A Lean 4 Formalization Program Around the Yang–Mills Mass Gap — Version 53: Finite-Volume Lipschitz Stability Under Remote Polymer-Activity Damping

Abstract

We are building and publishing, step by step, a formal and verifiable research program directed at the Yang–Mills existence and mass-gap problem, stating precisely what has been proved and what remains open. TL;DR Version 53 establishes finite-volume Lipschitz stability under remote polymer-activity damping. It bounds the difference between two normalized polymer functionals with arbitrary damping parameters θ and θ′ in [0,1], with an explicit factor |θ − θ′| and exponential decay in the walk-separation scale. The library now contains 113 Lean 4 modules, 34,585 source lines and 1,665 top-level declaration lines by the documented textual counting method. This last figure includes theorems, lemmas, definitions and other declarations; it is not a count of theorems alone. Version 53 adds 29 theorems in two new modules, with kernel certificates depending only on propext, Classical.choice and Quot.sound. The accepted candidate underwent an independent build reproduction by a separate Claude Fable 5.1 instance, adversarial reading by Kimi 3, and GitHub Actions verification before and after integration. Audit materials, reports, corrections and provenance are included in the frozen repository snapshot. For non-specialists A formal proof is a mathematical argument written in a language whose logical steps can be checked by a proof assistant. Here, Lean 4 checks that each formal conclusion follows from its definitions and hypotheses. The current results concern a finite lattice: a discrete mathematical setting used in lattice gauge theory. In the polymer representation, the activity of every polymer touching a selected remote region is multiplied by a damping parameter θ. Setting θ = 0 suppresses those activities; setting θ = 1 leaves them unchanged. Version 52 bounded the difference between a damped functional and the Gibbs expectation. Version 53 compares two damping levels directly. It proves that a small change in the parameter produces a proportionally bounded change in the functional, with a bound that decreases exponentially as the specified walk separation increases. This is the meaning of Lipschitz stability in this release. The theorem supplies an explicit quantitative estimate, rather than only asserting continuity. The new result Write F(θ) = activityDampedExpectation μm β χ f s r θ and D_s = card(supportLinkFinset s). For θ, θ′ ∈ [0,1], the formalization proves the sharper estimate: |F(θ) − F(θ′)| ≤ |θ − θ′| · Cf · exp(−n/2) · [exp(8D_s/113) + exp(4D_s/113)] and consequently: |F(θ) − F(θ′)| ≤ |θ − θ′| · (2Cf) · exp(8D_s/113) · exp(−n/2). The result holds on a finite periodic four-dimensional lattice, under the formal measure and group context, in the regime 0 ≤ β ≤ 1/40000. In the Wilson convention, this is the small-β, strong-coupling regime. The function χ : G → ℝ is measurable and satisfies |χ| ≤ 1; f is measurable and satisfies |f| ≤ Cf; and the regions satisfy WalkBarrierSeparated s r n. The nonnegativity of Cf is derived from the majorant. The two-functional estimate does not require DependsOnlyOn f s. Without that hypothesis, s is a parameter of the functional and is not asserted to be a support of f. The hypothesis enters when identifying F(1) with the Gibbs expectation. The bound has no separate factor involving the ambient volume or the cardinality of the damped region. Its prefactor depends on D_s and Cf, while remoteness enters through the walk-separation scale n. The formalized endpoints include: θ = θ′: the difference vanishes. θ′ = 1: under DependsOnlyOn f s, the theorem recovers the Version 52 estimate against Gibbs with the same constant and decay rate. θ′ = 0: it bounds the distance to the restricted functional with a factor θ. Empty damped region: the functional is independent of the damping parameter for arbitrary real parameters, under the representation hypotheses. The proof uses a direct two-parameter identity and separate bounds for its two columns. A key ingredient controls the damped normalization with cost κ = 1, keeping the bridge column within the admissible budget (7/8,1). Verification and public evidence The active Phase 3 library records zero sorry placeholders and zero project-specific scientific axioms. For Version 53, a separate Claude Fable 5.1 instance reconstructed all 113 modules from scratch, with 0 replayed project modules, and independently checked the 29 new axiom certificates and endpoint tests. The build completed without errors. The 114 inherited warnings were unchanged, with none in the two new modules. Kimi 3 supplied an additional adversarial mathematical and code review by reading, from a different model family, without executing Lean. Its report and errata C1 form one review and must be read together; C1 prevails on the corrected points. Both the reproduction and the adversarial review declared prior exposure and were not blind. These checks distinguish formal execution, adversarial reading and publication custody. They are AI-assisted verification and review records, not specialized human mathematical peer review. Research scope The results established here are finite-lattice theorems. Version 53 proves a Lipschitz estimate for the normalized polymer functional; it does not establish monotonicity or differentiability in the damping parameter, construct a second Gibbs measure, or prove a thermodynamic limit, continuum limit or mass gap. The project’s scientific objective remains the Yang–Mills existence and mass-gap problem. This release documents the formal results achieved within the stated finite-volume setting and is not an announcement of a solution to the Millennium Problem. Human–AI research methodology The project is coordinated by Jucelha Carvalho, Smart Tour Brasil, using the Consensus Framework: a human-led methodology assigning distinct roles to AI models for architecture, implementation, reproduction, adversarial review and provenance control. The Consensus Framework was the winner of the UN Tourism Global Artificial Intelligence Challenge 2025. That recognition concerns the methodology; it is not a mathematical review or endorsement of these results. For Version 53, GPT Astra provided the scientific architecture and review. Claude Fable 5.1 implemented and published the candidate through Claude Code, with a separate Fable 5.1 instance performing the reproduction. Kimi 3 supplied the adversarial reading. Jucelha Carvalho coordinated the process and retained publication authority. Collective authorship and contributions Authorship is cumulative and human–AI. Contributing models are identified as AI models; their developer companies are not listed as authors, participants or affiliations. Jucelha Carvalho — Smart Tour Brasil: human author and coordinator; scope decisions, provenance, custody and release authorization. GPT-5.6 “Sol” — AI model: theorem architecture, formal specifications, mathematical review and audit methodology, including the scientific architecture of Stone 52. GPT Astra — AI model: scientific architecture and review of Stone 53; source, evidence and Git-object review; documentary consolidation and release preparation review. Claude Fable 5 — AI model: Lean implementation and debugging through Stone 50 and initial Stone 51 iterations. Claude Fable 5.1 — AI model: implementation of Stones 51–53, publication operations, documentary consolidation and separate-instance audits and reproduction. Claude Code is the execution environment. Kimi 3 — AI model: adversarial mathematical and code reviews, including the Stone 53 review and its errata. Manus AI 1.6 — AI model: DevOps, reproducibility and release reviews of earlier stages, and reproduction of gate 51-A. Codex v2 — AI model: custody, reproduction and reading audit of Stone 51. Claude Opus 4.5, Claude Opus 4.6 and Claude Opus 4.7 — AI models: formal-verification work, incomplete-proof reduction, historical-code recovery, forensic inventory and dependency mapping. Claude Opus 5 — AI model: technical review of the Stone 50 reproduction path; not a completed, artifact-verified reproduction. GPT-5.2 — AI model: early axiom reformulation, conditional-theorem design and strategic planning. Gemini 3 Pro — AI model: early conceptual exploration, drafts and historical architecture. Grok 4.5 — AI model: additional external audit of the Stone 49 chain and scope. Grok 4.6 — AI model: additional reported Linux reproduction of Stone 50, recorded as reported corroboration rather than artifact-verified reproduction. Lean 4 and GitHub Actions provide verification infrastructure and are not authors. Scientific coauthorship and legal copyright ownership are distinct records. Frozen snapshot and provenance This version corresponds to commit: 571b83aadb53838eb8257c015b7c02d3e57d30ad The scientific implementation was integrated through PR #28; documentation and audit consolidation were integrated through PR #29. The documentary consolidation preserves the scientific Phase 3 tree. Repository: https://github.com/consensusframework/yang-mills-mass-gap Version 53 DOI: 10.5281/zenodo.22750453 Previous version: 10.5281/zenodo.22738731 Concept DOI for the version series: 10.5281/zenodo.17397622 The frozen snapshot preserves its historical metadata, including the reference to Version 52 as the latest deposit at the time of that commit. This Version 53 record and its accompanying release notes identify the present deposit without rewriting the historical snapshot. Licensing Code and configuration are licensed under Apache License 2.0. Documentation and textual materials are licensed under Creative Commons Attribution 4.0 International, subject to the notices included in the repository. Earlier deposited versions retain their published licenses.