Trace — 2607.19276v1

DECLARATION — Central claim: for every n ≥ 2, every red–blue coloring of the edges of the n-dimensional Boolean hypercube in which each edge has the opposite color from its antipodal edge contains a monochromatic graph path from some vertex x to its coordinatewise antipode. Declared target: R3 (core). I will first give a faithful Lean 4/mathlib statement (R1), then mechanically verify the paper-relevant finite/small decidable instances (R2), then formalize and prove the paper's abstract chain-level Borsuk–Ulam obstruction, Lemma 3.1, without sorry or added axioms (R3). I do not declare R4 because the paper's subdivision-invariant spherical polyhedral chain quotient and radial-chain construction would require a substantial new geometry library.

Pre-declaration reading completed: paper.txt has 1,420 lines / 8,757 words. Read the full extracted paper, including Theorem 1.1, the rook-labeling reduction, Lemma 3.1, the subdivision-invariant target chains, the rook-gallery rank lemma, radial cancellation, and the final forbidden-chain-map argument. No AGENTS.md or pre-existing trace.jsonl was present.

Inspected the local mathlib checkout. There is no prebuilt hypercube graph, but mathlib supplies hammingDist, SimpleGraph.edgeSet as unordered Sym2 pairs, List.IsChain, and the standard chain-complex infrastructure. An initial search was accidentally issued from the parent directory and reported Mathlib not found; rerunning inside repro located the needed APIs.

R1 achieved in repro/Repro.lean. Defined CubeVertex = Fin n → Bool, coordinatewise antipode, Q_n by Hamming distance one, CubeEdge as the actual unordered edge-set subtype, edge antipode by Sym2.map, antipodal red-blue colorings, and MonochromaticPath as a nodup list with colored consecutive edges. NorineConjecture faithfully states Theorem 1.1 for every n ≥ 2; no ordered-edge or walk weakening was used.

Ran `lake env lean Repro.lean` after R1; exit code 0 with no diagnostics.

R2 achieved. Added an executable finite formulation via reachability in each one-color spanning subgraph and proved reachability yields a genuine nodup monochromatic path. `native_decide` exhaustively verified all 16 edge-colorings of Q_2 and all 4096 edge-colorings of Q_3 (respectively 4 and 12 unordered edges), with the antipodal implication selecting 4 and 64 relevant colorings. Also formalized the paper's faithful rook labels Ω_K and exhaustively ruled out rook labelings for K=2 and K=3; the direct K=3 raw search considered 6^8 = 1,679,616 label functions and completed successfully.

Finite-check development dead ends and resolutions: generic Decidable inference initially failed because opaque graph predicates lacked explicit DecidableRel instances; supplied executable instances. A temporary Check.lean could not import a root-level unbuilt Repro module and was deleted. The first K=3 rook search was slow (~70 seconds) but completed; it was temporarily omitted during core-proof edit cycles and will be restored for the final audit.

R3 achieved. Formalized involutive augmented nonnegative chain complexes of ZMod 2 modules; equivariant augmentation-preserving chain maps; source exactness, target truncation, and norm kernel=image hypotheses; and proved the full abstract chain-level Borsuk--Ulam obstruction (paper Lemma 3.1). The proof constructs the ascending source tower and performs the descending target correction recursively, including d=0, with no sorry or added axioms.

Ran `lake env lean Repro.lean`; exit code 0. `#print axioms ChainBU.chainLevelBorsukUlam` reports only [propext, Classical.choice, Quot.sound], standard mathlib foundations. Search found no `sorry`, `admit`, or axiom declaration in Repro.lean (the only textual match is the #print axioms audit command).

Overshoot beyond declared R3: fully formalized the Section 2 reduction. Proved antipodal transport of colored adjacency; constructed the red-component rook labeling; enumerated components into Fin K; extended labels from Q_n to Q_K by coordinate restriction; proved edges restrict to an edge or collapse; obtained the full finite-extension form of Lemma 2.2; and proved NorineConjecture conditionally from the all-dimensional no-rook Theorem 2.3.

Investigated an R4 continuation by searching all local Mathlib Lean sources for spherical polytopes, polyhedral chains, face-to-face subdivisions/common refinements, Freudenthal triangulations, cubical cellular chains, and cellular cube complexes. Found only a passing comment about polyhedral cones and no reusable formal infrastructure for the paper's Sections 4–5 construction.

Stopped climbing after R3 plus the Section 2 overshoot. R4 would require de novo formalization of spherical pointed polyhedral cones and their face lattices, common fan refinements, a quotient by every subdivision relation, well-defined boundary and antipodal norm exactness, Freudenthal chains, the rook-gallery rank lemma, radial degenerate cancellation, and the resulting equivariant chain map. This is a genuine library-scale boundary, not a proof failure at the declared target.

Final full audit with the K=3 rook computation restored: `lake env lean Repro.lean` exited 0 and printed only the expected axiom audit [propext, Classical.choice, Quot.sound]. repro/Repro.lean is 573 lines. A final search found zero `sorry`, `admit`, or axiom declarations. A git-status check from the nested mathlib repository could not include ../trace.jsonl because that trace lies outside the nested repo; this diagnostic had no effect on files or verification.

Final trace audit: the first shell audit attempt used the reserved zsh variable `status` and failed without side effects. The corrected audit confirmed every trace.jsonl line is valid JSON, the temporary repro/Check.lean is absent, and repro/Repro.lean contains no sorry, admit, or axiom declaration.