[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"urOLpS2Y8F":3},"# lgt — Lattice Gauge Theory in Lean 4\n\n> **Starting here?** See [`docs/mass-gap-roadmap.md`](docs/mass-gap-roadmap.md)\n> for a short human-facing summary of the project's goal and architecture.\n>\n> **Per-file informal summaries** (auto-generated, English + LaTeX):\n> [`summary/`](summary/README.md) — declaration-by-declaration walk-through\n> of the core files, with proof dependencies cross-linked to source.\n\n**Status**: formalisation of the **d ≥ 2 lattice Yang–Mills mass gap\nat strong coupling**, via the Dobrushin uniqueness method\n(Chatterjee 2026, Ch. 16). Both headline theorems\n(`ym_mass_gap_exponential_decay`, `ym_mass_gap_rate_exists`) are\nfully proved as of PR #2 (2026-05-04) — zero sorries, zero project\naxioms.\n\n## Headline theorems\n\nBoth in `LGT/MassGap/StrongCoupling.lean`. For U(n) Wilson lattice\ngauge theory on (ℤ/Nℤ)^d with d ≥ 2, N ≥ 3, n ≥ 1, coupling\nβ \u003C 1/(32n(d−1)) (equivalently `β \u003C 1/(4n · maxNeighbors d)`):\n\n**1. Algebraic decay** (`ym_mass_gap_exponential_decay`):\n\n    |⟨Re Tr(U_p) · Re Tr(U_q)⟩_c|\n        ≤ 32 n² / (1 − α) · α^((latticePlaquetteDist p q − 1) / 2)\n\nwhere α = dobrushinAlpha(n, d, β) \u003C 1 and `latticePlaquetteDist` is\nthe periodic L¹ distance between plaquette anchor sites. The\nexponent uses `Nat` subtraction and division (saturating at 0 for\nclose-range plaquettes). The factor of `1/2` is forced by the\ngeometry: one shared-plaquette influence-graph step displaces a\nlink anchor by up to 2 L¹ site-units, so `α^k` decay in graph-step\ncount yields `(log α) / 2` rate in L¹ plaqDist.\n\n**2. Existential mass-gap rate** (`ym_mass_gap_rate_exists`,\ncompanion form, requires β > 0):\n\n    ∃ m > 0, |⟨Re Tr(U_p) · Re Tr(U_q)⟩_c|\n        ≤ 32 n² / (α (1 − α)) · exp(−m · latticePlaquetteDist p q)\n\nwith `m = (−log α) / 2`. This packages the algebraic bound as the\ncanonical exponential-decay statement of mass gap.\n\nThe theorems are stated for U(n); other compact gauge groups\nG ⊆ U(n) require supplying the `HasGaugeTrace` instance.\n\n## What's in the repository\n\n**Lean infrastructure (proved, axiom-clean):**\n\n* Wilson action, gauge invariance, the YM measure, the Gibbs\n  specification framework, DLR identity, Dobrushin-condition\n  verification, U(n) instances.\n* The distance-parameterised wrapper `ym_mass_gap_strong_coupling`\n  that takes ~10 hypotheses (continuity, `dobrushinAlpha \u003C 1`, a\n  caller-supplied distance with refl/triangle/nearest-neighbour\n  support) and discharges the ~28 hypotheses of\n  `ym_mass_gap_2pt_via_multisite` from first principles.\n* Periodic-torus distance machinery in\n  `LGT/Lattice/LatticeDistance.lean` (`ZMod.periodicDist`,\n  `latticeSiteDist`, `latticePlaquetteDist`, `linkAmbientAdj`,\n  `ambientLinkGraph`, `linkGraphDist`).\n* The geometric closure (this PR): `linkGraphDist_support` (Dobrushin\n  α^k decay carried through the shared-plaquette link graph),\n  `boundary_sum_bound` (16-term boundary-link sum bounded by\n  `16 · α^((plaqDist−1)/2) / (1−α)`), and the two headline theorems.\n\nSee [docs/mass-gap-roadmap.md](docs/mass-gap-roadmap.md) for a\ntwo-page human-facing summary,\n[docs/mass-gap-proof-outline.md](docs/mass-gap-proof-outline.md)\nfor the full proof outline, and\n[docs/codex-review.txt](docs/codex-review.txt) for independent\nreview records.\n\n## Axiom footprint\n\nVerified by `#print axioms` on a fresh build:\n\n```\n#print axioms ym_mass_gap_strong_coupling\n-- propext, Classical.choice, Quot.sound\n\n#print axioms ym_mass_gap_exponential_decay\n-- propext, Classical.choice, Quot.sound\n\n#print axioms ym_mass_gap_rate_exists\n-- propext, Classical.choice, Quot.sound\n```\n\nOnly Lean foundationals — no project axioms, no Mathlib analytic\naxioms beyond standard. The wider repository (Wilson action, YM\nmeasure, Gibbs spec, DLR, Dobrushin verification) is also fully\nsorry-free and axiom-clean as of PR #2.\n\n## Proof architecture\n\nEight layers (1–7 are the main Dobrushin path; 8 is an independent\nbridge to tensor-network models):\n\n1. **Lattice geometry** (`Lattice/CellComplex.lean`) — sites, links,\n   plaquettes on (ℤ/Nℤ)^d with boundary links and shift operations.\n\n2. **Gauge fields** (`GaugeField/`) — G-valued connections on links,\n   plaquette holonomy, gauge covariance, U(n) instantiation with\n   trace bounds |Re Tr(U)| ≤ n.\n\n3. **Wilson action** (`WilsonAction/`) — plaquette cost, Wilson action\n   S(U) = β Σ(n − Re Tr U_p), Boltzmann weight, gauge invariance.\n\n4. **YM measure** (`MassGap/YMMeasure.lean`) — product Haar measure,\n   YM probability measure via withDensity, partition function Z > 0,\n   ymExpect bridge to integrals.\n\n5. **Gibbs specification** (`Gibbs/`) — YM as a GibbsSpec on link\n   lattice, DLR identity (`ymMeasure_isGibbs`), Dobrushin condition\n   at strong coupling (`ymDobrushinCondition`).\n\n6. **Dobrushin correlation decay** (in `markov-semigroups`) — canonical\n   maximal coupling with Giry measurability, Dobrushin coupling via\n   minimum-disagreement + Prokhorov compactness, single-site\n   disintegration, multi-site covariance bounds via condKernel\n   disintegration, Neumann series exponential decay.\n\n7. **Mass gap assembly** (`MassGap/StrongCoupling.lean`) — discharge\n   integrability/measurability from continuity, influence bounds via\n   action splitting, link distance, U(n) specialization.\n\n8. **EKR-Dobrushin bridge** (`Bridge/`) — connects tensor\n   renormalization group (TRG) convergence to Dobrushin mass gap.\n   For any nearest-neighbor spin model with an EKR-style certificate\n   (hat-tensor contraction λ \u003C 1), derives exponential correlation\n   decay on the original lattice. Zero sorries, zero axioms.\n   Applicable to Ising, XY, O(3) NLSM, Potts, or any model with a\n   tensor RG certificate. See `docs/mass-gap-blueprint.md` for details.\n\nAlso: 2D mass gap via Doeblin (`MassGap2D.lean`, partially complete).\n\n## File structure\n\n```\nLGT/\n  Lattice/CellComplex.lean           -- cell complex on (ℤ/Nℤ)^d\n  GaugeField/\n    Connection.lean                   -- connections, holonomy, gauge covariance\n    GaugeGroup.lean                   -- HasGaugeTrace typeclass\n    UnitaryGroup.lean                 -- U(n) instantiation + trace bounds\n  WilsonAction/\n    PlaquetteAction.lean              -- Wilson action, Boltzmann weight\n    GaugeInvariance.lean              -- S(g·U) = S(U)\n  Gibbs/\n    YMSpec.lean                       -- YM as GibbsSpec\n    YMDobrushin.lean                  -- Dobrushin condition verification\n    YMIsGibbs.lean                    -- DLR identity (ymMeasure is Gibbs)\n  MassGap/\n    YMMeasure.lean                    -- YM probability measure\n    DobrushinVerification.lean        -- influence bound algebra\n    GaugeFixing.lean                  -- correlation bound wiring\n    MassGap3D.lean                    -- d≥3 mass gap theorem\n    StrongCoupling.lean               -- final assembly + U(n)\n    MassGap2D.lean                    -- 2D mass gap (Doeblin path)\n  Bridge/\n    TensorGibbsSpec.lean              -- Gibbs spec from tensor network\n    TensorDobrushin.lean              -- influence bounds + Dobrushin condition\n    ScaleTransfer.lean                -- RG scale transfer (coarse → fine)\n    O3MassGap.lean                    -- assembly for O(3) / general models\ndocs/\n  mass-gap-roadmap.md                 -- short human-facing summary (START HERE)\n  mass-gap-completion-plan.md         -- detailed record of the geometric closure\n  mass-gap-proof-outline.md           -- proof outline\n  mass-gap-blueprint.md               -- full math blueprint\n  mass-gap-blueprint.tex              -- LaTeX version\n  mass-gap-proof.md                   -- 2D Doeblin path notes\n  codex-review*.txt                   -- historical review record\n```\n\n## Dependencies\n\n- **Lean 4**: `leanprover/lean4:v4.29.0` (pinned in `lean-toolchain`;\n  installed automatically by `elan` when you enter the directory).\n- **[Mathlib](https://github.com/leanprover-community/mathlib4)** v4.29.0\n- **[markov-semigroups](https://github.com/mrdouglasny/markov-semigroups)** —\n  Dobrushin uniqueness theory, canonical maximal coupling, covariance\n  bounds, Neumann series, single-site disintegration, condKernel wiring.\n- **[gaussian-field](https://github.com/mrdouglasny/gaussian-field)** —\n  lattice site types.\n\nAll resolved via `lakefile.toml`; `lake-manifest.json` pins the\nexact commits.\n\n## Building\n\n```bash\ngit clone https://github.com/mrdouglasny/lgt.git\ncd lgt\nlake build          # fetches deps on first run via the pinned manifest\n```\n\nFull build takes ~15–30 min on first run (Mathlib cache download +\nlocal compilation). Incremental rebuilds are seconds.\n\nTo verify the main result builds and the axiom footprint is clean:\n\n```bash\nlake build LGT.MassGap.StrongCoupling\n```\n\n## Contributing\n\n**New collaborator?** Read in this order:\n\n1. **[`docs/mass-gap-roadmap.md`](docs/mass-gap-roadmap.md)** —\n   two-page human summary: goal, status, approach, and what was built.\n2. **[`docs/mass-gap-completion-plan.md`](docs/mass-gap-completion-plan.md)** —\n   the detailed, phase-by-phase record of the geometric closure, with\n   concrete lemma signatures.\n3. **[`docs/mass-gap-proof-outline.md`](docs/mass-gap-proof-outline.md)** —\n   deeper math context for the proofs.\n\n**The strong-coupling mass gap is complete** (zero sorries, zero\nproject axioms). Natural directions to extend: the 2D Doeblin path\n(`MassGap2D.lean`, partially complete) and the RG scale-transfer\nbridge (`Bridge/`, several fields still stated as hypotheses).\n\n**Conventions**: this project follows Mathlib-style naming and layout\n(see `CLAUDE.md` in the root for the working-methods summary; detailed\nrules in `~/Documents/GitHub/mathlib-ready/docs/` for those with\naccess). Before writing a new lemma, check whether it exists upstream\nor in sibling projects — the `markov-semigroups` and `gaussian-field`\nlibraries cover most of the Dobrushin / lattice-geometry machinery\nalready. Axioms are budgeted: see `research-dev/library/lean/AXIOM_MANAGEMENT.md`\nfor the vetting protocol if a new one is truly needed.\n\n**How to propose a change**: fork, branch, PR to `main`. For anything\nnon-trivial, open an issue first to coordinate. Commit messages:\nconcise, describe the *why*.\n\n## References\n\n- S. Chatterjee, *Gauge Theory Lecture Notes* (2026), Ch 15–16\n- R. L. Dobrushin, \"Description of a random field by means of\n  conditional probabilities\" (1968)\n- H.-O. Georgii, *Gibbs Measures and Phase Transitions* (1988), §8\n- K. G. Wilson, \"Confinement of quarks,\" Phys. Rev. D 10 (1974) 2445\n\n## Authors\n\nMichael R. Douglas (CMSA, Harvard), with Fred Rajasekaran.\n\n## Independent verification (Lean FRO comparator)\n\nBoth headline theorems (`ym_mass_gap_exponential_decay` and\n`ym_mass_gap_rate_exists`) have been checked at commit\n[`ec975bd`](https://github.com/mrdouglasny/lgt/commit/ec975bd) by the\n[Lean FRO comparator](https://github.com/leanprover/comparator) — an\nindependent judge that re-exports the kernel terms via `lean4export`,\nverifies the solution's elaborated theorem statement matches the\nchallenge's, and replays the proof through Lean's kernel under a\npermitted-axiom whitelist. The run reported:\n\n```\nLean default kernel accepts the solution\nYour solution is okay!\n```\n\nwith the whitelist `{propext, Quot.sound, Classical.choice}`. The macOS\nrun used `comparator/scripts/fake-landrun.sh` (the official insecure\nshim), so the kernel-acceptance and axiom-whitelist guarantees hold but\nthe sandbox guarantee does not — for adversarial use one would re-run\non Linux with real `landrun`.\n\n## License\n\nCopyright (c) 2026 Michael R. Douglas. Released under the Apache 2.0 license.\n",1786349530052]