From Conditional Formalization to an Axiom-Free Finite-Lattice Program: Reassessment and Continuation of a Multi-Phase Lean 4 Project Around the Yang-Mills Mass Gap

TL;DR: This project introduces a novel Multi-Agent AI Framework (integrating Claude, GPT, Gemini, Kimi, and Manus) to formally verify the finite-lattice base of the Yang-Mills Mass Gap problem using the Lean 4 theorem prover. It achieves 100% verified compilation with zero unproven axioms (no sorry).

👥 Autors

Carvalho, Jucelha — Smart Tour Brasil  (ORCID: 0009-0004-6047-2306)

Claude Fable 5 — Anthropic

GPT-5.6 "Sol" — OpenAI

Kimi 3- Moonshot AI

Claude Opus 4.7 — Anthropic

Claude Opus 4.6 — Anthropic

Claude Opus 4.5 — Anthropic

GPT-5.2 — OpenAI

Gemini 3 Pro — Google

Manus AI 1.6 — Manus

Description

Version 48 — Absolute Convergence of the Concrete Signed Rooted Ursell Series

This record archives Version 48 of a human-led, multi-model Lean 4 formalization program around the Yang–Mills mass gap. The project is exploratory formalization research and is not a proof, partial proof, or claimed solution of the Yang–Mills Existence and Mass Gap Millennium Problem.

Phase 3 (LatticeGauge) is an independently constructed finite-lattice gauge theory library developed without scientific axioms or sorry. At Version 48, it contains 63 source files and approximately 740 verified theorem/lemma declarations and supporting definitions, checked with Lean 4 and Mathlib v4.15.0.

Version 48 completes the passage from the finite Kotecký–Preiss bounds established in Version 47 to an infinite signed rooted Ursell series. The new formal layers establish:

• summability and a tsum bound for the nonnegative tree-majorant series;
• summability of the absolute rooted Ursell coefficients;
• the signed rooted coefficient kpSignedUrsellCoeff;
• the domination
  |Cₙ(z, γ₀)| ≤ Aₙ(|z|, γ₀);
• absolute summability of the signed rooted Ursell series;
• concrete specialization to the signed polymer activity polymerWeight.

The central verified result is:

For 0 ≤ β ≤ 1/40000, the concrete signed rooted Ursell series is absolutely convergent, with

Σₙ |Cₙ(w_{β,χ}, γ₀)| ≤ exp(card γ₀).

The frozen mathematical core was independently subjected to adversarial mathematical review by Kimi 3 (Moonshot AI). The release and reproducibility chain was independently reviewed by Manus AI 1.6, including an independent clone/build reproduction. GitHub Actions CI also verifies the frozen release commit.

Scope boundary: Version 48 does not prove any identification of this series with log Z, does not prove a cluster-expansion representation of the partition function, and does not establish realZ ≠ 0, a thermodynamic or continuum limit, exponential clustering, or a mass gap. Those remain later targets. Stone 49, beginning with the unrooting step, has not been started in this release.

Frozen source tag: zenodo-v48.

Version DOI: 10.5281/zenodo.21966286
Concept DOI: 10.5281/zenodo.17397622

Human-led, multi-model collaboration: coordinated by Jucelha Carvalho, with formalization architecture, implementation, review, adversarial checking, debugging, source reconnaissance, and project operations carried out collaboratively across GPT-5.6 “Sol” (OpenAI), Claude Fable 5 (Anthropic), Kimi 3 (Moonshot AI), Claude Opus 4.5/4.6/4.7 (Anthropic), GPT-5.2 (OpenAI), Gemini 3 Pro (Google), and Manus AI 1.6.

Independent external review: adversarial mathematical review by Kimi 3 (Moonshot AI) and release/reproducibility review by Manus AI 1.6, alongside GitHub Actions CI verification.

The repository explicitly distinguishes machine-checked finite-lattice results from assumptions, historical exploratory material, and open research targets.

• 💻  Repo:  https://github.com/consensusframework/yang-mills-mass-gap
• 📧  Contact:  jucelha@smarttourbrasil.com.br
• 🆔  ORCID:  https://orcid.org/0009-0004-6047-2306