• Method: Laminar Narayana recurrences and a three-variable Lorentzian lift.
• Dated versions: Our draft commit, 4 November 2026 (UTC); Defant’s preprint, 10 November 2026, Theorem 1.1 gives the corresponding full-distribution ULC result. The draft’s trimmed statement implies that conclusion; the preprint also develops a separate multivariate matroid structure.
• Review: Independently checked by AI-assisted mathematical reviews; human peer review and historical priority are not established.

• Proof: The degree-12 argument and degree-13 extension establish the initial-difference Kruskal–Katona conclusion, not merely Macaulay growth or the separate original-coefficient question.
• Reproducibility: 10,445,449 finite q values across 66 index cases leave 15 local candidate tuples, all excluded by exact algebraic arguments. The infinite tail is proved, not truncated at a search limit.
• Boundary: Newton inequalities alone suffice through degree 11 and fail in degree 12. That obstruction is not real-rooted and does not refute Question 4.4. Primary sources and prior-art distinctions

• Proof: A path-matrix determinant and Gauss–Lucas locate the transformed roots in (−2,2); the classical Grace–Walsh–Szegő theorem and reciprocal symmetry extend the conclusion to all positive integer capacities. The recurrence, generating function and classical tools are credited to prior work.
• Checks: 404 exact unit-circle certifications accompany the analytic argument. Finite checks and plotted examples are not the all-parameter proof. This project concerns Conjecture 4.5 only, not the separate h* real-rootedness or interlacing conjectures.

Both full original targets have October 2026 Lean source-replay and independent AI-assisted semantic-review records; this publication and presentation pass did not recompile Lean. AIM23's exact 4→6 uses the independent rational/Sturm certificate; Lean proves input at most four and output exactly six. AIM34 proves a finite polynomial witness exists for every parameter without a uniform degree bound. The original sources are a book chapter / AIM problem list, not the four leading mathematics journals. No human peer review, journal acceptance or global-priority claim is made. The fixed published versions retain sources, historical evidence, reproduction instructions and the historical-source-link limitations.

• Stronger trinomial structure: For every integer n ≥ 4004, the fourth-iterate polynomial has 2n distinct negative real roots.
• Proof and reproduction: The general review checks the analytic closure and independently rebuilds all 2,047 Poisson certificates by two exact routes. The trinomial manuscript, code and certificates combine an unbounded analytic argument with finite invariant-cone certificates; its 2 November review distinguishes replay from fresh enclosure reconstruction.
• Status: Computer-assisted proof drafts with independent AI-assisted mathematical and computational review. Trinomial PR #2 and general-audit PR #3 remain unmerged drafts. The general audit reviews an existing theorem, not a new result or family; no human peer review, formal verification or historical-priority certification is claimed.

• Original source: Andrew Howroyd, OEIS A297557 (2018).
• Prior work: John Machacek, Procedia Computer Science 195 (2021), proved the 3 ∣ d branch. Combining the two results completes the counting formula for d ≥ 9.
• Method: An exact weighted-cover certificate and a coordinate-reduction argument reduce the classification to three backgrounds with nine corner extensions each.
• Further results & materials: Explicit classification of all 27 sets, beyond the conjectured count.
• Verification: Lean 4 proof of the full d ≥ 10, 3 ∤ d statement; 89 rational certificate entries.

• Original source: Lukas Katthän, The Lecture Hall Cone as a toric deformation (2018), prepared for the Osaka lattice-polytopes workshop.
• Prior work: The original paper proposed the polynomiality and algebraic-structure statements and checked lengths at most 12. Flajolet’s continued-fraction theory and Sokal’s exposition provide credited classical tools.
• Method: A signed Dyck-path identification, Toeplitz–Hankel identities and the defining recurrence give the all-index polynomiality proof.
• Further results & materials: Composition coefficient formulas and exact monomial counts; a minimal monic SAGBI basis, a Hilbert-basis correspondence and flat toric degeneration over every field, under the manuscript’s disclosed square-minor interpretation.
• Verification: Lean formalizes the genuine rational-function polynomiality and signed-path core. SAGBI and degeneration extensions have written mathematical proofs.

• Original source: Francesco Brenti, Gordon F. Royle and David G. Wagner, Canadian Journal of Mathematics (1994), Conjecture 6.5. The graph-join assertion is the separate Conjecture 6.6.
• Structural theorem: An incomparability graph partitioned into k nonempty independent classes, with one anchor per class adjacent to every vertex outside that class, has real-rooted Bell chromatic quotients for every clique size 0 ≤ j ≤ k. Ordinary attachment along any clique to any τ-real graph therefore preserves τ-reality. This includes connected Ferrers/chain graphs with an edge.
• Complete theta family: For every a,b,c ≥ 2, the Bell edge quotient of Θ(a,b,c) has only strictly negative real zeros. Ordinary attachment along an edge to any τ-real graph therefore preserves τ-reality. An effective root-radius bound and Bernstein-discriminant certificates cover the analytic range; exact rational root certificates close all 2,460 remaining sorted triples, with 83,842 root brackets checked by independent integer-arithmetic replays.
• Uniform cycle theorem: Proof at the published version. An ordinary proof by root monotonicity and an enlarged arrowhead invariant covers every cycle length and every input degree, with strict interlacing for simple negative inputs and weak interlacing on the boundary. It subsumes C₅/C₇ and the special cases with k length-two paths plus one length-m path, not the independent all-three-path theorem or arbitrary-path-count long-minimum condition. No arbitrary simultaneous substitution or unrestricted larger-clique closure is claimed.
• Proofs & checks: Structural manuscript, complete theta proof and certificates, and the earlier uniform eventual theta addendum. The theta result is a computer-assisted proof combining an analytic theorem with exhaustive exact certification of its finite remainder, not finite sampling alone. AI-assisted work, including independent internal review of the uniform proof; no human peer review, proof-assistant verification or historical-priority claim; the literature comparison remains incomplete.

Peak polynomials: the last real root · _{Experimental Mathematics · 2016 journal version · 10 years ago · Real roots}

Partial interval results → the full last-real-root statement. An integral sign argument covers the missing noninteger interval, without bounds on peak count or spacing.

• Original source: Sara Billey, Matthew Fahrbach and Alan Talmage, Experimental Mathematics 25(2), 165–175 (2016); preprint first posted in 2014.
• Prior work: Billey–Fahrbach–Talmage supplied the recurrence and symmetry identities. Li’s coefficient theorem already implies right-side positivity and several local sign properties; the additional argument covers the remaining noninteger interval.
• Method: A self-contained integral representation and sign analysis across the full interval beyond the penultimate peak.
• Further results & materials: Signs on both sides of the largest peak, its simplicity, and the singleton case are included; overlapping consequences of Li’s theorem are credited in the manuscript.
• Verification: All-parameter mathematical proof, with supplementary exact finite checks and a separately recorded small-case reproduction.

K₆ chains: a false 22-colour bound replaced by the sharp bound 23 · _{Conjecture posed in 2025 · Graph colouring}

A conjectured bound of 22 → the correct sharp bound of 23. Refuted 22 with a 450-vertex graph, then established the uniform bound for the entire family, backed by a 16.6-million-record exact certificate.

• Original source: Furmańczyk, Gözüpek and Özkan, Packing coloring of graphs with long paths (2025), Conjecture 2.
• Prior work: The proposed common upper bound was 22 for the path-aligned K₆ clique chains.
• Method: A finite-state scheduling relaxation and decreasing-rank certificate exclude 22; an explicit periodic colouring supplies the upper bound 23.
• Further results & materials: Sharp uniform bound 23 for the full chain family and exact value 23 for every t ≥ 75. The minimum counterexample length and all shorter-chain exact values are not determined.
• Verification: A 16,646,562-record exact certificate with Python and C++ checkers. This is computational certificate verification, not a Lean formalization.

Fixed reproduction evidence · 2 November 2026 includes rank traces, source hashes and JSON-only oracle replay. The 18 jobs were run in an independent validation environment, not upstream CI. The diagram illustrates scheduling, not measured throughput. Scope: fixed two-rank CPU/Gloo, ordinary map-style sampling and deterministic resume fixtures; no GPU/NCCL, arbitrary stochastic-resume or performance claim. AI-assisted implementation, tests and evidence review.

• _2026-10-31 · claude-trading-skills #596 — merged a fix aligning reported risk with the selected position-sizing method; maintainer tradermonty’s follow-up #602 added stop validation, Decimal risk rounding and documentation.
• _2026-10-15 · Developer Roadmap #10451 — merged.
• _2026-10-15 · Free Programming Books #13533 — merged.
• _2026-10-15 · Published the Lean supplement for Hardinian asymptotics at r = 2.
• _2026-10-14 · KOReader #16220 — merged.
• _2026-10-14 · REA #3224 — Kotlin source inventory support adopted upstream with co-author credit.