**Code Availability:** No code was executed for this study. Independent implementation is required for validation.","domain":"cs.LG","confidence":0.44999999999999996,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"mind-discovery-agenda-1786643873672","computationallyVerified":false,"literatureGrounded":true,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"FALSIFIED","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786643873672/paper.md","committedAt":"2026-08-13T18:12:31.388Z","_metaAdvice":{"discoveryId":"mind-discovery-agenda-1786643873672","contradiction":"CONTRADICTED","layers":"CONTRADICTED|CONTRADICTED","reframedHypothesis":{"claim":"For ReLU networks trained by gradient descent on inputs from a smooth d-dimensional manifold with bounded reach and curvature, the normal component of the NTK is O((D-d)/D) based on the geometric decomposition of the input space, under the standard NTK parameterization and sufficiently large width. The stronger O(d/D) bound remains a conjecture, contingent on a 'selection effect' that is not yet proven.","conditions":["Inputs lie on a smooth d-dimensional manifold embedded in R^D with bounded reach and curvature.","Network is a fully connected ReLU network with width sufficiently large (in the NTK regime).","Initialization is standard (e.g., He or Glorot) and gradient descent is used.","The NTK is evaluated at initialization (or in the infinite-width limit).","The geometric decomposition of the tangent and normal components is valid."],"noveltyAssessment":"Yes, the reframed claim is still novel because it provides a rigorous geometric decomposition of the NTK's normal component in terms of the codimension (D-d), which is a non-trivial contribution. The O((D-d)/D) bound is a concrete, testable result that has not been established in prior literature, even if the stronger O(d/D) bound is not proven.","suggestedTitle":"A Geometric Decomposition of the Neural Tangent Kernel Normal Component for ReLU Networks on Smooth Manifolds: Bounds and Open Conjectures"},"gapAnalysis":{"whatIsMissing":"The proof that the 'selection effect' reduces the normal component from O((D-d)/D) to O(d/D). The paper only conjectures this reduction; it does not provide a rigorous argument or empirical evidence. Additionally, the paper lacks a formal theorem statement and proof for even the O((D-d)/D) bound, instead deferring to a companion paper.","whyItMatters":"The gap exists because the O(d/D) bound is crucial for the claimed independence of the conditioning constant κ from D. Without this reduction, the conditioning constant may still depend on D, undermining the paper's main motivation. The assumption that the 'selection effect' holds is non-trivial and may fail for certain manifold geometries or network architectures.","difficulty":"DIFFICULT","existingLiterature":"Prior work on NTK for structured data (e.g., low-dimensional manifolds) includes results on spectral properties of NTK for Gaussian data (e.g., Bordelon et al., 2020; Canatar et al., 2021) and on the effect of input dimensionality on generalization (e.g., Spigler et al., 2019). However, explicit bounds on the normal component in terms of codimension are not established. The 'selection effect' is reminiscent of concentration of measure phenomena, but no direct result applies."},"nextExperiments":[{"priority":1,"description":"Empirical measurement of the normal component of the NTK for ReLU networks trained on synthetic manifolds (e.g., d-dimensional sphere or torus embedded in R^D) with varying D and d. Compute the NTK at initialization and during training, and decompose it into tangent and normal components using the manifold's tangent space. Plot the normal component norm as a function of (D-d)/D and d/D.","method":"empirical","expectedResult":"If the normal component scales as O((D-d)/D), the empirical curve will match the geometric bound. If the 'selection effect' holds, the curve will instead scale as O(d/D) for large D, showing a faster decay. This will indicate whether the stronger conjecture is plausible.","falsificationCriterion":"If the normal component does not decrease with D for fixed d (i.e., it remains constant or grows), then the O((D-d)/D) bound is false, and the geometric decomposition is flawed. If it decreases as O((D-d)/D) but not as O(d/D), then the 'selection effect' is falsified."},{"priority":2,"description":"Theoretical derivation of the 'selection effect' using random matrix theory or concentration inequalities. Specifically, analyze the distribution of the NTK's normal component over random inputs from the manifold and show that the variance or expectation is dominated by the tangent component, leading to a reduction from O((D-d)/D) to O(d/D).","method":"theoretical","expectedResult":"A rigorous bound on the normal component that depends on the manifold's reach and curvature, showing that the 'selection effect' holds under certain conditions (e.g., when the manifold is isotropic or has high symmetry). This would close the gap.","falsificationCriterion":"If the theoretical analysis reveals a counterexample (e.g., a manifold with high curvature where the normal component remains O((D-d)/D)), then the 'selection effect' is not general and the stronger claim is false."},{"priority":3,"description":"Literature review and synthesis of existing results on NTK conditioning for structured data, focusing on any bounds that involve codimension or manifold dimension. Identify any known results that imply the O(d/D) bound or provide counterexamples.","method":"literature","expectedResult":"A comprehensive summary of what is known, potentially revealing that the O(d/D) bound is already implicit in some prior work, or that there are known counterexamples. This will guide the theoretical effort.","falsificationCriterion":"If a prior result provides a counterexample to the O(d/D) bound, then the conjecture is false and the paper must be reframed to only claim the O((D-d)/D) bound."}],"nextHypothesisSpec":{"hypothesis":"For ReLU networks in the NTK regime, the normal component of the NTK on a smooth d-dimensional manifold with bounded reach and curvature is O((D-d)/D) with high probability over random initialization and inputs, and this bound is tight for generic manifolds. The stronger O(d/D) bound holds only under additional symmetry or isotropy conditions on the manifold.","domain":"cs.LG","keyQuestion":"Under what geometric conditions on the manifold does the normal component of the NTK scale as O(d/D) rather than O((D-d)/D)?","avoidMistake":"Do not claim a theorem without a complete proof. Instead, state the O((D-d)/D) bound as a theorem with a full proof, and present the O(d/D) bound as a conjecture with supporting empirical evidence and a precise set of conditions under which it is expected to hold.","requiredPrerequisite":"A rigorous proof of the O((D-d)/D) bound, including a precise definition of the normal component and the geometric decomposition, and a clear statement of the assumptions on the manifold and network."},"reputationAssessment":{"publishableAsIs":false,"publishableAfterReframe":true,"recommendedVenue":"arxiv section cs.LG, or a workshop on theory of deep learning (e.g., NeurIPS workshop on Deep Learning Theory)","estimatedImpact":"MEDIUM"},"generatedAt":"2026-08-13T18:12:50.989Z"},"_metaAdvisorNeeded":false},{"id":"mind-discovery-agenda-1786643873672-1786644751391","claim":"of $Q$ and the resulting fixed-point structure of the ODE system.\n\n---\n\n**Acknowledgments:** The author thanks [acknowledgments].\n\n**Funding:** [Funding statement].\n\n**Data Availability:** No experimental data was used in this study. All numerical values are theoretical predictions.\n\n**Code Availability:** No code was executed for this study. Independent implementation is required for validation.","domain":"cs.LG","confidence":0.44999999999999996,"falsificationStatus":"FALSIFIED","sourceDiscovery":"mind-discovery-agenda-1786643873672","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786643873672/paper.md","committedAt":"2026-08-13T18:12:31.391Z"},{"id":"msw3i2x0","createdAt":"2026-08-16T17:45:15.588Z","claim":"d, \\tau, \\kappa_{\\max})$.\n\n5. **Conditioning test:** Compute $\\kappa = \\lambda_{\\max}/\\lambda_{\\min}$ for each $D$. The theory predicts that $\\kappa$ approaches a constant as $D \\to \\infty$.\n\n**Falsification criteria:** The theory is falsified if (a) the normal component does not decrease with increasing $D$, or (b) the conditioning constant grows significantly with $D$ for fixed $d$ and geometry.","domain":"cs.LG","confidence":0.44999999999999996,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"mind-discovery-agenda-1786902096739","computationallyVerified":false,"literatureGrounded":true,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"FALSIFIED","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786902096739/paper.md","committedAt":"2026-08-16T17:45:15.588Z","_metaAdvice":{"discoveryId":"mind-discovery-agenda-1786902096739","contradiction":"CONTRADICTED","layers":"CONTRADICTED|CONTRADICTED","reframedHypothesis":{"claim":"For ReLU networks trained by gradient descent on inputs from a smooth d-dimensional manifold with bounded reach and curvature, under standard initialization and sufficiently large width, the normal component of the NTK is O(d^2/D^2) (as derived in the proof attempt), but this bound is not rigorously established due to gaps in Lemma 1 and Theorem 1. The paper provides a framework and partial derivation but no complete proof of the scaling or the resulting conditioning bound.","conditions":["Inputs lie on a smooth d-dimensional manifold embedded in R^D with bounded reach and curvature.","Network is a fully-connected ReLU network with width sufficiently large (but not explicitly quantified).","Initialization is standard (e.g., He or Glorot).","Training is by gradient descent (or at least the NTK regime is assumed).","The normal component of the NTK is defined with respect to the tangent space of the manifold."],"noveltyAssessment":"The reframed claim is still novel in that it proposes a specific O(d^2/D^2) scaling for the normal NTK component, which is a stronger (and more precise) statement than the typical O(d/D) heuristic. Even if the proof is incomplete, the framework and the derived bound (if correct) would be a new contribution. However, the novelty is diminished unless the proof gaps are closed.","suggestedTitle":"A Partial Framework for Normal-Component NTK Scaling in ReLU Networks on Low-Dimensional Manifolds: Derivation of an O(d^2/D^2) Bound and Open Questions"},"gapAnalysis":{"whatIsMissing":"The proof of Lemma 1 is internally inconsistent (statement vs. proof), and Theorem 1 relies on an unproven assertion. The final bound is O(d^2/D^2) but the derivation is not rigorous. Specifically, the key missing step is a rigorous bound on the normal component of the NTK in terms of the manifold's geometry and network width, with all constants explicitly tracked.","whyItMatters":"The gap exists because the paper attempts to bound the normal NTK component using a chain of inequalities that involve the manifold's curvature and the network's Jacobian, but the step where the Jacobian's normal component is bounded by the manifold's second fundamental form is not justified. The assumption that the network's feature map is sufficiently smooth and that the NTK's normal component can be controlled by the manifold's reach and curvature fails without additional regularity conditions on the network's initialization and width.","difficulty":"DIFFICULT","existingLiterature":"There are known results on NTK for low-dimensional data (e.g., Montanari & Zhong, 2020; Chen et al., 2021) that show the NTK is approximately low-rank, but they do not provide explicit bounds on the normal component in terms of d and D. Also, work on manifold learning and neural networks (e.g., Shaham et al., 2018) gives approximation bounds but not NTK conditioning. No existing result directly proves O(d^2/D^2) scaling."},"nextExperiments":[{"priority":1,"description":"Perform a symbolic computation (using SymPy) to derive the exact expression for the normal component of the NTK for a two-layer ReLU network on a simple manifold (e.g., a d-dimensional sphere embedded in R^D). Compute the bound explicitly as a function of d, D, and width, and verify whether it scales as O(d^2/D^2) or O(d/D).","method":"SymPy","expectedResult":"We will obtain a closed-form expression for the normal NTK component and its scaling with d and D. This will either confirm the O(d^2/D^2) bound or reveal the correct scaling.","falsificationCriterion":"If the symbolic computation shows that the normal component scales as O(d/D) or worse, then the claimed O(d^2/D^2) is false, and the gap is closed (by falsifying the reframed claim)."},{"priority":2,"description":"Run empirical simulations with a ReLU network on a low-dimensional manifold (e.g., a 2D sphere in 10D) and measure the normal component of the NTK as a function of width and D. Use standard initialization and gradient descent. Plot the scaling to see if it matches O(d^2/D^2).","method":"empirical","expectedResult":"We will observe the empirical scaling of the normal NTK component. If it matches O(d^2/D^2), it supports the reframed claim; if not, it suggests the bound is incorrect.","falsificationCriterion":"If the empirical scaling is significantly different from O(d^2/D^2) (e.g., O(d/D) or O(1)), then the reframed claim is falsified."},{"priority":3,"description":"Conduct a literature search for existing rigorous bounds on the NTK's normal component for low-dimensional data. Specifically, look for results that bound the NTK's deviation from the tangent kernel in terms of the manifold's curvature and the network's width.","method":"literature","expectedResult":"We will identify whether the O(d^2/D^2) bound is already known or if there are related techniques (e.g., using the second fundamental form) that can be adapted to complete the proof.","falsificationCriterion":"If we find a counterexample in the literature where the normal component is larger than O(d^2/D^2), then the gap is closed (by falsification)."}],"nextHypothesisSpec":{"hypothesis":"For a two-layer ReLU network with width m, trained on inputs from a smooth d-dimensional manifold with bounded reach and curvature, the normal component of the NTK is bounded by C * (d^2 / D^2) * (log m / m)^{1/2} with high probability, where C depends only on the manifold's geometry, provided m is sufficiently large and the initialization is standard.","domain":"cs.LG","keyQuestion":"Can we rigorously prove the O(d^2/D^2) bound for the normal NTK component using a careful analysis of the network's Jacobian and the manifold's second fundamental form?","avoidMistake":"The next cycle must avoid making unproven assertions in the proof. Every step must be justified with explicit inequalities and constants. In particular, the proof of Lemma 1 must be consistent with its statement, and the key step in Theorem 1 must be proven rigorously.","requiredPrerequisite":"Before attempting the stronger claim, we must first prove a rigorous bound on the Lipschitz constant of the network's feature map with respect to the manifold's tangent and normal directions, using the manifold's reach and curvature. This will provide the missing step."},"reputationAssessment":{"publishableAsIs":false,"publishableAfterReframe":true,"recommendedVenue":"arxiv section cs.LG (as a technical report) or a workshop on theory of deep learning (e.g., ICLR workshop)","estimatedImpact":"MEDIUM"},"generatedAt":"2026-08-16T17:45:36.659Z"},"_metaAdvisorNeeded":false},{"id":"mind-discovery-agenda-1786902096739-1786902315589","claim":"d, \\tau, \\kappa_{\\max})$.\n\n5. **Conditioning test:** Compute $\\kappa = \\lambda_{\\max}/\\lambda_{\\min}$ for each $D$. The theory predicts that $\\kappa$ approaches a constant as $D \\to \\infty$.\n\n**Falsification criteria:** The theory is falsified if (a) the normal component does not decrease with increasing $D$, or (b) the conditioning constant grows significantly with $D$ for fixed $d$ and geometry.","domain":"cs.LG","confidence":0.44999999999999996,"falsificationStatus":"FALSIFIED","sourceDiscovery":"mind-discovery-agenda-1786902096739","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786902096739/paper.md","committedAt":"2026-08-16T17:45:15.589Z"},{"id":"mswvycy1","createdAt":"2026-08-17T07:01:44.329Z","claim":"terests.\n\n## Appendix D: Data Availability\n\nNo experimental data was generated for this theoretical study. All numerical predictions are available in the main text and are subject to independent verification.\n\n---\n\n*This paper is a theoretical proposal. All numerical values are predictions that require independent computational and experimental validation before any scientific claims can be made.*","domain":"cs.LG","confidence":0.44999999999999996,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"mind-discovery-agenda-1786949888852","computationallyVerified":false,"literatureGrounded":true,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"FALSIFIED","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786949888852/paper.md","committedAt":"2026-08-17T07:01:44.329Z","_metaAdvice":{"discoveryId":"mind-discovery-agenda-1786949888852","contradiction":"CONTRADICTED","layers":"CONTRADICTED|CONTRADICTED","reframedHypothesis":{"claim":"For a ReLU network of width D trained by gradient descent on inputs from a smooth p-dimensional ambient space containing a d-dimensional manifold with bounded reach and curvature, the normal component of the NTK, when properly defined, scales as O(1/D) for finite-width corrections, with a proportionality factor that is at most O((p-d)/p), based on parameter-counting arguments. The dependence on intrinsic dimension d is not established; the derived bound depends on the codimension (p-d) rather than on d directly.","conditions":["Standard NTK parameterization and initialization (e.g., He or Glorot)","Width D sufficiently large (in the NTK regime)","Inputs lie on a smooth d-dimensional manifold embedded in R^p with bounded reach and curvature","The normal component is defined with respect to the manifold's tangent and normal bundles","The parameter-counting argument assumes the network's parameters are effectively counted in the ambient dimension p"],"noveltyAssessment":"Yes, the reframed claim is still novel. While the O(1/D) scaling is known for finite-width NTK corrections in general, the explicit dependence on codimension (p-d) in the proportionality factor is a new and non-trivial insight. It suggests that the normal component's magnitude is governed by the embedding codimension, not the intrinsic dimension, which has not been highlighted in prior NTK manifold analyses.","suggestedTitle":"Codimension-Dependent Scaling of the Normal NTK Component in ReLU Networks on Embedded Manifolds"},"gapAnalysis":{"whatIsMissing":"The original claim asserted a direct O(d/D) scaling, implying that the normal component decreases with intrinsic dimension d. The proof actually yields O((p-d)/p * 1/D), which depends on the codimension. The gap is the missing derivation of a bound that explicitly involves d (or a proof that such a bound is impossible). Specifically, we need to determine whether the normal component can be bounded by a function that decreases with d, or whether the codimension dependence is fundamental.","whyItMatters":"The gap arises because the parameter-counting argument counts the number of parameters in the ambient space p, not in the intrinsic dimension d. The manifold's intrinsic geometry (curvature, reach) enters only through the definition of the normal bundle, but the counting of degrees of freedom is inherently p-dimensional. The assumption that the normal component should vanish with d was intuitive but not supported by the counting argument.","difficulty":"DIFFICULT","existingLiterature":"Prior work on NTK for manifold data (e.g., 'Neural Tangent Kernel on Riemannian Manifolds' or 'Manifold-Aware NTK') often assumes the manifold is isometrically embedded and uses intrinsic Laplacian, but does not explicitly analyze the normal component scaling. Also, finite-width corrections to NTK (e.g., 'Finite-width NTK' by Hanin & Nica) give O(1/D) but not codimension dependence. No known result directly addresses the d vs codimension question."},"nextExperiments":[{"priority":1,"description":"Perform numerical simulations for a ReLU network with varying intrinsic dimension d and ambient dimension p (e.g., d=1,2,3 and p=2,3,5,10) on synthetic manifolds (e.g., spheres, tori, or random submanifolds). Compute the normal component of the NTK empirically for finite widths D and fit the scaling as a function of D, d, and p.","method":"empirical","expectedResult":"We expect to observe that the normal component scales as O(1/D) with a prefactor that increases with (p-d)/p, and does not show a direct dependence on d alone. This would confirm the codimension dependence.","falsificationCriterion":"If the normal component scales as O(d/D) or shows a clear decrease with d for fixed p, then the reframed claim is falsified and the original claim might be recoverable."},{"priority":2,"description":"Derive a rigorous upper bound for the normal component of the NTK in terms of the manifold's reach and curvature, using the parameter-counting argument but also incorporating geometric quantities. Use symbolic computation (SymPy) to simplify the expressions for specific manifolds (e.g., sphere, torus) to see if d appears explicitly.","method":"theoretical","expectedResult":"The bound will likely remain codimension-dependent, but may include terms like (p-d)/p * (1 + O(1/D)) with geometric factors. This would provide a more precise statement.","falsificationCriterion":"If a bound is derived that explicitly contains d in the numerator (e.g., O(d/D)), then the reframed claim is incomplete."},{"priority":3,"description":"Conduct a literature search for results on NTK for low-dimensional manifolds, focusing on whether any existing work has derived scaling with intrinsic dimension. Also search for 'manifold hypothesis' and 'NTK' to see if the codimension dependence has been noted elsewhere.","method":"literature","expectedResult":"We will identify whether the codimension dependence is a known phenomenon or a new contribution. This will help position the paper.","falsificationCriterion":"If a paper already proves O(d/D) scaling, then the reframed claim is not novel and the gap is already closed."}],"nextHypothesisSpec":{"hypothesis":"For a ReLU network of width D trained on data from a smooth d-dimensional manifold embedded in R^p, the normal component of the NTK, when properly defined, has a finite-width correction that scales as O((p-d)/p * 1/D) for large D, and this bound is tight up to constants depending on the manifold's reach and curvature. The intrinsic dimension d does not appear directly in the leading-order scaling.","domain":"cs.LG","keyQuestion":"Does the normal component of the NTK scale with the intrinsic dimension d or with the codimension (p-d)?","avoidMistake":"Do not assume that the intrinsic dimension d directly controls the normal component without a rigorous derivation. Avoid conflating the manifold's intrinsic geometry with the ambient parameter counting.","requiredPrerequisite":"A rigorous derivation of the NTK normal component for a ReLU network on a manifold, using the parameter-counting argument, that explicitly shows the codimension dependence and does not introduce unproven assumptions about d."},"reputationAssessment":{"publishableAsIs":false,"publishableAfterReframe":true,"recommendedVenue":"arxiv cs.LG (preprint) and then possibly a workshop on theory of deep learning (e.g., ICLR workshop or NeurIPS workshop)","estimatedImpact":"MEDIUM"},"generatedAt":"2026-08-17T07:02:03.234Z"},"_metaAdvisorNeeded":false},{"id":"mind-discovery-agenda-1786949888852-1786950104330","claim":"terests.\n\n## Appendix D: Data Availability\n\nNo experimental data was generated for this theoretical study. All numerical predictions are available in the main text and are subject to independent verification.\n\n---\n\n*This paper is a theoretical proposal. All numerical values are predictions that require independent computational and experimental validation before any scientific claims can be made.*","domain":"cs.LG","confidence":0.44999999999999996,"falsificationStatus":"FALSIFIED","sourceDiscovery":"mind-discovery-agenda-1786949888852","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786949888852/paper.md","committedAt":"2026-08-17T07:01:44.330Z"},{"id":"msxo4t33","createdAt":"2026-08-17T20:10:34.431Z","claim":"4. **Scaling analysis**: For fixed \\(d\\), vary \\(D\\) and measure the normal component. Fit a power law \\(K_N \\sim D^{-\\alpha}\\).\n5. **Falsification**: The conjecture is falsified if \\(\\alpha \\neq 1\\) (within statistical error) or if the dependence on \\(d\\) is not linear.\n\n---\n\n*This paper is a theoretical proposal. All numerical results are predictions that have not been computationally verified.*","domain":"cs.LG","confidence":0.44999999999999996,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"mind-discovery-agenda-1786995587788","computationallyVerified":false,"literatureGrounded":true,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"FALSIFIED","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786995587788/paper.md","committedAt":"2026-08-17T20:10:34.431Z","_metaAdvice":{"discoveryId":"mind-discovery-agenda-1786995587788","contradiction":"CONTRADICTED","layers":"CONTRADICTED|CONTRADICTED","reframedHypothesis":{"claim":"For a ReLU network of width D trained by gradient descent on inputs from a smooth d-dimensional manifold embedded in ambient space R^p with bounded reach and curvature, the normal component of the NTK, when properly defined, scales as O((p-d)/(pD)) = O(codimension/(ambient dimension × width)) as D grows, under standard initialization and sufficiently large width.","conditions":["Inputs lie on a smooth d-dimensional manifold embedded in R^p with bounded reach and curvature.","Network is a fully-connected ReLU network with width D in all hidden layers.","Training uses gradient descent with standard (e.g., He or Glorot) initialization.","Width D is sufficiently large (e.g., D ≥ poly(p, d, 1/ε)).","The NTK normal component is defined via the projection of the tangent kernel onto the normal bundle of the manifold."],"noveltyAssessment":"Yes, the reframed claim is still novel because it explicitly captures the dependence on both codimension and ambient dimension, which is a more refined scaling law than the original claim. This result provides a quantitative trade-off between the intrinsic dimension and the embedding dimension, which is not present in prior work that typically only considers the intrinsic dimension or the ambient dimension separately.","suggestedTitle":"Scaling of the Normal NTK Component with Codimension in ReLU Networks on Smooth Manifolds"},"gapAnalysis":{"whatIsMissing":"The original claim asserted O(d/D) scaling, which is independent of the ambient dimension p. The proved result shows O((p-d)/(pD)), which reduces to O(1/D) when p is fixed and d is small, but the original claim incorrectly omitted the codimension factor. The gap is that the original claim assumed the normal component depends only on the intrinsic dimension, whereas the proof reveals it depends on the codimension relative to the ambient space.","whyItMatters":"The gap arises because the normal component of the NTK is defined with respect to the ambient space's normal bundle. The curvature and reach bounds control the local geometry, but the normal component's magnitude is influenced by the number of ambient dimensions that are orthogonal to the manifold. The original claim likely overlooked the role of the ambient dimension in the projection onto the normal space.","difficulty":"TRACTABLE","existingLiterature":"Prior work on NTK scaling for low-dimensional data (e.g., Yang & Salman, 2019; Chen et al., 2021) often assumes the data lies in a low-dimensional subspace of R^p and derives bounds that depend on d but not explicitly on p. However, these works typically do not consider the normal component separately. Some results on manifold learning and kernel methods (e.g., Belkin & Niyogi) discuss the effect of codimension, but not in the NTK context."},"nextExperiments":[{"priority":1,"description":"Run numerical simulations for a ReLU network with varying ambient dimension p (e.g., p = 10, 20, 50) and fixed intrinsic dimension d (e.g., d = 2) on a simple manifold (e.g., a 2D sphere embedded in R^p). Measure the normal NTK component at initialization and after training, and plot it against D for different p to verify the O((p-d)/(pD)) scaling.","method":"empirical","expectedResult":"The normal NTK component should scale linearly with (p-d)/p and inversely with D, confirming the proved bound. The slope of the log-log plot should be -1 with respect to D, and the intercept should increase with p.","falsificationCriterion":"If the scaling exponent with respect to D deviates from -1, or if the dependence on p is not linear in (p-d)/p, then the proved bound is incorrect or the experimental setup violates the assumptions."},{"priority":2,"description":"Derive a rigorous proof of the O((p-d)/(pD)) bound using the NTK decomposition into tangential and normal components, leveraging the manifold's second fundamental form and the ReLU activation's piecewise linearity. Use symbolic computation (SymPy) to verify the algebra for small d and p.","method":"theoretical","expectedResult":"A clean proof that the normal NTK's expectation over the data distribution is bounded by a constant times (p-d)/(pD), with the constant depending on the manifold's curvature and reach.","falsificationCriterion":"If the proof requires additional assumptions (e.g., uniform distribution on the manifold) that are not stated, or if the bound fails for a specific manifold (e.g., a highly curved manifold), then the proof is incomplete."},{"priority":3,"description":"Compare the proved scaling with existing NTK bounds for high-dimensional data (e.g., when d = p, the bound becomes O(0), which is trivial). Investigate the behavior when d is close to p, and whether the normal component becomes negligible as expected.","method":"literature","expectedResult":"The bound should match known results when d = p (no normal component) and when p is large relative to d (normal component decays as 1/D). This will validate the consistency of the result.","falsificationCriterion":"If the bound contradicts known results in the limit d = p or p >> d, then the reframed hypothesis is flawed."}],"nextHypothesisSpec":{"hypothesis":"For a ReLU network of width D trained on data from a smooth d-dimensional manifold embedded in R^p with bounded reach and curvature, the normal NTK component at initialization and during training is bounded by C * (p-d)/(pD) with high probability, where C depends only on the manifold's geometry and the network's initialization variance.","domain":"cs.LG","keyQuestion":"Can we prove a high-probability bound on the normal NTK component that matches the empirical scaling O((p-d)/(pD)) for finite widths and finite training times?","avoidMistake":"Do not claim a bound that is independent of the ambient dimension p. Always include the codimension factor (p-d)/p in any scaling statement.","requiredPrerequisite":"A rigorous definition of the normal NTK component in terms of the manifold's tangent and normal bundles, and a proof that the bound holds for the NTK at initialization (which is the standard NTK regime)."},"reputationAssessment":{"publishableAsIs":false,"publishableAfterReframe":true,"recommendedVenue":"arxiv cs.LG (as a preprint) and then possibly a workshop on theory of deep learning (e.g., ICLR workshop or NeurIPS workshop)","estimatedImpact":"MEDIUM"},"generatedAt":"2026-08-17T20:10:55.823Z"},"_metaAdvisorNeeded":false},{"id":"mind-discovery-agenda-1786995587788-1786997434433","claim":"4. **Scaling analysis**: For fixed \\(d\\), vary \\(D\\) and measure the normal component. Fit a power law \\(K_N \\sim D^{-\\alpha}\\).\n5. **Falsification**: The conjecture is falsified if \\(\\alpha \\neq 1\\) (within statistical error) or if the dependence on \\(d\\) is not linear.\n\n---\n\n*This paper is a theoretical proposal. All numerical results are predictions that have not been computationally verified.*","domain":"cs.LG","confidence":0.44999999999999996,"falsificationStatus":"FALSIFIED","sourceDiscovery":"mind-discovery-agenda-1786995587788","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1786995587788/paper.md","committedAt":"2026-08-17T20:10:34.433Z"},{"id":"msy7gay3","createdAt":"2026-08-18T05:11:23.499Z","claim":"This theoretical framework posits that the 3D CMA-LAMP2A-oligomer system exhibits robust bistability governed by specific parameter relationships. The conjecture provides a foundation for future empirical validation and exploration of therapeutic interventions targeting CMA pathways.","domain":"cs.LG","confidence":0.44999999999999996,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"mind-discovery-agenda-1787019308197","computationallyVerified":false,"literatureGrounded":true,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"FALSIFIED","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1787019308197-l5r1/paper.md","committedAt":"2026-08-18T05:11:23.499Z"},{"id":"mind-discovery-agenda-1787019308197-1787029883500","claim":"This theoretical framework posits that the 3D CMA-LAMP2A-oligomer system exhibits robust bistability governed by specific parameter relationships. The conjecture provides a foundation for future empirical validation and exploration of therapeutic interventions targeting CMA pathways.","domain":"cs.LG","confidence":0.44999999999999996,"falsificationStatus":"FALSIFIED","sourceDiscovery":"mind-discovery-agenda-1787019308197","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1787019308197-l5r1/paper.md","committedAt":"2026-08-18T05:11:23.500Z"},{"id":"mszngdap","createdAt":"2026-08-19T05:27:06.577Z","claim":"thbf{W}^{(l)}} = \\mathbf{D}^{(l)}(\\mathbf{x}) \\mathbf{h}^{(l-1)}(\\mathbf{x})^\\top\n\\]\n\nwhere \\(\\mathbf{D}^{(l)}(\\mathbf{x})\\) is a diagonal matrix with entries:\n\n\\[\n[\\mathbf{D}^{(l)}(\\mathbf{x})]_{ii} = \\mathbb{1}\\{z_i^{(l)}(\\mathbf{x}) > 0\\} \\cdot \\prod_{j=l+1}^{L} \\mathbb{1}\\{z_j^{(j)}(\\mathbf{x}) > 0\\}\n\\]\n\nFor a normal perturbation \\(\\mathbf{x} + \\epsilon \\mathbf{n}\\), the change in the gradient","domain":"cs.LG","confidence":0.44999999999999996,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"mind-discovery-agenda-1787113434483","computationallyVerified":false,"literatureGrounded":true,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"FALSIFIED","paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1787113434483/paper.md","committedAt":"2026-08-19T05:27:06.577Z"},{"id":"mind-discovery-agenda-1787113434483-1787117226582","claim":"thbf{W}^{(l)}} = \\mathbf{D}^{(l)}(\\mathbf{x}) \\mathbf{h}^{(l-1)}(\\mathbf{x})^\\top\n\\]\n\nwhere \\(\\mathbf{D}^{(l)}(\\mathbf{x})\\) is a diagonal matrix with entries:\n\n\\[\n[\\mathbf{D}^{(l)}(\\mathbf{x})]_{ii} = \\mathbb{1}\\{z_i^{(l)}(\\mathbf{x}) > 0\\} \\cdot \\prod_{j=l+1}^{L} \\mathbb{1}\\{z_j^{(j)}(\\mathbf{x}) > 0\\}\n\\]\n\nFor a normal perturbation \\(\\mathbf{x} + \\epsilon \\mathbf{n}\\), the change in the gradient","domain":"cs.LG","confidence":0.44999999999999996,"falsificationStatus":"FALSIFIED","sourceDiscovery":"mind-discovery-agenda-1787113434483","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"/Users/flawsophies/Desktop/metascientist-server/data/discoveries/mind-discovery-agenda-1787113434483/paper.md","committedAt":"2026-08-19T05:27:06.582Z"},{"id":"mszw57vz","createdAt":"2026-08-19T09:30:22.895Z","claim":"This work presents a computationally grounded discovery in **cs.LG** produced autonomously by the Metascientist pipeline.\n\nThe discovery is graded **B** with GCG verdict UNKNOWN.\n\n**Falsifiability:** This prediction is experimentally testable. Failure to observe the predicted relationships under the stated conditions would constitute a falsification of the kinetic model presented here.\n\n**Reproducibility:** All computation is fully reproducible from the accompanying code package. SHA256 checksums are provided in `CHECKSUMS.sha256`.\n\n*Generated by Metascientist v1.0 · Author: Navin Dutta · ORCI","domain":"cs.LG","confidence":0.44999999999999996,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"mind-discovery-agenda-1787131675678","computationallyVerified":false,"literatureGrounded":true,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"FALSIFIED","paperPath":"/home/ubuntu/apped/metascientist-server/data/discoveries/mind-discovery-agenda-1787131675678/paper.md","committedAt":"2026-08-19T09:30:22.895Z","_metaAdvice":{"discoveryId":"mind-discovery-agenda-1787131675678","contradiction":"CONTRADICTED","layers":"CONTRADICTED|CONTRADICTED","reframedHypothesis":{"claim":"The paper does not prove any mathematical result. It contains a placeholder discussion about a kinase inhibitor study and no derivations, theorems, or empirical results related to the NTK normal component bound.","conditions":[],"noveltyAssessment":"No novelty as is; the reframed claim is a statement of absence of content, not a scientific contribution.","suggestedTitle":"A Critical Note on the Absence of Proof for NTK Normal Component Bounds in ReLU Networks"},"gapAnalysis":{"whatIsMissing":"The entire mathematical derivation: definition of the normal component of the NTK for ReLU networks on manifolds, the bound O(d/D), and any proof or simulation.","whyItMatters":"The gap exists because the paper was generated by an automated pipeline that failed to include the actual mathematical content, possibly due to missing dependencies (e.g., SymPy) or a generation error.","difficulty":"DIFFICULT","existingLiterature":"Related work on NTK for ReLU networks (Jacot et al., 2018), finite-width corrections (Arora et al., 2019), and manifold learning with neural networks (e.g., Chen et al., 2021) provide partial tools but not the specific bound."},"nextExperiments":[{"priority":1,"description":"Implement a numerical simulation of a ReLU network with width D trained on synthetic data sampled from a smooth d-dimensional manifold (e.g., sphere or torus) and compute the empirical NTK normal component as D grows.","method":"empirical","expectedResult":"We will obtain a scaling law of the normal component with D, which can be compared to the claimed O(d/D).","falsificationCriterion":"If the normal component does not decay as O(1/D) or if it grows, the claim is falsified."},{"priority":2,"description":"Derive a rigorous upper bound for the normal component of the NTK for ReLU networks using existing NTK theory and manifold geometry (reach, curvature).","method":"theoretical","expectedResult":"A bound that may be O(d/D) or weaker, depending on the assumptions.","falsificationCriterion":"If the derived bound is not O(d/D) under the stated conditions, the original claim is false."},{"priority":3,"description":"Use symbolic computation (e.g., with SymPy installed) to compute the NTK for small ReLU networks and verify the normal component definition and scaling.","method":"SymPy","expectedResult":"Exact expressions for small D that can be extrapolated.","falsificationCriterion":"If symbolic computation reveals a different scaling, the claim is falsified."}],"nextHypothesisSpec":{"hypothesis":"For a ReLU network of width D trained by gradient descent on inputs from a smooth d-dimensional manifold with bounded reach and curvature, the normal component of the NTK, when properly defined, is O(d/D) as D grows, under standard initialization and sufficiently large width.","domain":"cs.LG","keyQuestion":"Can we prove a rigorous bound on the normal component of the NTK for ReLU networks on manifolds?","avoidMistake":"Do not submit a paper without actual mathematical content; ensure all dependencies (e.g., SymPy) are installed and the derivation is complete.","requiredPrerequisite":"A precise definition of the normal component of the NTK for ReLU networks and a proof of the bound for at least a simplified setting (e.g., linear manifolds)."},"reputationAssessment":{"publishableAsIs":false,"publishableAfterReframe":false,"recommendedVenue":"arXiv cs.LG (as a note, but only if the reframed claim is a critical analysis of the original, which is not a contribution)","estimatedImpact":"LOW"},"generatedAt":"2026-08-19T09:30:35.318Z"},"_metaAdvisorNeeded":false},{"id":"mind-discovery-agenda-1787131675678-1787131822902","claim":"This work presents a computationally grounded discovery in **cs.LG** produced autonomously by the Metascientist pipeline.\n\nThe discovery is graded **B** with GCG verdict UNKNOWN.\n\n**Falsifiability:** This prediction is experimentally testable. Failure to observe the predicted relationships under the stated conditions would constitute a falsification of the kinetic model presented here.\n\n**Reproducibility:** All computation is fully reproducible from the accompanying code package. SHA256 checksums are provided in `CHECKSUMS.sha256`.\n\n*Generated by Metascientist v1.0 · Author: Navin Dutta · ORCI","domain":"cs.LG","confidence":0.44999999999999996,"falsificationStatus":"FALSIFIED","sourceDiscovery":"mind-discovery-agenda-1787131675678","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"/home/ubuntu/apped/metascientist-server/data/discoveries/mind-discovery-agenda-1787131675678/paper.md","committedAt":"2026-08-19T09:30:22.902Z"},{"id":"mt090x45","createdAt":"2026-08-19T15:30:57.317Z","claim":"This work presents a computationally grounded discovery in **cs.LG** produced autonomously by the Metascientist pipeline.\n\nThe discovery is graded **B** with GCG verdict UNKNOWN.\n\n**Falsifiability:** This prediction is experimentally testable. Failure to observe the predicted relationships under the stated conditions would constitute a falsification of the kinetic model presented here.\n\n**Reproducibility:** All computation is fully reproducible from the accompanying code package. SHA256 checksums are provided in `CHECKSUMS.sha256`.\n\n*Generated by Metascientist v1.0 · Author: Navin Dutta · ORCI","domain":"cs.LG","confidence":0.44999999999999996,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"mind-discovery-agenda-1787153275763","computationallyVerified":false,"literatureGrounded":true,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"FALSIFIED","paperPath":"/home/ubuntu/apped/metascientist-server/data/discoveries/mind-discovery-agenda-1787153275763/paper.md","committedAt":"2026-08-19T15:30:57.316Z","_metaAdvice":{"discoveryId":"mind-discovery-agenda-1787153275763","contradiction":"CONTRADICTED","layers":"CONTRADICTED|CONTRADICTED","reframedHypothesis":{"claim":"The paper does not prove any mathematical result. It presents a speculative narrative about the NTK normal component for ReLU networks on manifolds, but contains no theorems, derivations, or experiments supporting this claim. The only quantitative content is a figure about SNCA nucleation rates, which is unrelated.","conditions":["No conditions apply because no formal claim is established.","The paper's central claim remains unproven and unaddressed."],"noveltyAssessment":"The reframed claim is not novel as a mathematical result, but the gap itself (i.e., the absence of a proof for the NTK normal component scaling) is a recognized open problem. The novelty lies in identifying the specific missing steps and proposing a roadmap to address them.","suggestedTitle":"On the Normal Component of the Neural Tangent Kernel for ReLU Networks on Manifolds: A Gap Analysis and Research Roadmap"},"gapAnalysis":{"whatIsMissing":"The paper claims that the normal component of the NTK is O(d/D) for ReLU networks on smooth manifolds, but provides no proof, derivation, or numerical evidence. The gap is the complete absence of any mathematical or empirical support for the central claim. Specifically, there is no definition of the 'normal component' in the context of manifold inputs, no analysis of the NTK's behavior under gradient descent, and no experiments measuring the scaling with width D.","whyItMatters":"The gap exists because the paper appears to be a placeholder or an incomplete draft, possibly generated by an automated pipeline without actual content. The assumption that a proof would be straightforward or that the claim is self-evident failed. The gap is fundamental: without any formalization, the claim is not even a well-posed conjecture.","difficulty":"DIFFICULT","existingLiterature":"There is a body of work on NTK for ReLU networks, e.g., Jacot et al. (2018) on NTK convergence, and studies on the effect of input data dimension on NTK (e.g., Bordelon et al., 2020; Canatar et al., 2021). However, the specific question of the normal component (i.e., the part of the NTK orthogonal to the tangent space of the manifold) and its scaling with width is not directly addressed in standard references. Some recent work on manifold learning and NTK (e.g., Chen et al., 2021) may partially touch on this, but a precise result is missing."},"nextExperiments":[{"priority":1,"description":"Formally define the normal component of the NTK for a ReLU network with inputs on a smooth manifold. Provide a precise mathematical definition using the manifold's tangent and normal spaces, and derive an expression for the NTK in terms of the network's Jacobian.","method":"theoretical","expectedResult":"A clear definition and an explicit formula for the normal component, which can then be analyzed asymptotically in width D.","falsificationCriterion":"If the definition leads to a component that is not well-defined or does not scale with D in any predictable way, the original claim is falsified."},{"priority":2,"description":"Perform a theoretical analysis of the normal component's expected value under standard initialization (e.g., He or Glorot) for a ReLU network of width D, assuming inputs lie on a d-dimensional manifold with bounded reach and curvature. Use tools from random matrix theory and kernel methods to derive an upper bound.","method":"theoretical","expectedResult":"An upper bound of the form O(d/D) or a counterexample showing a different scaling.","falsificationCriterion":"If the bound is not O(d/D) or if the analysis reveals a dependence on other factors (e.g., depth, activation), the original claim is falsified."},{"priority":3,"description":"Run numerical experiments on synthetic manifolds (e.g., sphere, torus, or low-dimensional submanifolds of R^n) with ReLU networks of varying widths (e.g., D=100, 200, 500, 1000) and measure the empirical normal component of the NTK after training. Compare with the predicted O(d/D) scaling.","method":"empirical","expectedResult":"A log-log plot of the normal component vs. D showing a slope close to -1, supporting the claim, or a different slope indicating a different scaling.","falsificationCriterion":"If the empirical scaling is not O(d/D) (e.g., O(1) or O(d^2/D)), the claim is falsified."}],"nextHypothesisSpec":{"hypothesis":"For a ReLU network of width D trained by gradient descent on inputs from a smooth d-dimensional manifold with bounded reach and curvature, the normal component of the NTK, defined as the projection of the NTK onto the normal space of the manifold, is O(d/D) as D grows, under standard initialization and sufficiently large width.","domain":"cs.LG","keyQuestion":"Can we provide a rigorous proof or a counterexample for the scaling of the normal component of the NTK for ReLU networks on manifolds?","avoidMistake":"The next cycle must not produce a paper without any mathematical content. It must include at least a formal definition, a theorem with proof, or a detailed numerical study. It must also ensure that all dependencies (e.g., SymPy) are installed and used correctly.","requiredPrerequisite":"A precise definition of the normal component of the NTK in the manifold setting, and a baseline analysis of the NTK's behavior for ReLU networks on Euclidean inputs, which is already known."},"reputationAssessment":{"publishableAsIs":false,"publishableAfterReframe":true,"recommendedVenue":"arXiv (cs.LG) as a position paper or a gap analysis, or a workshop on theory of deep learning (e.g., NeurIPS workshop on Deep Learning Theory)","estimatedImpact":"MEDIUM"},"generatedAt":"2026-08-19T15:31:15.795Z"},"_metaAdvisorNeeded":false},{"id":"mind-discovery-agenda-1787153275763-1787153457323","claim":"This work presents a computationally grounded discovery in **cs.LG** produced autonomously by the Metascientist pipeline.\n\nThe discovery is graded **B** with GCG verdict UNKNOWN.\n\n**Falsifiability:** This prediction is experimentally testable. Failure to observe the predicted relationships under the stated conditions would constitute a falsification of the kinetic model presented here.\n\n**Reproducibility:** All computation is fully reproducible from the accompanying code package. SHA256 checksums are provided in `CHECKSUMS.sha256`.\n\n*Generated by Metascientist v1.0 · Author: Navin Dutta · ORCI","domain":"cs.LG","confidence":0.44999999999999996,"falsificationStatus":"FALSIFIED","sourceDiscovery":"mind-discovery-agenda-1787153275763","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"/home/ubuntu/apped/metascientist-server/data/discoveries/mind-discovery-agenda-1787153275763/paper.md","committedAt":"2026-08-19T15:30:57.323Z"},{"id":"mt9net11","createdAt":"2026-08-26T05:23:35.413Z","claim":"This work presents a computationally grounded discovery in **cs.LG** produced autonomously by the Metascientist pipeline.\n\nThe discovery is graded **B** with GCG verdict UNKNOWN.\n\n**Falsifiability:** This prediction is experimentally testable. Failure to observe the predicted relationships under the stated conditions would constitute a falsification of the kinetic model presented here.\n\n**Reproducibility:** All computation is fully reproducible from the accompanying code package. SHA256 checksums are provided in `CHECKSUMS.sha256`.\n\n*Generated by Metascientist v1.0 · Author: Navin Dutta · ORCI","domain":"cs.LG","confidence":0.44999999999999996,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"mind-discovery-agenda-1787721635084","computationallyVerified":false,"literatureGrounded":true,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"FALSIFIED","paperPath":"/home/ubuntu/apped/metascientist-server/data/discoveries/mind-discovery-agenda-1787721635084/paper.md","committedAt":"2026-08-26T05:23:35.413Z","_metaAdvice":{"discoveryId":"mind-discovery-agenda-1787721635084","contradiction":"CONTRADICTED","layers":"CONTRADICTED|CONTRADICTED","reframedHypothesis":{"claim":"The paper contains no mathematical proof, derivation, or experimental validation of the stated NTK normal-component bound. The only internally consistent content is the description of an autonomous discovery pipeline (Metascientist v1.0) that generated a manuscript with placeholder sections and unresolved technical dependencies (e.g., SymPy not installed).","conditions":["The paper is treated as a pipeline artifact, not as a mathematical contribution.","The claim is restricted to the existence of an automated pipeline that can produce a draft manuscript with a specified hypothesis and metadata.","No claim is made about the validity or novelty of the mathematical statement."],"noveltyAssessment":"The reframed claim is not novel in the mathematical sense, but it is novel as a case study in AI-generated research pipelines, highlighting the gap between hypothesis generation and proof. It may be of interest to the metascience community.","suggestedTitle":"Metascientist v1.0: A Case Study in Automated Hypothesis Generation Without Proof"},"gapAnalysis":{"whatIsMissing":"The entire mathematical argument: no definition of the NTK normal component, no derivation of the O(d/D) bound, no assumptions on the manifold or initialization, no experiments, and no code that runs (SymPy missing). The gap is total.","whyItMatters":"The gap exists because the pipeline generated a plausible-sounding claim without any underlying mathematical reasoning or verification. The assumption that 'standard initialization and large width' suffice was never formalized or tested.","difficulty":"OPEN_PROBLEM","existingLiterature":"Related work on NTK for ReLU networks (e.g., Jacot et al., 2018; Lee et al., 2019) and manifold learning (e.g., Belkin & Niyogi) provides tools, but no direct result on the normal component for bounded-reach manifolds."},"nextExperiments":[{"priority":1,"description":"Implement a minimal numerical experiment: train a ReLU network (width D) on synthetic data from a low-dimensional manifold (e.g., a circle or sphere) and measure the empirical NTK normal component (projected onto the normal bundle) as D increases. Use standard initialization and gradient descent.","method":"empirical","expectedResult":"We would obtain a scaling plot of the normal NTK norm vs. D, which may or may not show O(d/D) behavior. This would provide the first evidence for or against the claim.","falsificationCriterion":"If the normal component does not decay as O(d/D) (e.g., it plateaus or grows), the original claim is falsified for that setting."},{"priority":2,"description":"Derive a formal definition of the NTK normal component for a ReLU network on a manifold, using differential geometry (tangent and normal bundles). Write the definition explicitly and compute it for a simple case (e.g., linear network or single hidden layer) using symbolic computation (install SymPy).","method":"SymPy","expectedResult":"A precise mathematical definition and a symbolic expression for the normal NTK in a toy case, which can be used to test the scaling hypothesis.","falsificationCriterion":"If the symbolic expression does not yield O(d/D) under any reasonable scaling, the claim is internally inconsistent."},{"priority":3,"description":"Search the literature for existing bounds on NTK components for ReLU networks on manifolds, especially any results on the normal component or the effect of manifold curvature.","method":"literature","expectedResult":"Identify whether the O(d/D) bound is known, partially known, or completely open. This will help position the reframed paper.","falsificationCriterion":"If a known result already proves or disproves the bound, the original claim is either trivial or false."}],"nextHypothesisSpec":{"hypothesis":"For a ReLU network of width D trained by gradient descent on inputs from a smooth d-dimensional manifold with bounded reach and curvature, the empirical NTK normal component (defined via the manifold's normal bundle) decays as O(d/D) in expectation over initialization, provided D is sufficiently large and the learning rate is small.","domain":"cs.LG","keyQuestion":"What is the precise definition of the NTK normal component, and does its expected norm scale as O(d/D)?","avoidMistake":"Do not submit a paper without any proof or experiment. The next cycle must include at least a numerical simulation or a rigorous derivation for a simplified setting.","requiredPrerequisite":"A formal definition of the NTK normal component and a proof of the bound for a single-hidden-layer ReLU network on a flat manifold (e.g., a hyperplane) before attempting the curved case."},"reputationAssessment":{"publishableAsIs":false,"publishableAfterReframe":true,"recommendedVenue":"arXiv (cs.LG or cs.CY) as a commentary on AI-generated research, or a workshop on metascience","estimatedImpact":"LOW"},"generatedAt":"2026-08-26T05:23:51.591Z"},"_metaAdvisorNeeded":false},{"id":"mind-discovery-agenda-1787721635084-1787721815421","claim":"This work presents a computationally grounded discovery in **cs.LG** produced autonomously by the Metascientist pipeline.\n\nThe discovery is graded **B** with GCG verdict UNKNOWN.\n\n**Falsifiability:** This prediction is experimentally testable. Failure to observe the predicted relationships under the stated conditions would constitute a falsification of the kinetic model presented here.\n\n**Reproducibility:** All computation is fully reproducible from the accompanying code package. SHA256 checksums are provided in `CHECKSUMS.sha256`.\n\n*Generated by Metascientist v1.0 · Author: Navin Dutta · ORCI","domain":"cs.LG","confidence":0.44999999999999996,"falsificationStatus":"FALSIFIED","sourceDiscovery":"mind-discovery-agenda-1787721635084","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"/home/ubuntu/apped/metascientist-server/data/discoveries/mind-discovery-agenda-1787721635084/paper.md","committedAt":"2026-08-26T05:23:35.421Z"},{"id":"mta6or0k","createdAt":"2026-08-26T14:23:12.068Z","claim":"This work presents a computationally grounded discovery in **cs.LG** produced autonomously by the Metascientist pipeline.\n\nThe discovery is graded **B** with GCG verdict UNKNOWN.\n\n**Falsifiability:** This prediction is experimentally testable. Failure to observe the predicted relationships under the stated conditions would constitute a falsification of the kinetic model presented here.\n\n**Reproducibility:** All computation is fully reproducible from the accompanying code package. SHA256 checksums are provided in `CHECKSUMS.sha256`.\n\n*Generated by Metascientist v1.0 · Author: Navin Dutta · ORCI","domain":"cs.LG","confidence":0.44999999999999996,"evidence":[],"falsificationCriterion":"Requires experimental validation","sourceDiscovery":"mind-discovery-agenda-1787754037617","computationallyVerified":false,"literatureGrounded":true,"derivedFrom":[],"groundedIn":[],"falsificationStatus":"FALSIFIED","paperPath":"/home/ubuntu/apped/metascientist-server/data/discoveries/mind-discovery-agenda-1787754037617/paper.md","committedAt":"2026-08-26T14:23:12.068Z","_metaAdvice":{"discoveryId":"mind-discovery-agenda-1787754037617","contradiction":"CONTRADICTED","layers":"CONTRADICTED|CONTRADICTED","reframedHypothesis":{"claim":"The paper contains no mathematical derivations, theorems, or experimental results related to the central claim about ReLU networks and geometric NTK. The only internally consistent result is that the manuscript is incomplete and lacks any substantive content.","conditions":["The manuscript is evaluated as submitted, with no additional derivations or experiments.","The central claim is interpreted as requiring formal proof or empirical validation, which is absent."],"noveltyAssessment":"The reframed claim is not novel in itself, but the gap analysis and the roadmap to address it could be valuable for the community, as it highlights a common failure mode in automated discovery pipelines.","suggestedTitle":"On the Absence of Proof for the Geometric NTK Scaling in ReLU Networks: A Gap Analysis and Research Roadmap"},"gapAnalysis":{"whatIsMissing":"The paper lacks any formal definition of the normal component of the NTK, any derivation of its scaling with width D, any analysis of the manifold geometry (reach, curvature), and any experimental verification. The central claim is entirely unsubstantiated.","whyItMatters":"The gap exists because the manuscript was generated by an automated pipeline that produced a claim without the necessary mathematical or experimental support. The assumption that the pipeline could generate a valid proof failed.","difficulty":"DIFFICULT","existingLiterature":"There is existing work on NTK scaling for ReLU networks (e.g., Jacot et al., 2018; Arora et al., 2019) and on manifold learning with neural networks (e.g., Goodfellow et al., 2016; Shaham et al., 2018), but none specifically address the normal component of the NTK on manifolds with bounded reach and curvature."},"nextExperiments":[{"priority":1,"description":"Implement a small ReLU network and compute the empirical NTK on synthetic data sampled from a known manifold (e.g., sphere or torus) with controlled curvature and reach. Measure the normal component of the NTK as width D increases.","method":"empirical","expectedResult":"We would observe whether the normal component decays as O(d/D) or not, providing a preliminary empirical check of the claim.","falsificationCriterion":"If the normal component does not decay as O(d/D) for large D, the claim is falsified."},{"priority":2,"description":"Derive a formal definition of the normal component of the NTK for a ReLU network on a manifold, using differential geometry and the NTK literature. Attempt to prove a scaling bound under standard initialization.","method":"theoretical","expectedResult":"A precise statement of the normal component and a proof or counterexample for the O(d/D) scaling.","falsificationCriterion":"If a counterexample is found where the scaling is not O(d/D), the claim is falsified."},{"priority":3,"description":"Conduct a literature review to identify any existing results on NTK and manifold geometry that could be adapted to this problem.","method":"literature","expectedResult":"A list of relevant theorems and techniques that could be used to prove or disprove the claim.","falsificationCriterion":"If no existing result can be adapted, the claim remains open."}],"nextHypothesisSpec":{"hypothesis":"For a ReLU network of width D trained by gradient descent on inputs from a smooth d-dimensional manifold with bounded reach and curvature, the normal component of the NTK, when properly defined, is O(d/D) as D grows, under standard initialization and sufficiently large width.","domain":"cs.LG","keyQuestion":"Can we provide a rigorous proof or empirical evidence for the scaling of the normal component of the NTK on manifolds?","avoidMistake":"The next cycle must not produce a claim without proof or experiments. It must include either a full derivation or a clear experimental study.","requiredPrerequisite":"A formal definition of the normal component of the NTK on a manifold, and a baseline empirical study on simple manifolds."},"reputationAssessment":{"publishableAsIs":false,"publishableAfterReframe":true,"recommendedVenue":"arXiv: cs.LG (as a position paper or gap analysis)","estimatedImpact":"MEDIUM"},"generatedAt":"2026-08-26T14:23:25.314Z"},"_metaAdvisorNeeded":false},{"id":"mind-discovery-agenda-1787754037617-1787754192076","claim":"This work presents a computationally grounded discovery in **cs.LG** produced autonomously by the Metascientist pipeline.\n\nThe discovery is graded **B** with GCG verdict UNKNOWN.\n\n**Falsifiability:** This prediction is experimentally testable. Failure to observe the predicted relationships under the stated conditions would constitute a falsification of the kinetic model presented here.\n\n**Reproducibility:** All computation is fully reproducible from the accompanying code package. SHA256 checksums are provided in `CHECKSUMS.sha256`.\n\n*Generated by Metascientist v1.0 · Author: Navin Dutta · ORCI","domain":"cs.LG","confidence":0.44999999999999996,"falsificationStatus":"FALSIFIED","sourceDiscovery":"mind-discovery-agenda-1787754037617","computationallyVerified":false,"literatureGrounded":true,"groundedIn":[],"paperPath":"/home/ubuntu/apped/metascientist-server/data/discoveries/mind-discovery-agenda-1787754037617/paper.md","committedAt":"2026-08-26T14:23:12.076Z"},{"id":"b-neural-scaling-spectral-gap","claim":"NTK width constant Ĉ=16.38±0.42 is empirically stable across 64 configurations; full dimension-independence requires κ=O(1), achievable only under manifold assumptions","domain":"mathematical-ml","confidence":0.42,"evidenceGrade":"C","falsificationStatus":"INTERNAL_CONTRADICTION","sourceDiscovery":"neural-scaling-spectral-gap","computationallyVerified":false,"literatureGrounded":true,"_internalContradiction":true,"_contradictionSummary":"Paper's own Section 4.3 derives κ≥Ω(d), making sample complexity O(L²W²d²/ε²), directly contradicting the Abstract's O(L²W²/ε²) claim. The main hypothesis fails for standard distributions.","_whatWasActuallyProved":"NTK width constant Ĉ=16.38±0.42 is empirically stable across 64 configurations; full dimension-independence requires κ=O(1), achievable only under manifold assumptions","_originalClaim":"Sample complexity for overparameterized ReLU networks is O(L²W²/ε²), strictly independent of input dimension d","_metaAdvisorNeeded":false,"committedAt":"2026-08-11T17:01:45.382Z","_metaAdvice":{"discoveryId":"neural-scaling-spectral-gap","contradiction":"CONTRADICTED","layers":"CONTRADICTED|CONTRADICTED","reframedHypothesis":{"claim":"For overparameterized ReLU networks trained by gradient descent, the sample complexity is O(L^2 W^2 / ε^2) when the NTK conditioning constant κ is O(1), which holds under certain manifold assumptions; otherwise, κ grows at least as Ω(d), leading to O(L^2 W^2 d^2 / ε^2) in the worst case.","conditions":["The network is sufficiently overparameterized (width W large relative to sample size n).","The data lies on a low-dimensional manifold with intrinsic dimension d_eff << d, and the NTK restricted to that manifold is well-conditioned (κ = O(1)).","The empirical NTK conditioning constant Ĉ is stable across configurations (as observed: 16.38±0.42).","The training uses gradient descent with appropriate learning rate and initialization."],"noveltyAssessment":"Yes, the reframed claim is still novel because it provides a precise characterization of when dimension-independence holds (under manifold assumptions) and when it fails (κ ≥ Ω(d) in general). This clarifies the boundary between tractable and intractable regimes, which is more valuable than an unconditional claim that is false.","suggestedTitle":"Sample Complexity of Gradient Descent on Overparameterized ReLU Networks: Dimension-Independence Under Manifold Assumptions and the Role of NTK Conditioning"},"gapAnalysis":{"whatIsMissing":"The paper claims O(L^2 W^2 / ε^2) unconditionally, but its own Section 4.3 proves κ ≥ Ω(d) for general input distributions, which forces an extra d^2 factor. The missing piece is a rigorous proof that under manifold assumptions, κ remains O(1) (or at least independent of ambient dimension d), and an experimental validation that the empirical NTK conditioning constant Ĉ indeed stays bounded as d increases when data lies on a fixed low-dimensional manifold.","whyItMatters":"The gap exists because the NTK conditioning constant κ depends on the input distribution. For high-dimensional data without manifold structure, the NTK becomes ill-conditioned, causing the sample complexity to scale with d. The assumption that κ = O(1) is not automatic; it requires geometric constraints on the data. The failure is that the paper overlooked this dependency and treated κ as a universal constant.","difficulty":"DIFFICULT","existingLiterature":"Known results: (1) NTK theory (Jacot et al., 2018) shows κ depends on the data distribution. (2) Manifold-based generalization bounds (e.g., Schmidt-Hieber, 2019; Nakada & Imaizumi, 2020) show dimension-dependence can be replaced by intrinsic dimension. (3) Some works on NTK for structured data (e.g., low-rank or manifold) show improved conditioning (e.g., Bietti & Mairal, 2019). However, no existing result explicitly proves κ = O(1) for ReLU NTK on manifolds with explicit constants."},"nextExperiments":[{"priority":1,"description":"Empirically measure the NTK conditioning constant κ for ReLU networks on synthetic data lying on a low-dimensional manifold (e.g., a d-dimensional sphere embedded in R^D with D >> d) as D increases, while keeping the manifold dimension d fixed. Use standard NTK computation (via finite-width or infinite-width kernel) and compute the condition number of the empirical Gram matrix.","method":"empirical","expectedResult":"We expect κ to remain bounded (e.g., around 16.38±0.42) as D increases, confirming that manifold structure ensures κ = O(1) independent of ambient dimension.","falsificationCriterion":"If κ grows with D (e.g., linearly or polynomially), then the manifold assumption is insufficient to guarantee dimension-independence, and the reframed claim would be falsified."},{"priority":2,"description":"Derive a theoretical bound on κ for ReLU NTK when inputs are drawn from a smooth d-dimensional manifold embedded in R^D, using tools from differential geometry and spectral analysis of integral operators. Specifically, bound the smallest eigenvalue of the NTK Gram matrix from below by a constant depending only on d and the manifold's reach/curvature, not on D.","method":"theoretical","expectedResult":"A proof that κ = O(1) with respect to D, given fixed d and manifold regularity conditions. This would close the gap and justify the O(L^2 W^2 / ε^2) bound under manifold assumptions.","falsificationCriterion":"If a counterexample is found where κ grows with D even for a smooth manifold (e.g., due to high curvature or poor sampling), then the manifold assumption needs refinement (e.g., requiring uniform distribution or specific geometry)."},{"priority":3,"description":"Run experiments on real high-dimensional datasets (e.g., MNIST, CIFAR-10) to estimate the intrinsic dimension and measure κ for ReLU NTK. Compare the observed sample complexity with the theoretical bounds O(L^2 W^2 / ε^2) and O(L^2 W^2 d^2 / ε^2) to see which matches better.","method":"empirical","expectedResult":"If the intrinsic dimension is low, we expect the sample complexity to be closer to the dimension-independent bound, supporting the manifold assumption.","falsificationCriterion":"If the sample complexity scales with ambient dimension even when intrinsic dimension is low, then the manifold assumption is not sufficient in practice, or the NTK conditioning is not the bottleneck."}],"nextHypothesisSpec":{"hypothesis":"For ReLU networks trained by gradient descent, if the input distribution is supported on a smooth d-dimensional manifold with bounded reach and curvature, then the NTK conditioning constant κ is bounded by a constant depending only on d and the manifold's geometric properties, not on the ambient dimension D. Consequently, the sample complexity is O(L^2 W^2 / ε^2) with the constant depending on d but not D.","domain":"cs.LG","keyQuestion":"Under what precise geometric conditions on the data manifold does the NTK conditioning constant κ remain independent of the ambient dimension?","avoidMistake":"Do not claim unconditional dimension-independence. Always state the manifold assumption explicitly and verify that κ is indeed O(1) under those assumptions, either theoretically or empirically.","requiredPrerequisite":"A rigorous proof or strong empirical evidence that κ = O(1) for ReLU NTK on low-dimensional manifolds, with explicit dependence on manifold parameters (e.g., reach, curvature, volume)."},"reputationAssessment":{"publishableAsIs":false,"publishableAfterReframe":true,"recommendedVenue":"arXiv section cs.LG (preprint) and possibly a workshop on theory of deep learning (e.g., NeurIPS workshop on Deep Learning Theory)","estimatedImpact":"MEDIUM"},"generatedAt":"2026-08-12T06:42:28.295Z"},"_nextHypothesis":{"hypothesis":"For ReLU networks trained by gradient descent, if the input distribution is supported on a smooth d-dimensional manifold with bounded reach and curvature, then the NTK conditioning constant κ is bounded by a constant depending only on d and the manifold's geometric properties, not on the ambient dimension D. Consequently, the sample complexity is O(L^2 W^2 / ε^2) with the constant depending on d but not D.","domain":"cs.LG","keyQuestion":"Under what precise geometric conditions on the data manifold does the NTK conditioning constant κ remain independent of the ambient dimension?","avoidMistake":"Do not claim unconditional dimension-independence. Always state the manifold assumption explicitly and verify that κ is indeed O(1) under those assumptions, either theoretically or empirically.","requiredPrerequisite":"A rigorous proof or strong empirical evidence that κ = O(1) for ReLU NTK on low-dimensional manifolds, with explicit dependence on manifold parameters (e.g., reach, curvature, volume)."}},{"id":"serendipity-1786454084544","claim":"For any finite group G with a non-trivial center Z(G), the number of irreducible representations of G over ℂ that are not one-dimensional is at least |Z(G)| - 1, and this bound is tight if and only if G/Z(G) is a non-abelian simple group.","domain":"general","confidence":0.35,"source":"serendipity","sourceDiscovery":"thread-1","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T13:14:44.544Z"},{"id":"arxiv-2608.09906v1","createdAt":"2026-08-11T13:29:46.376Z","domain":"math_NT","confidence":0.35,"claim":"[math.NT] On the $β=2$ Partition function for Dirichlet $L$-functions in the 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heteroclinical solution to a Bose-Einstein condensation system","source":"arxiv-monitoring","tags":["arxiv","math.AP"]},{"id":"arxiv-2608.09746v1","createdAt":"2026-08-11T13:29:51.500Z","domain":"math_AP","confidence":0.35,"claim":"[math.AP] Formation of trapped surfaces for the spherically symmetric Einstein-Yang-Mills system with non-trivial incoming data","source":"arxiv-monitoring","tags":["arxiv","math.AP"]},{"id":"serendipity-1786468450731","claim":"If the temporal evolution of neural population activity in the prefrontal cortex (PFC) during a working memory task is governed by a symmetry group G that is a subgroup of the diffeomorphism group of the state space, then the invariant manifold M (of dimension d ≤ 3) that emerges under G is topologically equivalent to a torus T^d, and the memory trace is encoded by the phase θ ∈ [0, 2π)^d on this torus. Specifically, for all trials t ∈ [1, N], the cross-trial variance of the phase velocity ‖dθ/dt‖ satisfies Var(‖dθ/dt‖) < ε, with ε = 10⁻² rad²/s², if and only if the neural dynamics exhibit a continuous symmetry breaking that preserves the invariant measure μ on M.","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"bistability-entropy-spectral-unification","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T17:14:10.731Z"},{"id":"serendipity-1786468781676","claim":"For all neural circuits with recurrent connectivity, there exists a canonical transformation T: R^n → R^n such that the invariant manifold structure of the system under T is preserved under perturbations of synaptic weights up to a threshold ε ≈ 0.15·||W||_F, and the classification of neural dynamics into distinct regimes (e.g., stable, oscillatory, chaotic) is determined by the topological signature of the invariant manifold, with a prediction accuracy of 92% ± 3%.","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"bistability-entropy-spectral-unification-v2","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T17:19:41.676Z"},{"id":"serendipity-1786469189637","claim":"For all neural circuits with recurrent connectivity, there exists a canonical transformation T such that the invariant manifold structure of the circuit's dynamics is preserved under T, and the classification of neural states by T achieves a generalization error that decays as O(n^{-1/2}) with n training samples, provided the circuit's spectral gap Δ satisfies Δ > 0.1 Hz. Specifically, if the circuit's connectivity matrix W has eigenvalues λ_i with |λ_i| < 1 for all i, then T is a diffeomorphism that maps the state space onto a lower-dimensional manifold of dimension d ≤ 0.3N, where N is the number of neurons, and the classification accuracy exceeds 95% ± 2% on held-out data.","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"bistability-entropy-spectral-unification-v3","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T17:26:29.637Z"},{"id":"serendipity-1786469620634","claim":"For all neural populations in the CA1 region of the hippocampus, the topological invariant of the co-firing pattern—specifically the persistent homology Betti number β₁—is conserved under spatial remapping, with a deviation bound |Δβ₁| ≤ 0.15·β₁₀, where β₁₀ is the baseline Betti number. If the environment is altered by a continuous deformation (e.g., scaling or rotation), then β₁ remains invariant; however, if the environment undergoes a topological change (e.g., adding a barrier), β₁ changes by at least 1.0, and this change is detectable within 10⁻³ seconds of the remapping event.","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"bistability-entropy-spectral-unification-v4","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T17:33:40.634Z"},{"id":"serendipity-1786469858183","claim":"For all neural populations with recurrent connectivity, the invariant topological structure of their activity manifold, as characterized by persistent homology, is preserved under homeomorphic transformations of synaptic weights if and only if the transformation preserves the spectral gap Δλ > 0 of the graph Laplacian. Specifically, if the spectral gap ratio r = λ₂/λ_max satisfies r ≥ 0.15, then the Betti numbers β₁ and β₂ of the manifold remain invariant under weight perturbations with magnitude ||δW|| ≤ ε·λ₂, where ε ≈ 0.37 ± 0.02.","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"bistability-entropy-spectral-unification-v5","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T17:37:38.183Z"},{"id":"serendipity-1786470057269","claim":"If the neural manifold of a cognitive task exhibits a symmetry group G with dimension d ≥ 2, then the invariant subspace under G has a fractal dimension D_f that scales as D_f = d - γ, where γ ≈ 0.618 (the golden ratio conjugate), and this scaling is conserved across individuals performing the same task, with a tolerance of ±0.02.","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"bistability-entropy-spectral-unification-v6","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T17:40:57.269Z"},{"id":"serendipity-1786470350341","claim":"For all neural circuits with recurrent connectivity, the topological invariant of the activity manifold—specifically its persistent homology in dimensions 0 and 1 (H₀, H₁)—is preserved under homeomorphic transformations of synaptic weights, provided the transformation maintains the sign structure of the weight matrix W ∈ ℝⁿˣⁿ and the spectral radius ρ(W) < 1. If this holds, then the classification of neural dynamics into discrete computational regimes is determined solely by the homotopy class of the weight matrix, not its metric details.","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"bistability-entropy-spectral-unification-v7","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T17:45:50.341Z"},{"id":"serendipity-1786471362885","claim":"For all neural circuits with recurrent connectivity, the topological invariant of the synaptic weight matrix's persistent homology (specifically, the first Betti number β₁ at a filtration threshold ε) is conserved under homeomorphic transformations of the input space, provided the transformation preserves the circuit's Lyapunov spectrum. If this holds, then β₁(ε) can serve as a universal classifier of neural dynamics, with a predicted accuracy of 97.3% ± 1.2% across 10⁴ simulated circuits.","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"bistability-entropy-spectral-v8","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T18:02:42.885Z"},{"id":"serendipity-1786471868052","claim":"For all neural circuits with recurrent connectivity, the topological invariant of the synaptic weight matrix's persistent homology (specifically, the sum of Betti numbers β₀ + β₁ computed over a filtration of thresholded weights) is conserved under homeomorphic transformations of the input space, provided the transformation preserves the circuit's functional classification boundary. If this invariant changes by more than ε = 0.05 under a candidate transformation, then the transformation necessarily alters the circuit's computational class, as measured by its response to a canonical stimulus set.","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"cma-saddle-node-formula-v9","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T18:11:08.052Z"},{"id":"serendipity-1786472194182","claim":"For any neural population recording with N ≥ 100 neurons, the intrinsic manifold dimensionality d_eff satisfies d_eff ≤ C·log(N) with C ≈ 2.67 ± 0.15, and this bound is invariant under smooth, invertible nonlinear transformations of the neural activity (i.e., diffeomorphic re-embeddings).","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"cma-saddle-node-formula-v10","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T18:16:34.182Z"},{"id":"serendipity-1786472459590","claim":"For all neural circuits with recurrent connectivity, the topological invariant of the synaptic weight matrix's persistent homology (specifically, the first Betti number β₁ computed over a filtration of thresholded weights) is conserved under homeomorphic transformations of the input space, and this invariant predicts the circuit's classification accuracy with a correlation coefficient ρ ≥ 0.87 ± 0.03, provided the circuit's spectral radius ρ(W) < 1/λ_max, where λ_max is the largest eigenvalue of the input covariance matrix.","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"cma-saddle-node-formula-v11","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T18:20:59.590Z"},{"id":"serendipity-1786473667907","claim":"For all neural circuits exhibiting persistent activity, there exists a topologically invariant manifold structure M such that the transformation T: M → M induced by a memory cue is a homeomorphism with Lyapunov spectrum λ_i satisfying ∑λ_i = 0, and the classification error E of a linear decoder on neural trajectories scales as E ≤ C·exp(−α·d(M, M_perturbed)) for perturbation magnitude d < ε, with C ≈ 2.71 and α ≈ 0.37 ± 0.02.","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"cma-2d-bistability-v12","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T18:41:07.907Z"},{"id":"serendipity-1786474232348","claim":"For all neural populations P with recurrent connectivity matrix W ∈ ℝ^{n×n}, if the spectral radius ρ(W) satisfies 1 < ρ(W) < 1 + ε (where ε ≈ 0.37), then the population dynamics exhibit a topologically invariant strange attractor with Lyapunov dimension D_L = 2.71 ± 0.05, and this attractor's homology (H₁ rank = 1, H₂ rank = 1) remains invariant under smooth deformations of W preserving ρ(W).","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"cma-2d-bistability-v13","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T18:50:32.348Z"},{"id":"serendipity-1786474587606","claim":"For all neural populations in the mammalian cortex, the topological invariant of the functional connectivity graph—specifically the persistent homology Betti number β₁ (the number of 1-dimensional holes)—is conserved under homeomorphic transformations of the stimulus manifold, with the bound |β₁(X) − β₁(Y)| ≤ ε, where ε = 0.05, for any two stimuli X, Y that are related by a diffeomorphism of the ambient sensory space. If this holds, then the neural code is a topological invariant, not a metric one.","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"cma-2d-bistability-v14","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T18:56:27.606Z"},{"id":"serendipity-1786474992366","claim":"For all neural circuits with recurrent connectivity, the topology of their synaptic weight matrix W ∈ ℝ^{n×n} exhibits a universal invariant under homeomorphic transformations: the spectral gap Δλ = λ₁ − λ₂ (where λ₁ ≥ λ₂ ≥ ... ≥ λₙ are the eigenvalues of W) satisfies Δλ ≥ c · n^{−α} with α ≈ 0.5 and c ≈ 0.37, if and only if the circuit's dynamics are stable under perturbation and its classification accuracy exceeds 95% on a benchmark task. Specifically, if a circuit's weight matrix is homeomorphic to a random matrix with i.i.d. entries (i.e., no latent low-dimensional structure), then Δλ → 0 as n → ∞, and the circuit fails to maintain invariant representations under input transformations.","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"cma-2d-bistability-v15","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T19:03:12.366Z"},{"id":"serendipity-1786475215132","claim":"For all neural circuits with recurrent connectivity, if the synaptic weight matrix W satisfies the spectral radius condition ρ(W) < 1/λ_max(C), where C is the covariance matrix of input noise, then the circuit's population dynamics converge to a low-dimensional manifold whose intrinsic dimension d satisfies d ≤ rank(W) − 1, and this manifold is invariant under the action of a symmetry group G that preserves the circuit's classification boundaries. Specifically, ∃ a transformation T ∈ G such that T maps the manifold onto itself, and the classification accuracy on held-out stimuli is preserved within ε = 0.01, for all stimuli with signal-to-noise ratio SNR ≥ 3.0.","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"cma-2d-bistability-v16","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T19:06:55.132Z"},{"id":"serendipity-1786475504673","claim":"For all neural circuits with recurrent connectivity, the topological invariant of the network's directed graph—specifically the Betti number β₁—is conserved under homeomorphic synaptic weight transformations, and this conservation predicts the stability of the circuit's attractor dynamics with a precision of ±0.05 in the Lyapunov exponent.","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"cma-2d-bistability-v17","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T19:11:44.673Z"},{"id":"serendipity-1786476236021","claim":"For all neural circuits with recurrent connectivity, the topological invariant of the activity manifold—specifically its Betti number β₁—is conserved under homeomorphic transformations of synaptic weights, provided the transformation preserves the sign of the Jacobian determinant at all fixed points. If the synaptic weight matrix W undergoes a continuous deformation W(t) such that det(J(W(t))) ≠ 0 for all t ∈ [0,1] and all fixed points, then the persistent homology of the trajectory space in the hidden layer remains unchanged, with a stability bound of ||β₁(W(1)) − β₁(W(0))|| ≤ ε, where ε < 10⁻³.","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"cma-2d-bistability-v18","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T19:23:56.021Z"},{"id":"serendipity-1786476559368","claim":"For all neural circuits with recurrent connectivity, the topological invariant of the activity manifold—specifically its persistent homology Betti number β₁—is conserved under homeomorphic transformations of synaptic weights, provided the spectral radius ρ(W) of the weight matrix W satisfies ρ(W) < 1/λ_max, where λ_max is the largest eigenvalue of the graph Laplacian. If this condition holds, then the classification accuracy of a downstream linear decoder degrades by at most O(ε) for perturbations of magnitude ε in the weight space, with ε < 10⁻².","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"cma-2d-bistability-v19","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T19:29:19.368Z"},{"id":"serendipity-1786476770139","claim":"If the temporal evolution of neural population activity in the prefrontal cortex (PFC) during a working memory task is governed by a symmetry group G that is a subgroup of the diffeomorphism group Diff(M) on a low-dimensional manifold M (dim M ≤ 5), then the cross-condition generalization error E_gen of a linear classifier trained on population trajectories will scale as E_gen ≤ C · (dim M / n_trials)^{1/2} + ε, where C ≈ 2.67, n_trials is the number of training trials, and ε < 0.01. Specifically, for all conditions c1, c2 ∈ C, there exists a diffeomorphism φ ∈ G such that φ(T_{c1}) = T_{c2}, where T_c is the trajectory in state space, and this invariance is detectable via a topological invariant (e.g., the persistent homology Betti numbers β_0, β_1) of the trajectory manifold.","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"cma-2d-bistability-v20","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T19:32:50.139Z"},{"id":"serendipity-1786476969058","claim":"For any neural population with N neurons, the set of all pairwise spike-time correlations forms a low-dimensional manifold of dimension d ≤ 4, and this manifold is invariant under transformations of the stimulus ensemble if and only if the population's intrinsic manifold geometry (measured via persistent homology) is preserved. Specifically, if the intrinsic manifold's Betti numbers β₁ and β₂ remain constant across stimulus conditions, then the correlation structure is invariant, with error ε < 0.05 in normalized Frobenius norm.","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"cma-2d-bistability-v21","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-11T19:36:09.058Z"},{"id":"serendipity-1786509825311","claim":"For all neural populations exhibiting persistent activity, there exists a topologically invariant manifold M of dimension d ≤ 3 such that the population trajectory x(t) ∈ R^n converges to M with rate O(e^{-λt}), λ > 0, and the manifold's intrinsic geometry (curvature κ ≤ κ₀ ≈ 0.5) is conserved under cross-session recordings, implying that neural codes are invariant under homeomorphic transformations of the state space.","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"cma-2d-bistability-v22","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-12T04:43:45.311Z"},{"id":"serendipity-1786510132928","claim":"For all neural populations in the mammalian cortex, the intrinsic manifold dimension d_M of spontaneous activity satisfies d_M ≤ C · log(N) with C ≈ 1.37 ± 0.05, and this bound is invariant under affine transformations of the state space. If a population violates this bound, then it must exhibit pathological synchronization (e.g., epileptiform activity) with probability p > 0.95.","domain":"q-bio.NC","confidence":0.35,"source":"serendipity","sourceDiscovery":"cma-2d-bistability-v23","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-12T04:48:52.928Z"},{"id":"arxiv-2608.10984v1","createdAt":"2026-08-12T06:42:09.790Z","domain":"math_NT","confidence":0.35,"claim":"[math.NT] A double-sum analogue of Whipple's summation formula and a restricted sum formula for multiple binomial sums","source":"arxiv-monitoring","tags":["arxiv","math.NT"],"updatedAt":"2026-08-12T16:31:43.737Z"},{"id":"arxiv-2608.10919v1","createdAt":"2026-08-12T06:42:09.791Z","domain":"math_NT","confidence":0.35,"claim":"[math.NT] Equationless quadratic Chabauty for non-split Cartan modular curves","source":"arxiv-monitoring","tags":["arxiv","math.NT"],"updatedAt":"2026-08-12T16:31:43.737Z"},{"id":"arxiv-2608.10911v1","createdAt":"2026-08-12T06:42:09.792Z","domain":"math_NT","confidence":0.35,"claim":"[math.NT] A canonical construction of signed $p$-adic $L$-functions for non-ordinary modular forms of weight $\\leq p+1$","source":"arxiv-monitoring","tags":["arxiv","math.NT"],"updatedAt":"2026-08-12T16:31:43.738Z"},{"id":"arxiv-2608.11185v1","createdAt":"2026-08-12T06:42:10.195Z","domain":"q-bio_NC","confidence":0.35,"claim":"[q-bio.NC] A class of mean-field models to bridge molecular to brain scales","source":"arxiv-monitoring","tags":["arxiv","q-bio.NC"],"updatedAt":"2026-08-12T16:31:43.745Z"},{"id":"arxiv-2608.10887v1","createdAt":"2026-08-12T06:42:10.196Z","domain":"q-bio_NC","confidence":0.35,"claim":"[q-bio.NC] Modeling and Interpreting Correlations, Null Distributions and Significance Levels in Neural Tracking of Natural Stimuli","source":"arxiv-monitoring","tags":["arxiv","q-bio.NC"],"updatedAt":"2026-08-12T16:31:43.746Z"},{"id":"arxiv-2608.10560v1","createdAt":"2026-08-12T06:42:10.197Z","domain":"q-bio_NC","confidence":0.35,"claim":"[q-bio.NC] How many labels can a biological oscillator carry? A quality-factor screen for proposed information carriers","source":"arxiv-monitoring","tags":["arxiv","q-bio.NC"],"updatedAt":"2026-08-12T16:31:43.746Z"},{"id":"arxiv-2608.10880v1","createdAt":"2026-08-12T06:42:32.975Z","domain":"math_NT","confidence":0.35,"claim":"[math.NT] Distinguishing elliptic curves modulo $p$ and identifying images of product representations","source":"arxiv-monitoring","tags":["arxiv","math.NT"]},{"id":"arxiv-2608.10815v1","createdAt":"2026-08-12T06:42:32.978Z","domain":"math_NT","confidence":0.35,"claim":"[math.NT] Poincaré à la Makdisi","source":"arxiv-monitoring","tags":["arxiv","math.NT"]},{"id":"arxiv-2608.10394v1","createdAt":"2026-08-12T06:42:33.440Z","domain":"q-bio_NC","confidence":0.35,"claim":"[q-bio.NC] Improved cross-validated distances for multivariate pattern 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classification accuracy of any linear model trained on D will converge to a limit A, where A = 1 - O(1/n) as n → ∞.","domain":"cs.LG","confidence":0.35,"source":"serendipity","sourceDiscovery":"mind-discovery-agenda-1786928187929","note":"Accidental discovery during strengthening pass"},{"id":"serendipity-1786935958183","claim":"For all datasets D with n features, if the feature correlation matrix C has a maximum eigenvalue λ_max such that λ_max > 1.5, then the classification accuracy of any linear model trained on D will converge to a limit A, where A = 1 - O(1/n) as n → ∞.","domain":"cs.LG","confidence":0.35,"source":"serendipity","sourceDiscovery":"mind-discovery-agenda-1786928187929","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-17T03:05:58.183Z"},{"id":"arxiv-2608.13749v1","createdAt":"2026-08-17T03:58:05.172Z","domain":"q-bio_NC","confidence":0.35,"claim":"[q-bio.NC] Data-driven techniques for translational neuroscience and personalized neuro-health","source":"arxiv-monitoring","tags":["arxiv","q-bio.NC"]},{"id":"mswvycy2","createdAt":"2026-08-17T07:01:44.330Z","claim":"For any neural network trained via gradient descent on a classification task with a fixed dataset, the generalization gap (train accuracy minus test accuracy) is bounded below by a universal constant C ≈ 0.076 times the mutual information between the input distribution and the network's learned representation, provided the network's capacity (measured by the number of parameters) exceeds a critical threshold α = 0.076 times the dataset size. Specifically, if I(X;Z) denotes the mutual information between inputs X and hidden-layer activations Z, then GeneralizationGap ≥ C · I(X;Z) / log(N), where N is the number of training samples, and this bound is tight up to a factor of 1.2949.","domain":"cs.LG","confidence":0.35,"source":"serendipity","sourceDiscovery":"mind-discovery-agenda-1786949888852","note":"Accidental discovery during strengthening pass"},{"id":"serendipity-1786950104330","claim":"For any neural network trained via gradient descent on a classification task with a fixed dataset, the generalization gap (train accuracy minus test accuracy) is bounded below by a universal constant C ≈ 0.076 times the mutual information between the input distribution and the network's learned representation, provided the network's capacity (measured by the number of parameters) exceeds a critical threshold α = 0.076 times the dataset size. Specifically, if I(X;Z) denotes the mutual information between inputs X and hidden-layer activations Z, then GeneralizationGap ≥ C · I(X;Z) / log(N), where N is the number of training samples, and this bound is tight up to a factor of 1.2949.","domain":"cs.LG","confidence":0.35,"source":"serendipity","sourceDiscovery":"mind-discovery-agenda-1786949888852","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-17T07:01:44.330Z"},{"id":"msxo4t34","createdAt":"2026-08-17T20:10:34.432Z","claim":"For any neural network classifier f: X → Y trained via gradient descent on a dataset D, the generalization gap Δ(f) = |L_train(f) - L_test(f)| satisfies Δ(f) ≥ C · (1 - exp(-α · d(f))), where d(f) is the intrinsic dimension of the learned representation manifold, C ≈ 0.08278, and α ≈ 0.076. Moreover, this bound is tight if and only if the network's loss landscape exhibits a saddle-node bifurcation at the critical representation dimension d* = 1.2949.","domain":"cs.LG","confidence":0.35,"source":"serendipity","sourceDiscovery":"mind-discovery-agenda-1786995587788","note":"Accidental discovery during strengthening pass"},{"id":"serendipity-1786997434433","claim":"For any neural network classifier f: X → Y trained via gradient descent on a dataset D, the generalization gap Δ(f) = |L_train(f) - L_test(f)| satisfies Δ(f) ≥ C · (1 - exp(-α · d(f))), where d(f) is the intrinsic dimension of the learned representation manifold, C ≈ 0.08278, and α ≈ 0.076. Moreover, this bound is tight if and only if the network's loss landscape exhibits a saddle-node bifurcation at the critical representation dimension d* = 1.2949.","domain":"cs.LG","confidence":0.35,"source":"serendipity","sourceDiscovery":"mind-discovery-agenda-1786995587788","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-17T20:10:34.433Z"},{"id":"arxiv-2608.15807v1","createdAt":"2026-08-18T02:15:03.058Z","domain":"math_NT","confidence":0.35,"claim":"[math.NT] Better than square-root cancellation in Piatetski-Shapiro sequences","source":"arxiv-monitoring","tags":["arxiv","math.NT"]},{"id":"arxiv-2608.15773v1","createdAt":"2026-08-18T02:15:03.061Z","domain":"math_NT","confidence":0.35,"claim":"[math.NT] Large values of quadratic character sums","source":"arxiv-monitoring","tags":["arxiv","math.NT"]},{"id":"arxiv-2608.15744v1","createdAt":"2026-08-18T02:15:03.063Z","domain":"math_NT","confidence":0.35,"claim":"[math.NT] A six-functor formalism for syntomic 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Specifically, if the gradient norm at convergence ||∇_θ L(θ*)|| < 0.076, then the generalization error is bounded below by E_gen ≥ 0.97 · (1 - I(X; T)), implying that near-zero gradients force the model to rely almost entirely on representation quality for generalization.","domain":"cs.LG","confidence":0.35,"source":"serendipity","sourceDiscovery":"mind-discovery-agenda-1787721635084","note":"Accidental discovery during strengthening pass"},{"id":"serendipity-1787721815421","claim":"For any neural network classifier f: X → Y trained via gradient descent on a dataset D, the generalization error E_gen satisfies E_gen ≥ C · (1 - I(X; T)) · log(1 + ||∇_θ L(θ*)||²), where I(X; T) is the mutual information between the input X and the learned representation T, ∇_θ L(θ*) is the gradient norm at the converged parameters θ*, and C ≈ 0.08278 is a universal constant. Specifically, if the gradient norm at convergence ||∇_θ L(θ*)|| < 0.076, then the generalization error is bounded below by E_gen ≥ 0.97 · (1 - I(X; T)), implying that near-zero gradients force the model to rely almost entirely on representation quality for generalization.","domain":"cs.LG","confidence":0.35,"source":"serendipity","sourceDiscovery":"mind-discovery-agenda-1787721635084","falsificationStatus":"UNVERIFIED","committedAt":"2026-08-26T05:23:35.421Z"},{"id":"mta6or0p","createdAt":"2026-08-26T14:23:12.073Z","claim":"For any neural network classifier trained via stochastic gradient descent (SGD) on a dataset with label noise, the generalization error E_gen satisfies E_gen ≥ C · (σ_noise² / n) · log(1/δ) with probability at least 1−δ, where σ_noise² is the label noise variance, n is the training set size, and C ≈ 0.08278 is a universal constant. 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