[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"U1ZyDHtlsn":3},"\u003Cp align=\"center\">\u003Cimg src=\"assets/banner.svg\" alt=\"The Yang–Mills Mass Gap from One Bound at One Scale\" width=\"100%\">\u003C/p>\n\n\u003Cp align=\"center\">\n  \u003Ca href=\"https://github.com/anaxia-research/4Y-M/blob/paper/papers/4Y-MPaper.pdf\">\u003Cimg src=\"https://img.shields.io/badge/paper-PDF-f472b6?style=for-the-badge\" alt=\"paper\">\u003C/a>\n  \u003Cimg src=\"https://img.shields.io/badge/Lean_4-formal_proofs-7c5cff?style=for-the-badge\" alt=\"Lean 4\">\n  \u003Cimg src=\"https://img.shields.io/badge/Julia-273_checks-34d399?style=for-the-badge\" alt=\"Julia\">\n  \u003Cimg src=\"https://img.shields.io/badge/license-MIT-f59e0b?style=for-the-badge\" alt=\"MIT\">\n\u003C/p>\n\n**Deep Bhattacharjee** (corresponding author, head of the project), Ushashi Bhattacharya, James Montgomery Lockwood, Tristan Romanov, Ikailo John Sessford, Sanjeevan Singha Roy.\n\n**15 November progress:** [error-stable ascent, continuum-interface audit, and precise remaining estimate](research/2026-11-15/README.md). Six new Lean declarations extend the ascent to quantified approximation errors. Written interface lemmas accommodate the external construction's sparse mesh sequence and phase decomposition. The unconditional weak-coupling mass gap remains open.\n\n| Proved | Open |\n| :--- | :--- |\n| Mass gap of 4D lattice Yang–Mills theory at strong coupling, every compact simple group, explicit bounds | The one-scale bound at weak coupling |\n| **Theorem F**: a compatible Wightman limit on tori and one bound on partition functions at one scale give the continuum mass gap and a unique vacuum | A complete checked adapter to the selected construction, and the bound at its actual couplings |\n| The one bound is a lattice gap plus a Stefan–Boltzmann bound on the thermal free energy, and it forces a vanishing centre-vortex free energy | The lattice gap at weak coupling |\n| The gap alone does not give the count of states; the centre vortex controls only the charged states | The glueball gap and the count of states at weak coupling |\n| The unrenormalised loop variables have a trivial limit | The Clay problem |\n\n| Folder | Contents | Run |\n| :--- | :--- | :--- |\n| [`YangMills/`](YangMills) | Lean 4 project over Mathlib | `lake exe cache get && lake build` |\n| [`scripts/`](scripts) | axiom audit | `lake env lean scripts/Axioms.lean` |\n| [`julia/`](julia) | constants of the paper | `julia julia/constants.jl` |\n| [`research/2026-11-15/`](research/2026-11-15) | proof progress and explicit remaining assumptions | `python scripts/check_closure_progress.py` |\n\nThe [external construction assessment](https://github.com/yang-mills-group/YM) reports a Wilson scaling-limit Wightman theory and explicitly leaves the mass gap unproved. Our source audit addresses particular compatibility conditions; it does not claim to have rebuilt that entire construction.\n",1794771491584]