Convex geometry · formal theorem
Polygonal Reflection-Allowed Worm Lower Bound
Four Lean-checked pointwise area inequalities cover finite polygonal worms and all unit worms, with direct motions or reflections allowed. Their strict rational corollaries give the lower bound 0.23743658226923856768; the exact optimum remains open.
Scope: Every compact convex cover universal for finite polygonal worms or for all unit worms has area strictly greater than 0.23743658226923856768, both with direct motions and when reflections are allowed.
Exact formal proposition
Hypotheses and conclusion
theorem polygonalReflection_area_ge_THat (C : ConvexCover) (hC : PlacesEveryUnitPolyline C .reflectionAllowed) : THat ≤ coverArea CResult boundary
What this theorem does—and does not—establish
Every compact convex cover universal for finite polygonal worms or for all unit worms has area strictly greater than 0.23743658226923856768, both with direct motions and when reflections are allowed.
- The four formal endpoints are pointwise cover-area inequalities; the paper separately takes infima to state the bounds for M₊ and M±.
- The result is a lower-bound improvement and does not determine the optimal cover or solve Moser's convex worm problem.
- ProofAtlas accepted-result status, independent statement alignment, specialist peer review, historical priority, and unrelated replication remain open.
- The explanatory infographic and adversarial manuscript audit are not Lean proof evidence.
- The disclosed Lean foundations are standard logical foundations, not geometric assumptions.
Companion research paper
A Mixed-Area Lower Bound for Moser's Convex Worm Problem
The September 2 paper proves that both the direct and reflection-allowed convex-worm covering infima exceed 0.23743658226923856768, improving the prior 0.232239 lower bound. The exact optimum remains open. A separate Lean development checks stronger pointwise versions for each universal convex cover.
Author-authorized early release. This fixed paper package first appeared while independent presentation review was pending; hosting it does not mark the Lean result accepted.
Paper, rights, and source relationship
Hosting authorized by the rightsholder. ProofAtlas metadata cover; not a reproduction of a paper page or part of the proof.
Lech Mazur, “A Mixed-Area Lower Bound for Moser's Convex Worm Problem,” September 2, 2026.
- The four Lean endpoints are pointwise inequalities for each universal convex cover. The paper separately takes infima to state bounds for M_+ and M_±; that infimum packaging is not a Lean declaration in this release.
- The source release ledger reports successful targeted checks, aggregate replay, no admissions, and the standard foundation closure propext, Classical.choice, and Quot.sound. ProofAtlas accepted-result status remains separate because the retained evidence is not the complete hash-bound build transcript required by the current contract.
- The paper's adversarial audit predates the final 21:22 build. Its mathematical and expository findings F2–F5 are repaired in the matching final TeX; F1 is resolved by the permanent ProofAtlas routes published with this package.
- Neither the paper nor the Lean result determines the exact optimal cover, establishes optimality of the lower bound, or closes Moser's convex worm problem.
Paper overview
Moser's convex worm problem and the improved lower bound
Author-supplied explanation of the covering problem and the new lower bound. The scaled interval shows about 18.1% of the previously open gap closed; the image is explanatory, not proof or formal-verification evidence.
Line counts exclude blank lines; comments and documentation count. The total is the commit-pinned first-party Lean import closure; Mathlib and other third-party dependencies are excluded.
Continue the mathematics
Open questions and extensions
The pinned theorem and its complete local import closure let people and AI agents inspect the proof, compare another route, isolate reusable lemmas, or formulate a stronger exact statement. Lean checks each proposed extension against its own exact statement.
What the source ZIP contains
The ZIP contains the checked first-party Lean import closure, exact statements and boundaries, license, notice, evidence, source-footprint manifest, and continuation data. Mathlib and other third-party dependencies are not bundled.
Technical evidence, source identity, and assumptions
- Main Lean declaration
polygonalReflection_area_ge_THat- Source commit
c9254797b42c
Mechanical evidence
Lean verification
These fields support the exact Lean declaration, not a broader informal claim.
Why ProofAtlas acceptance is still pending: the supplied release audit records a successful one-worker rehash build, no unfinished proof commands, and the reported axiom closure. It does not retain the complete command/output transcript and clean-source collection record required for accepted-result status.
- Artifact ID
artifact.moser-worm.polygonal-reflection-area.lean-package.v001- Declarations covered by recorded evidence
MoserWorm.polygonalReflection_area_ge_THatMoserWorm.polygonalReflection_area_gt_published- Lean build
- passed
- Recorded build time
- Not retained in the supplied release audit
- Unfinished proof check
- passed
- Lean toolchain
leanprover/lean4:v4.30.0-rc2- Recorded source commit
c9254797b42c2a89acb00281c0822c7d6fc21923- Source SHA-256
sha256:2f873beecb35c02bf40af8516eaef6393bb34533d9271e79eb8085b76c1e263f- Statement alignment
- under review
Lean foundations
Standard foundations used by the proof
Lean reports the logical foundations below through Mathlib. They are standard proof-system foundations, not conjectural mathematical assumptions about this theorem. The recorded closure stays within the approved classical_mathlib_standard profile, with no unexpected axiom or unfinished-proof placeholder.
Classical.choiceQuot.soundpropext
Independent review
Review open
Review remains open for the exact statement, scope, evidence, and public wording. See the remaining review questions.
Credit for this release
Contributors and roles
Authorship and other contributions are recorded separately for this version.
Contributions
- Direct contributionOpenAI Codex, OpenAI GPT-5.6 Pro
OpenAI GPT-5.6 Pro and OpenAI Codex have source-reported roles in mathematical exploration, proof development, computational testing, adversarial auditing, formalization, and exposition.
- Computation
- Exposition
- Formalization
- Gap or error discovery
- Proof strategy
- Direct contributionLech Mazur
Lech Mazur is the paper author and accountable editor and directed the AI-assisted research, verification process, output selection, and reconciliation.
- Exposition
- Proof strategy
- Research direction
Citation
Cite this release
Use the version-specific citation so the authorship, scope, and public record remain attached.
Release history
Versions and public record
- Release
- Paper release
- Version
- public version of September 2, 2026
- Published
- Canonical page
- Open canonical page
- Earlier release
credit.release.moser-worm-mixed-area-lower-bound.v001
Release chronology
This version is part of the release history. No independent external timestamp is claimed.
Scope of this release
- No accepted-result record exists for the Moser worm artifacts in this snapshot.
- No independent external priority anchor, specialist review, or independent statement-alignment review is recorded.
- The paper's infimum notation is not a Lean declaration in this release; the checked endpoints are stronger pointwise cover inequalities.
- The paper, bundle, supplied infographic, and ProofAtlas covers are authorized for ProofAtlas distribution but have no general open-content license.
- The result improves a lower bound and does not determine the optimal cover or solve Moser's convex worm problem.