A Lean 4 Formalization Program Around the Yang–Mills Mass Gap — Version 58: Finite-Volume Lipschitz Stability Under Additive Multiregion Cover

TL;DR

This release advances a formal, machine-checked research program directed at the Yang–Mills existence and mass-gap problem, developed through human–AI collaboration under the Consensus Framework.

Version 58 — Finite-Volume Lipschitz Stability Under Additive Multiregion Cover

Version 58 broadens the coverage hypothesis of Version 57. Several regions may now share the budget for changing the effective activity factor of a polymer. The stability estimate retains the same constant, exponential decay rate and small-β regime. The advance is in the range of changes covered by the theorem; it is not a universal numerical improvement of the bound.

The release archives the repository at commit 52cc0915547a13c0b738b0b506d572d00b011a12, including the formal proofs, documentary consolidation, independent reproduction evidence, adversarial review with its corrective errata, and provenance records.

For non-specialists

Imagine calculating a quantity associated with one part of a finite network while changing mathematical weights in several other regions. Each region has a budget limiting how much it can change a contribution, and its effect on the bound is weighted by its separation from the region being studied.

In Version 57, each changed polymer needed at least one region capable of covering its entire effective-factor change. Version 58 allows the regions that the polymer touches to combine their budgets.

For example, a change of 1/2 may be covered by two budgets of 1/4, even though neither region can cover it alone. The formal development and the auditor’s tests demonstrate this distinction under the stated contact hypotheses. No concrete lattice simulation is claimed.

Here, polymers are mathematical objects used to organize the expansion of the lattice model. Their activity factors are mathematical weights. The result bounds the change in the normalized polymer functional; it does not describe a physical switch that turns interactions off.

What Version 58 proves

Let F_R(a) denote the normalized profile-damped polymer functional. R is the damping region, a and a′ are profiles taking values in [0,1], and D_s = card(supportLinkFinset s).

For a polymer η, its effective factor b(η) is a(η) if it touches R, and 1 otherwise. Define b′ from a′ in the same way. Raw profile values outside R do not change these effective factors.

Consider a finite family of change regions r_i. Each region has a walk-barrier separation scale n_i and a nonnegative budget δ_i. Define:

B = sum_i δ_i · exp(−n_i/2).

The additive coverage condition requires, for every polymer η:

|b(η) − b′(η)| ≤ sum of δ_i over the indices i for which η touches r_i.

Under this condition and the analytic hypotheses below, the formalized estimates are:

|F_R(a) − F_R(a′)| ≤ Cf · [exp(6D_s/113) + exp(4D_s/113)] · B.

|F_R(a) − F_R(a′)| ≤ (2Cf) · exp(6D_s/113) · B.

The hypotheses include the formalized group and probability-measure assumptions, 0 ≤ β ≤ 1/40000, a measurable real function χ with |χ(g)| ≤ 1, a measurable observable f with |f(U)| ≤ Cf, profile values in [0,1], and walk-barrier separation between s and each change region r_i. Nonnegativity of Cf is derived from the absolute majorant. Separation refers to walks in the plaquette graph, not Euclidean distance.

The regions may overlap or repeat. No disjointness, inclusion of r_i in R, separation of s from all of R, unit common background, or upper bound δ_i ≤ 1 is required. The theorem introduces no extra factor of the number of regions or the lattice volume outside B.

The public interface requires neither an assignment of each polymer to one region nor a supplied decomposition of its change. It also does not require DependsOnlyOn f s. Without that additional condition, s is a regional parameter of the functional and is not asserted to support f.

Proof structure and special cases

The proof constructs intermediate profiles using a truncated scalar move towards the target. Each region contributes capacity at its own step; a polymer may therefore change in several steps. The move stops at the target when the accumulated capacity covers the required change.

At every step, the proof applies the published two-term localized estimate of Version 56. Finite telescoping from Version 57 combines the estimates. Additive coverage ensures that the final effective factors equal those of the target profile, even when raw values outside R differ.

Nothing is divided by a budget, an accumulated capacity or a profile difference. Divisions by the fixed constants 2, 113 and 40000 remain in the formulas and regime.

• The Version 57 coverage condition implies additive coverage, recovering both Version 57 estimates by specialization.

• One region recovers the Version 56 bound through a compatibility interface. Version 56 remains an input to the construction.

• An empty family and zero budgets yield exact equality under additive coverage. The corresponding algebraic interfaces allow arbitrary real profiles.

• Budgets greater than one are allowed; the analytic estimates still require profile values in [0,1].

• Overlapping regions, repeated regions, zero factors and a common non-unit background are admitted.

The Version 57 capstone is not used to prove the additive theorem. No optimal choice of budgets or ordering of the steps is established. Uniformity of the bound over admissible common backgrounds does not mean that the functional is independent of the background or equal to the original Gibbs expectation.

Formal verification and review

The snapshot contains:

• 119 Lean modules;

• 39,181 Lean source lines;

• 1,834 top-level declaration lines, counted by the published textual method;

• 28 new theorems and 4 new definitions in the Version 58 module;

• 317 in-file #print axioms commands, including 26 in the new module.

The declaration-line count includes definitions and other declaration types; it is not a theorem count. The certificate-command count is not a global inventory of every declaration’s axioms.

The new module’s 26 in-file certificates cover 22 theorems and its four definitions. The separate QA1 reproduction checked all 32 declarations, including the six auxiliary theorems without an in-file certificate command. These checks reported exactly propext, Classical.choice and Quot.sound, with no sorryAx.

A separate Claude Fable 5.1 instance rebuilt all 119 project modules, with zero replayed project modules and no errors, using dependencies at the nine revisions of the versioned manifest. It performed its own tests and subsequently re-executed the constructor’s tests. The 114 inherited project warnings matched the baseline; the new module had none. The verdict was PASS NO ESCOPO (PASS within scope). This used a separate instance and environment of the same model as the constructor; prior exposure was declared and the audit was not blind.

Kimi (Moonshot AI), as identified in the review documents, performed an adversarial review by reading and mathematical analysis with custody checks, without executing Lean. Its PASS NO ESCOPO verdict was maintained after its own errata C1. The report and errata must be read together. Numerical sampling mentioned in the original report has no preserved reproducible record and is not used as evidence. No complementary Manus execution is documented for Version 58.

The audit index preserves the distinctions between the constructor’s tests, the auditor’s tests and the coordination’s reporting clarifications. In particular, the auditor’s Q2 supplies global additive coverage for its parametric profiles, and the expected negative test fails at a demonstrably impossible coverage obligation. This is a failure to apply the theorem outside its hypotheses, not a counterexample to its conclusion.

Scientific integration and documentary consolidation proceeded through pull requests #42 and #43. CI runs 511–514 succeeded. Each reports 119 project modules built and zero replayed, with 320 standard axiom outputs covering 318 distinct names: 317 in-file commands plus three workflow checks, two of which repeat names. Run 514 checks the frozen commit itself.

Lean 4.15.0 and Mathlib are pinned, and dependencies are resolved from the versioned manifest. Cached dependency artifacts are distinguished from a full dependency rebuild. Audit packages, reports, matrices, clarifications and integrity manifests are included in the snapshot. No specialized human mathematical review of the proofs is documented for this stage.

Scientific scope

The lattice-gauge results concern finite periodic four-dimensional lattices in the stated small-β, strong-coupling regime. Supporting results include general analytic and combinatorial lemmas.

Version 58 establishes finite-volume stability under additive coverage of effective polymer-activity changes. The thermodynamic and continuum limits, and the construction of a Yang–Mills theory with a mass gap, remain open. This release does not claim a solution or partial solution of the Millennium Problem.

No new Gibbs measure, boundary condition or modified action is constructed. No monotonicity or differentiability of the functional, spatial-mixing theorem, optimal constant, optimal budget allocation or bibliographic priority is claimed.

Human–AI collaboration and contributions

Jucelha Carvalho, founder of Smart Tour Brasil, coordinates the project under the Consensus Framework. Human coordination includes defining objectives, transporting instructions and evidence between instances, preserving provenance, making integration decisions and managing publication. Separate AI instances perform architecture, implementation, reproduction and adversarial review, with the scope of each contribution recorded.

The Consensus Framework methodology received recognition in the UN Tourism Global Artificial Intelligence Challenge 2025. That recognition concerns the methodology, not an assessment or endorsement of these mathematical results.

Direct contributions to Version 58:

• Jucelha Carvalho — Smart Tour Brasil: human coordination, provenance, decisions, DOI reservation and manual publication.

• GPT Astra / Can: scientific architecture and review; documentary and packaging review; checks of Git objects, evidence and logs. No Lean execution is attributed to this role.

• Claude Fable 5.1, in Claude Code: feasibility, construction, integration, documentary consolidation and packaging.

• A separate Claude Fable 5.1 instance: QA1 reproduction and tests in Lean.

• Kimi (Moonshot AI): adversarial mathematical and code review by reading, without Lean execution, with errata C1 to its own report.

Cumulative project coauthors, identified as AI models where applicable: Jucelha Carvalho; GPT-5.6 “Sol”; GPT Astra; Claude Fable 5; Claude Fable 5.1; Kimi 3; Manus AI 1.6; Codex v2; Claude Opus 4.5; Claude Opus 4.6; Claude Opus 4.7; Claude Opus 5; GPT-5.2; Gemini 3 Pro; Grok 4.5; and Grok 4.6.

Historical credits and their scopes remain documented in the repository. The historical Kimi designation does not establish the precise build used for the Version 58 review. Earlier Manus contributions are preserved in their original scope. Claude Code is an execution environment. Developer companies are not listed as authors or participating institutions.

Files and reproducibility

This deposit contains four files:

• yang-mills-mass-gap-zenodo-v58.zip — the repository snapshot at the frozen commit, containing 220 tracked files.

• RELEASE_NOTES_PEDRA58.md — scientific details, verification scope, credits, provenance and the Version 58 DOI.

• MANIFEST_v58.txt — inventory of snapshot paths, Git objects, modes, sizes and SHA-256 hashes.

• SHA256SUMS.txt — SHA-256 checksums of the other three assets.

The snapshot preserves the repository exactly as committed. Statements inside it about the absence of a Version 58 DOI or deposit describe the historical state at documentary consolidation. The external release notes record the subsequent DOI reservation and the status at packaging time.

Identifiers and licenses

Version 58 DOI: https://doi.org/10.5281/zenodo.23439473

Previous version: https://doi.org/10.5281/zenodo.23305142

Concept DOI, all versions: https://doi.org/10.5281/zenodo.17397622

Repository: https://github.com/consensusframework/yang-mills-mass-gap

Frozen commit: 52cc0915547a13c0b738b0b506d572d00b011a12

Code and configuration: Apache License 2.0.

Documentation and textual materials: Creative Commons Attribution 4.0 International.