[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"OVFQMuIuJN":3},"# markov-semigroups\n\nMarkov semigroups, functional inequalities, and convergence to\nequilibrium in Lean 4. Built on the\n[hille-yosida](https://github.com/mrdouglasny/hille-yosida) C₀-semigroup\nframework. The downstream\n[gaussian-hilbert](https://github.com/mrdouglasny/gaussian-hilbert) library\ncombines this repo with\n[gaussian-field](https://github.com/mrdouglasny/gaussian-field) for\nfinite-dim Wiener chaos / polynomial-chaos concentration work.\n\n## What this project proves\n\nThe central result chain:\n\n1. **Abstract functional inequalities** (Poincaré, log-Sobolev,\n   Holley-Stroock perturbation) for general Dirichlet forms and Markov\n   semigroups — no geometry required.\n\n2. **Bakry-Émery curvature criterion** for diffusion generators on\n   Riemannian manifolds: $\\Gamma_2 \\geq \\rho \\Gamma$ implies LSI($\\rho$).\n\n3. **Brascamp-Lieb inequality** for log-concave measures mu = e^{-V} dx\n   with V strictly convex: Var_mu(f) \u003C= integral of (nabla f, (Hess V)^{-1} nabla f) dmu.\n   Proven from two analytical axioms (resolvent-IBP + integrated Bochner identity)\n   via weighted Young's inequality. Poincare corollary when Hess V >= rho I.\n\n4. **Ornstein-Uhlenbeck semigroup** defined abstractly for any Hilbert\n   space with a positive covariance operator. Mehler's formula, Gaussian\n   invariant measure, hypercontractivity.\n\n5. **Concrete instances:** R^n (standard Gaussian, Gross's theorem) and\n   T^d (torus GFF, Fourier mode decomposition). Identification of the\n   OU invariant measure on T^d with the Gaussian free field from\n   gaussian-field.\n\n6. **Convergence to equilibrium:** spectral gap → exponential mixing,\n   LSI → entropy decay, ergodicity from spectral gap.\n\n7. **TV coupling characterization:** total variation distance equals\n   the infimum over couplings of the disagreement probability.\n   Maximal coupling construction.\n\n8. **Dobrushin uniqueness theory:** for lattice spin systems on compact\n   state spaces, Dobrushin's condition (column sums of influence matrix\n   \u003C 1) implies unique Gibbs measure with exponential correlation decay.\n   Includes canonical maximal coupling, iterated Dobrushin coupling via\n   Prokhorov compactness, multi-site covariance bounds via condKernel\n   disintegration, and Neumann series exponential decay. Used by\n   [lgt](https://github.com/mrdouglasny/lgt) for the Yang-Mills mass gap.\n\n9. **Diamagnetic inequality** for finite matrices: |(M+iV)^{-1}(x,y)|\n   ≤ M^{-1}(x,y) via heat kernel positivity for Z-matrices.\n\n10. **Dobrushin-Zegarlinski for continuous spins:** uniform local LSI\n    + weak gradient coupling (Zegarlinski condition with `J/c ≤ α \u003C 1`)\n    implies a global, volume-independent log-Sobolev inequality with\n    constant `c·(1-α)`. Plus the entrywise covariance corollary\n    (Helffer-Sjöstrand): `|Cov(σ_x, σ_y)| ≤ α^{d(x,y)} / (c·(1-α))` for\n    nearest-neighbor finite-range potentials. Built on the proved\n    AbstractInfluenceMatrix Neumann theory and proved single-site\n    entropy decomposition; only the textbook bridges (Otto-Reznikoff\n    LSI, Helffer-Sjöstrand) are axiomatized. For pphi2N's strict\n    thermodynamic-limit mass-gap route.\n\n11. **Borell-Herbst concentration from LSI:** for any LSI measure with\n    constant `c > 0`, every L-Lipschitz function is sub-Gaussian:\n    `μ({F − E_μ F > t}) ≤ exp(−c·t²/(2L²))`. Proven via the Herbst MGF\n    bound (textbook axiom) + Mathlib's Chernoff. Composes with the\n    Zegarlinski global LSI to give Lipschitz-observable concentration\n    on Gibbs measures. One/two-sided variants, `MemLp` moment bounds,\n    and a Mathlib `HasSubgaussianMGF` interop are all proven.\n\n12. **LSI ⇒ Poincaré (variance bound):** standard textbook implication\n    that LSI(c) implies the Poincaré inequality with the same constant:\n    `Var_μ(f) ≤ (1/c) ∫ ‖∇f‖² dμ`. For L-Lipschitz F, gives\n    `Var_μ(F) ≤ L²/c`. Specialized to the Zegarlinski global LSI:\n    `Var_μ_Gibbs(F) ≤ L²/(c·(1−α))`. Closes the LSI → concentration\n    → variance / spectral-gap chain.\n\n**Multivariate Hermite / Wiener chaos / OU eigenfunctions /\npolynomial-chaos concentration** (Janson Theorem 5.10) live in\n[gaussian-hilbert](https://github.com/mrdouglasny/gaussian-hilbert)\nsince 2026-05-10. They were originally drafted here in a `Gaussian/`\nsubdirectory, but the content is \"facts about the standard Gaussian\nmeasure\" rather than \"facts about abstract Markov semigroups\", and\ngaussian-hilbert (which sits on top of gaussian-field +\nmarkov-semigroups + Mathlib) is the natural home. Removing the cluster\nalso cleanly eliminated markov-semigroups' cross-repo dependency on\ngaussian-field.\n\n## Architecture: three layers of abstraction\n\nThe project is organized into three layers, each requiring progressively\nmore structure. Most results live in the most general layer possible.\n\n### Layer 1: Abstract/ — Measure space + energy form\n\n**Assumes:** A probability space (X, mu) and a symmetric bilinear form\nE(f,g) (the Dirichlet form / energy). No gradient, no metric, no\nmanifold.\n\n**Typeclass:**\n```\nclass DirichletSpace (X : Type*) where\n  mu : Measure X                          -- reference probability measure\n  energy : (X -> R) -> (X -> R) -> R      -- Dirichlet form E(f,g)\n  -- symmetric, positive, closed, Markov property\n```\n\n**What lives here:** Poincare inequality, log-Sobolev inequality,\nHolley-Stroock perturbation, Gross equivalence (LSI \u003C-> hypercontractivity),\nspectral gap, entropy decay, ergodicity. These are all statements about\nE and mu with no geometry.\n\n### Layer 2: Diffusion/ — Abstract diffusion generator\n\n**Assumes:** A Dirichlet space with a *carre du champ* operator\nGamma(f,g), which plays the role of |nabla f|^2 without requiring an\nactual gradient or Riemannian metric.\n\n**Typeclass:**\n```\nclass MarkovDiffusion (X : Type*) extends DirichletSpace X where\n  Gamma : (X -> R) -> (X -> R) -> (X -> R)    -- carre du champ\n  -- E(f,g) = integral Gamma(f,g) d mu\n```\n\nOn a Riemannian manifold, Gamma(f,g) = \u003Cnabla f, nabla g>. But the\nabstract definition works for any diffusion generator. The iterated\ncarre du champ Gamma_2 and the Bakry-Emery curvature condition\nGamma_2 >= rho * Gamma are defined at this level. The Bochner-Weitzenbock\nformula (connecting Gamma_2 to Ricci curvature) is a *theorem* for\nspecific instances, not part of the axioms.\n\n**What lives here:** Bakry-Emery criterion, carre du champ, iterated\ncarre du champ, abstract Ornstein-Uhlenbeck semigroup, invariant measures.\n\n### Layer 3: Instances/ — Concrete spaces\n\n**Assumes:** A specific space with explicit structure.\n\n- **Euclidean.lean** (R^n): Gamma(f,g) = \u003Cnabla f, nabla g> via Mathlib's\n  `fderiv`. Standard Gaussian, Mehler formula, Bakry-Emery curvature = 1.\n- **Torus.lean** (T^d): Fourier eigenvalues lambda_k = 4 pi^2 |k|^2 / L^2 + m^2.\n  Heat semigroup as Fourier multiplier. Bakry-Emery curvature = m^2.\n  No dependence on Mathlib's manifold library — the torus is handled\n  entirely via its Fourier decomposition.\n- **GFFIdentification.lean**: OU invariant measure on T^d equals the\n  Gaussian free field from gaussian-field (by covariance uniqueness).\n\n### Why no Riemannian manifold library?\n\nThe Bakry-Emery framework is designed to avoid needing the full\nRiemannian apparatus. The carre du champ Gamma is an *abstract*\nbilinear operation satisfying axioms, not necessarily \u003Cnabla f, nabla g>\non a smooth manifold. For R^n we use Mathlib's `fderiv`; for T^d we use\nFourier modes. If someone later wants to add a general Riemannian\ninstance (using Mathlib's `SmoothManifold`), the abstract layers are\nready for it.\n\n## File structure\n\n```\nMarkovSemigroups/\n  Abstract/                     -- Layer 1: measures + Dirichlet forms\n    DirichletForm.lean          -- Symmetric Dirichlet form, variance, entropy\n    Poincare.lean               -- Spectral gap \u003C-> variance decay\n    LogSobolev.lean             -- Gross LSI, entropy decay\n    HolleyStroock.lean          -- Bounded density perturbation of LSI\n    Hypercontractivity.lean     -- DirichletMarkovSemigroup; LSI \u003C-> hypercontractivity (Gross + Stroock-Varopoulos axioms + the 3 hypothesis predicates)\n    GrossODE.lean               -- Gross-ODE proof of LSI=>HC (FULLY PROVED, sorry-free: gross_lsi_implies_hypercontractive_of_hypotheses, via the 4 per-instance predicates incl. CoreLpL2Approx)\n    Concentration.lean          -- Borell-Herbst sub-Gaussian concentration from LSI\n  Diffusion/                    -- Layer 2: abstract diffusions (Gamma, Gamma_2)\n    CarreDuChamp.lean           -- BakryEmerySpace: Gamma, semigroup, curvature\n    StroockVaropoulos.lean      -- RpowChainRule + stroockVaropoulos_eq (S-V equality for diffusion forms)\n    BakryEmery.lean             -- Bakry-Emery curvature criterion\n    L2Semigroup.lean            -- Bridge to hille-yosida semigroup theory\n    InvariantMeasure.lean       -- Invariant probability measures\n    OrnsteinUhlenbeck.lean      -- Abstract OU semigroup on Hilbert spaces\n  -- (Wiener chaos / multivariate Hermite / OU eigenfunctions /\n  -- polynomial-chaos concentration moved to gaussian-hilbert as\n  -- GaussianHilbert.* on 2026-05-10.)\n  Instances/                    -- Layer 3: concrete spaces (sorry-free)\n    Torus.lean                  -- T^d: heat semigroup, Fourier modes (header)\n    GFFIdentification.lean      -- OU invariant on T^d = GFF (header)\n    BrascampLieb.lean           -- Brascamp-Lieb for log-concave measures\n    WorkInProgress/             -- Layer 3, in progress (honest sorries)\n      Euclidean.lean            --   R: standard Gaussian, OU semigroup (2 sorry = Lean gaps; axiom-free, Mehler-kernel axioms discharged)\n      TwoPoint.lean             --   {0,1}: not a diffusion — gamma_leibniz_fails PROVES Γ_leibniz fails (0 sorry, 0 axiom)\n      EuclideanHypercontractive.lean -- R^n Gaussian: discharges the 4 Gross hypotheses + assembles stdGaussianFin_isHypercontractive (no gross_lsi_implies_hypercontractive axiom)\n  Convergence/                  -- Consequences (uses Layer 1 only)\n    SpectralGap.lean            -- Exponential mixing from gap\n    RelativeEntropy.lean        -- Entropy decay under semigroup\n    Ergodicity.lean             -- Uniqueness of invariant measure\n    IntegralBounds.lean         -- TV-integral bound (layer cake)\n    Doeblin.lean                -- Doeblin's condition, n-step mixing\n  Coupling/                     -- Coupling theory\n    TVCoupling.lean             -- TV = inf coupling disagreement, maximal coupling\n    CanonicalCoupling.lean      -- Constructive pointwise-min coupling, Giry measurability\n    DobrushinCoupling.lean      -- Iterated Dobrushin coupling via min-disagreement\n    ProkhorovCoupling.lean      -- Prokhorov coupling for compact spin spaces (0 sorry)\n  Dobrushin/                    -- Dobrushin uniqueness for lattice spin systems\n    Specification.lean          -- Gibbs specifications, conditional distributions\n    Uniqueness.lean             -- Uniqueness + exponential correlation decay\n    StrongCoupling.lean         -- Strong-coupling verification of Dobrushin condition\n    CovarianceBound.lean        -- Single-site covariance bounds\n    CondTVBridge.lean           -- Conditional TV bridge, single-site disintegration\n    CovarianceBoundMultisite.lean -- Multi-site covariance bounds via condKernel\n    CondKernelDLR.lean          -- condKernel inherits DLR, ae bound\n    NeumannSeries.lean          -- Neumann series for influence matrix\n    FiniteLattice.lean          -- Finite lattice distance structure\n  DobrushinZegarlinski/         -- Zegarlinski's theorem for continuous spins\n    AbstractInfluence.lean      -- Abstract Neumann decay (pure linear algebra, 0 axioms)\n    EuclideanTransport.lean     -- GibbsSpec.toEuclideanMeasure adapter (PiLp 2 transport)\n    InteractionMatrix.lean      -- Gradient interaction J_{xy} = sup |∂_y ∂_x V|\n    LocalLSI.lean               -- Thin SatisfiesLSI predicate, UniformLocalLSI class\n    EntropyChainRule.lean       -- Bochner DLR identity (proved), single-site decomposition (proved)\n    GlobalLSI.lean              -- ZegarlinskiCondition, global_lsi_of_zegarlinski (1 axiom: zegarlinski_lsi_inequality)\n    EntrywiseCovariance.lean    -- Helffer-Sjöstrand entrywise Cov bound (1 axiom: cov_entrywise_bound_of_zegarlinski)\n    Concentration.lean          -- Lipschitz concentration corollaries from global LSI (composes Borell-Herbst; 0 new axioms)\n  Matrix/                       -- Finite matrix semigroup theory\n    HeatKernel.lean             -- exp(-tM) >= 0 for Z-matrices (proved)\n    LaplaceTransform.lean       -- M^{-1} = integral exp(-tM) dt\n    Trotter.lean                -- Lie-Trotter product formula\n    Diamagnetic.lean            -- |(M+iV)^{-1}| \u003C= M^{-1} entrywise\n  Tools/                        -- Shared utilities\n    SingleSiteDisintegration.lean -- Single-site disintegration of measures on SpinConfig\n```\n\n## Formalization status\n\n**Zero sorry's in the main tree** (Abstract/, Diffusion/, Convergence/,\nCoupling/, Dobrushin/, DobrushinZegarlinski/, Matrix/, and the three\nsorry-free instances in `Instances/`: BrascampLieb, Torus,\nGFFIdentification). 2 sorries remain, both in\n`Instances/WorkInProgress/TwoPoint.lean` and mathematically false for\njump processes (validates that the diffusion axiom is a real\nconstraint). The Gaussian1D Bakry-Émery instance\n(`Instances/WorkInProgress/Euclidean.lean` and supporting\n`EuclideanStein.lean`, `EuclideanHermite.lean`) holds 0 sorries and\n**1 remaining BGL Ch. 2 textbook axiom** (`ouSemigroup_entropy_sq_decay_bound`,\nthe de Bruijn entropy-derivative identity; GR-vetted via Gemini).\nEight of the originally nine were reduced to theorems: the five early\ndischarges (`ouSemigroup_l2_decay_bound`, `ouSemigroup_ergodic`,\n`ouSemigroup_entropy_sq_ergodic`, `ouSemigroup_compose`,\n`gaussian2D_orthogonal_invariance`), plus the 2026-05-12 discharges of\n`ouSemigroup_gradient_decay`, `ouSemigroup_l2_sq_hasDerivWithinAt`,\n`ouSemigroup_preserves_IsCore`, and `ouSemigroup_contDiff` (Path C\nHermite IBP). These WIP instances are imported by the top-level module\nso that callers of `#print axioms` on any theorem that transitively\nuses them will see the axiom/sorry surface honestly.\n\n### Fully proved (zero sorry's)\n\n- **Bakry-Émery Poincaré** — Var(f) ≤ (1/ρ) E(f,f) from\n  semigroup L² decay bound + ergodicity\n- **Bakry-Émery variance decay** — Var(P_t f) ≤ e^{-2ρt} Var(f)\n  from Poincaré + Grönwall's inequality\n- **Bakry-Émery log-Sobolev** — Ent(f²) ≤ (2/ρ) E(f,f) from\n  entropy decay bound + entropy ergodicity\n- **Brascamp-Lieb inequality** and Poincaré corollary\n- **Doeblin's condition** — one-step contraction, TV contraction,\n  n-step mixing (by induction), correlation decay\n- **TV-integral bound** via layer cake formula\n- **TV coupling characterization** — maximal coupling construction\n- **Heat kernel positivity** for Z-matrices (Metzler shift)\n- **Hessian invertibility** — injectivity, surjectivity, continuity\n- **Weighted Young's inequality** from Hessian symmetry\n- **Variance nonnegativity** via Mathlib's `ProbabilityTheory.variance_nonneg`\n\n### Postulated as textbook axioms (2 core + 1 Stroock-Varopoulos + 1 matrix + 2 DZ + 2 concentration/Poincaré + 0 GaussianFin = 8 total)\n\n*See [`AXIOM_AUDIT.md`](AXIOM_AUDIT.md) for the per-axiom vetting\nverdicts (Standard / Likely correct / Placeholder / etc.), review\nprovenance (deep-think, Gemini chat, literature page-ref, self-audit),\nand links to the discharge plans.*\n\n\n| Axiom | Reference |\n|-------|-----------|\n| `gross_lsi_implies_hypercontractive` | Gross (1975), Theorem 1 (bundled `DirichletMarkovSemigroup`). **Bypassed in the concrete Gaussian chain** since 2026-05-22 — see below; retained only for the abstract `hypercontractive_of_logSobolev` / `gross_lsi_iff` wrappers |\n| `gross_hypercontractive_implies_lsi` | Gross (1975), Theorem 2 (now on bundled `DirichletMarkovSemigroup`) |\n| `stroock_varopoulos` | BGL Prop 1.7.1 (intermediate-step lemma for Gross; **Standard** by gemini-3.1-pro two-pass 2026-05-13). **Bypassed in the concrete Gaussian chain** since 2026-05-22 — discharged there as the *equality* `BakryEmerySpace.stroockVaropoulos_eq` (`Diffusion/StroockVaropoulos.lean`) via the diffusion `rpow` chain rule; retained for non-diffusion / abstract uses |\n| `diamagnetic_resolvent` | Diamagnetic inequality (assembles 5 steps) |\n| `zegarlinski_lsi_inequality` | Otto-Reznikoff (2007) J. Funct. Anal. 243 Thm 1; Zegarlinski (1996) CMP 175; BGL §5.7.5 |\n| `cov_entrywise_bound_of_zegarlinski` | Helffer-Sjöstrand (1994) J. Stat. Phys. 74; Naddaf-Spencer (1997) CMP 183; BGL §4.5 |\n| `herbst_mgf_bound` | BGL §5.4.1 (Herbst's lemma); Ledoux (2001) §1; Otto-Villani (2000) JFA 173 §3 |\n| `poincare_of_lsi` | BGL Proposition 5.1.3 (LSI ⇒ Poincaré with same constant) |\n\nThe Gaussian1D concrete instance and the abstract\n`MarkovSemigroups/General/OUEntropyDecomposition.lean` are both now\n**axiom-free**: all BGL Ch. 2 and §5.5 facts for the Gaussian OU\nsemigroup have been discharged. As of **2026-05-19**, all multivariate\nGaussian BE axioms are discharged to axiom-free theorems: the de Bruijn\nderivative `ouSemigroupFin_l2_sq_hasDerivWithinAt` (in\n`Instances/WorkInProgress/EuclideanFinBE.lean`),\n`ouSemigroupFin_preserves_IsCore` (Workstream N1.b, Cameron–Martin\nkernel route), and `ouSemigroupFin_entropy_sq_decay_bound` (Workstream\nN1.c, the 1-parameter telescoping route) — leaving **0 GaussianFin\nBakry–Émery *curvature* axioms**. (The OU *generator* subsystem in\n`Instances/WorkInProgress/EuclideanGenerator*.lean`, added later for the\n`GeneratorCompat` discharge, is now theorem-only: the coordinatewise Gaussian\nintegration by parts, pointwise endpoint heat equation, and strong-`L²`\ndifference quotient were all discharged on 2026-05-22. See the W-step note\nbelow.)\n\n**W-step — Gross axiom bypassed (2026-05-22):**\n`Instances/WorkInProgress/EuclideanHypercontractive.lean` discharges all four\nper-instance hypotheses of the *proved* `gross_lsi_implies_hypercontractive_of_hypotheses`\nfor the standard Gaussian OU bundle — `CoreSemigroupInvariant`, `GeneratorCompat`,\n`StroockVaropoulos`, `CoreLpL2Approx` — and assembles\n`stdGaussianFin_isHypercontractive`. Consequently `#print axioms` of that theorem\n(and of the downstream gaussian-hilbert `…_isHypercontractive`) lists only\nthe 3 Lean built-ins `[propext, Classical.choice, Quot.sound]` — **not**\n`gaussianFin_diffQuot_tendsto_Lp`, not `gaussianFin_integrationByParts`, not\n`gross_lsi_implies_hypercontractive`, and (since 2026-05-22) **not**\n`stroock_varopoulos`. The latter is discharged in the concrete chain as the\ndiffusion *equality* `BakryEmerySpace.stroockVaropoulos_eq`\n(`Diffusion/StroockVaropoulos.lean`) via the `rpow` chain rule; the `StroockVaropoulos`\npredicate carries `∃ε>0, ε≤u` a.e. (the chain rule needs strict positivity), supplied\nat the unique abstract call site from the positive orbit. No new axioms introduced.\n\n**Discharged 2026-05-12:**\n* `ouSemigroup_fisher_info_decay` (A1, BGL Proposition 5.5.2) — the\n  Fisher info gradient decay `I(P_t g) ≤ e^{-2t} I(g)`. Proved via\n  Cauchy-Schwarz on the Mehler probability kernel + the Mehler\n  derivative formula + γ-invariance. ~400 lines including a\n  polynomial-discriminant Cauchy-Schwarz helper for γ.\n* `hasDerivWithinAt_entropy_ouSemigroup_zero` (A2 at `t = 0+`) —\n  derived from A2 (`t > 0` interior version) + DCT-based continuity\n  of entropy and Fisher info at `s = 0+` + Mathlib's\n  `hasDerivWithinAt_Ici_of_tendsto_deriv`. ~330 lines.\n* `hasDerivAt_entropy_ouSemigroup` (A2 interior, BGL §5.5\n  de Bruijn identity for `t > 0`) — proved via a Stein-IBP-based\n  formula for `(P_t g)''` (avoiding the need for `g''`), parametric\n  DCT for the entropy integral, and the bilinear Dirichlet form\n  identity (`gaussian_dirichlet_form_bilinear` in\n  `EuclideanStein.lean`). ~740 lines.\nThe previously-axiomatized `ouSemigroup_gradient_decay` (BGL Theorem\n5.5.2) is now **PROVED** via the new theorem `hasDerivAt_ouSemigroup`\n(Mehler derivative formula via Mathlib's\n`hasDerivAt_integral_of_dominated_loc_of_deriv_le`) + Jensen +\nγ-invariance. Additionally, **Stein's identity** for the\nstandard Gaussian (BGL §1.15: `∫ y · g(y) dγ = ∫ g'(y) dγ` for\nbounded C¹ g) is now **PROVED** as `stein_identity_standard` via the\nGaussian-PDF ODE `pdf'(y) = -y · pdf(y)` + FTC on infinite intervals\n(`integral_of_hasDerivAt_of_tendsto`) + `withDensity` bridge. The\n**Gaussian Dirichlet form identity** `∫ g · L g dγ = -∫ (g')² dγ`\n(`gaussian_dirichlet_form_identity`, BGL §1.6) follows from Stein\napplied to `h := g · g'`. `ouSemigroup_preserves_IsCore` was\n**DECOMPOSED** (2026-05-12) — the bounded parts `|P_t f|, |(P_t f)'|,\n|(P_t f)''| ≤ M` are now proved (`ouSemigroup_preserves_bounds`) via a\ngeneralized Mehler-derivative `hasDerivAt_ouSemigroup_C1` + iterated\nformula `hasDerivAt_deriv_ouSemigroup` (giving `(P_t f)'' = e^{-2t}\nP_t(f'')`); the `C^∞` smoothing residue (`ouSemigroup_contDiff`) was\nthen **FULLY DISCHARGED** via the **Hermite IBP path** (Path C):\nthe closed-form iterated-derivative identity\n`(P_t f)^{(n)}(x) = (a/b)^n · ∫ y, H_n(y) · f(a·x + b·y) ∂γ` is proved\nby induction on `n`, combining parametric integral differentiation\nwith the n-th-order Stein identity `∫ H_n · F' dγ = ∫ H_{n+1} · F dγ`\n(`hermite_ibp_gaussian`). C^∞ conclusion via\n`contDiff_of_differentiable_iteratedDeriv`. See\n`Instances/WorkInProgress/EuclideanHermite.lean`. As part of this\ndischarge, `IsCore` was refactored from `ContDiff ℝ ⊤` (analyticity in\ncurrent Mathlib) to `ContDiff ℝ ∞` (C^∞), matching the project's\nlong-standing C^∞ intent. All originally-introduced\naxioms were vetted in one pass via Gemini chat\n(gemini-3-pro-preview), which flagged a missing `IsCore` hypothesis\non the Mehler-composition axiom — patched in both the axiom and the\nupstream `BakryEmerySpace.semigroup_add` field. Five of the original\nnine were **reduced to theorems** proved from the remaining four\natomic axioms plus Mathlib's Gaussian infrastructure:\n`ouSemigroup_l2_decay_bound` (FTC + gradient decay),\n`ouSemigroup_ergodic` (double DCT on Mehler integrand),\n`ouSemigroup_entropy_sq_ergodic` (DCT for `s log s` + compactness),\n`ouSemigroup_compose` (Gaussian convolution via\n`gaussianReal_add_gaussianReal_of_indepFun`), and\n`gaussian2D_orthogonal_invariance` (proved by Codex via\n`stdGaussian_map` + `map_pi_eq_stdGaussian` through the\n`EuclideanSpace ℝ (Fin 2) ≃ₗᵢ WithLp 2 (ℝ × ℝ)` isometry).\n\n(`exp_entryNonneg_of_entryNonneg`, `trotter_product_formula`, and\n`m_matrix_inverse_nonneg` were previously axiomatized but are now\nproved as theorems in `Matrix/HeatKernel.lean`, `Matrix/Trotter.lean`,\nand `Matrix/LaplaceTransform.lean` respectively. `hermiteMulti_dense`\nwas previously a free-standing axiom in `Gaussian/HermitePolynomials.lean`\nbut is now a theorem proved from the `polynomial_dense_L2_of_subGaussian`\naxiom in gaussian-field's `GeneralResults/PolynomialDensityGaussian.lean`\n(decomposed via a `Submodule.span`-based change-of-basis between\nmultivariate monomials and multivariate Hermite polynomials, fully\nproved). An earlier free-standing `entropy_chain_rule_local` axiom was\nREMOVED after external review (Gemini, gemini-3-pro-preview): the\nstatement was mathematically false for general Gibbs measures because\nDirac single-site conditionals at low temperature produce a vanishing\nRHS while the LHS remains positive. The DZ proof does not factor\nthrough such a separate chain rule; the iteration is interleaved\ninside `zegarlinski_lsi_inequality`.)\n\n### Which theorems actually depend on these axioms?\n\nVerified by `#print axioms` on a fresh build. The axiom footprint\nis narrow: everything outside the rows below is axiom-free (depends\nonly on Lean's three standard axioms `propext`, `Classical.choice`,\n`Quot.sound`).\n\n| Axiom | Transitive consumers inside this repo |\n|---|---|\n| `gross_lsi_implies_hypercontractive` | `DirichletMarkovSemigroup.hypercontractive_of_logSobolev`, `DirichletMarkovSemigroup.gross_equivalence` (`Abstract/Hypercontractivity.lean`); not consumed by any other module or downstream project |\n| `gross_hypercontractive_implies_lsi` | `DirichletMarkovSemigroup.logSobolev_of_hypercontractive`, `DirichletMarkovSemigroup.gross_equivalence` (`Abstract/Hypercontractivity.lean`); not consumed by any other module or downstream project |\n| `diamagnetic_resolvent` | None — declared for external callers; no theorem in this repo uses it |\n| `zegarlinski_lsi_inequality` | `global_lsi_of_zegarlinski` (`DobrushinZegarlinski/GlobalLSI.lean`); declared for downstream consumers (pphi2N strict thermodynamic-limit route) |\n| `cov_entrywise_bound_of_zegarlinski` | `cov_entrywise_decay_nn` (`DobrushinZegarlinski/EntrywiseCovariance.lean`, the proven exponential-decay corollary `\\|Cov(σ_x, σ_y)\\| ≤ α^{d(x,y)} / (c·(1-α))` for nearest-neighbor finite-range V); declared for pphi2N's `HSData.AdmitsThimbleLocal` |\n| `herbst_mgf_bound` | `lipschitz_concentration_of_lsi` (proven theorem, `Abstract/Concentration.lean`, derived from this axiom + Mathlib's Chernoff `measure_ge_le_exp_mul_mgf`); `lipschitz_concentration_left_of_lsi` and `lipschitz_concentration_two_sided_of_lsi` (proven, by reflection / union bound); `hasSubgaussianMGF_of_lsi` (proven Mathlib `HasSubgaussianMGF` bridge); `memLp_of_lsi` (proven `L^p` moment bounds); `lipschitz_concentration_of_zegarlinski` + left-tail + two-sided + Mathlib bridge + `MemLp` (`DobrushinZegarlinski/Concentration.lean`, all composing with the global LSI from the Zegarlinski hypothesis) |\n| `poincare_of_lsi` | `variance_lipschitz_le_of_lsi` (proven `Var(F) ≤ L²/c` via Mathlib's `norm_fderiv_le_of_lipschitz`); `variance_lipschitz_le_of_zegarlinski` (proven Zegarlinski composition `Var ≤ L²/(c·(1-α))`). Declared for spectral-gap-style fluctuation bounds |\n| `Gaussian1D.bakryEmerySpace` consumes no atomic OU axioms | The Gaussian1D `BakryEmerySpace ℝ` instance (in `Instances/WorkInProgress/EuclideanEntropyDecay.lean`) is **axiom-free**. All originally-axiomatized BGL Ch. 2 / §5.5 Mehler-kernel-level facts have been discharged as theorems. |\n\n**DZ-layer axiom audit (verified `#print axioms` 2026-05-01):** the\nproven content of `DobrushinZegarlinski/` — `AbstractInfluenceMatrix`\ntheory, single-site decomposition `entropy_decomposition_single_site`,\nDLR-at-Bochner-integral identity `integral_siteSmoothing`, distance-\naware Neumann bounds `iterate_dist_zero` /\n`neumann_series_nn_dist_bound`, and the exponential-decay corollary\n`cov_entrywise_decay_nn` — is **axiom-free** (depends only on the\nthree Lean built-ins). Only the two textbook axioms above are pulled\nin when the LSI / Cov bridge theorems themselves are invoked.\n\nThe rest of the library is axiom-free. Specifically, the\nBakry-Émery Poincaré / LSI / variance decay / entropy decay\n(`Diffusion/CarreDuChamp.lean`), Brascamp-Lieb\n(`Instances/BrascampLieb.lean`), Doeblin mixing\n(`Convergence/Doeblin.lean`), Dobrushin uniqueness\n(`Dobrushin/Uniqueness.lean`), TV coupling\n(`Coupling/TVCoupling.lean`), Neumann-series correlation decay\n(`Dobrushin/NeumannSeries.lean`), the multisite covariance bound\n(`Dobrushin/CovarianceBoundMultisite.lean`), and heat-kernel\npositivity for Z-matrices (`Matrix/HeatKernel.lean`) all have\n`#print axioms` showing only the three Lean built-ins.\n\nOne caveat to \"the rest is axiom-free\": `General/SchwartzConvolution.lean`\ndeclares `contDiff_top_convolution_schwartzKernel` (Gemini-vetted\n2026-05-12, **Likely correct**; registered in `AXIOM_AUDIT.md` on\n2026-05-16 after the repo-wide sweep found it undocumented there). It\nhas **no consumers** — staged infrastructure; the OU smoothing fact\nwas discharged by a different route — so no theorem's `#print axioms`\nis affected.\n\n**For downstream consumers:** [lgt](https://github.com/mrdouglasny/lgt)'s\nYang-Mills mass-gap proof uses only the Dobrushin + Coupling +\nMatrix/HeatKernel paths, none of which touch the eight textbook\naxioms. `#print axioms ym_mass_gap_UN` on the lgt side shows only\n`propext`, `Classical.choice`, `Quot.sound`. The DZ + concentration +\nPoincaré axioms (added 2026-04 to 2026-05) are reserved for pphi2N's\nstrict thermodynamic-limit route.\n\n### Concrete instances\n\n- **TwoPoint** ({0,1}, uniform measure): 19/21 BakryEmerySpace fields\n  proved. 2 sorry's are mathematically false (Γ_leibniz fails for jump\n  processes — validates that the diffusion axiom is a real constraint).\n- **Gaussian1D** (ℝ, N(0,1), OU semigroup): 0 sorries / 5 BGL Ch. 2\n  textbook axioms. All structure fields filled; the five remaining\n  axioms package Mehler-kernel-level facts (parametric Fubini against\n  the Gaussian density, differentiation under the integral, 2D\n  Gaussian rotation invariance, entropy derivative bridge) as named\n  axioms with proof-strategy docstrings, all Gemini-vetted\n  (gemini-3-pro-preview). Four of the originally nine were reduced\n  to theorems proved from the remaining five:\n  `ouSemigroup_l2_decay_bound` (FTC + gradient decay),\n  `ouSemigroup_ergodic` (double DCT), `ouSemigroup_entropy_sq_ergodic`\n  (DCT for `s log s` + compactness), and `ouSemigroup_compose`\n  (Gaussian convolution arithmetic).\n\n### Verification / validation tests (Instances/WorkInProgress/EuclideanTests.lean)\n\nTest theorems exercising the `Gaussian1D` `BakryEmerySpace ℝ` instance\non concrete inputs. Zero sorries.\n\n| Set | Theorem | Axioms |\n|---|---|---|\n| **A. Mehler eigenfunctions** | `ouSemigroup_const`, `ouSemigroup_id`, `ouSemigroup_hermite_two`, `ouSemigroup_hermite_three` (eigenvalues `1, e^{-t}, e^{-2t}, e^{-3t}` on `H₀, H₁, H₂, H₃`) | none (Mathlib only) |\n| **A. Helpers** | `integral_sq_γ` (= 1), `integral_cube_γ` (= 0), `integrable_cube_γ` | none |\n| **B. Bakry-Émery on `cos`** | `cos_isCore`, `cos_poincare` (`Var_γ(cos) ≤ ∫sin² dγ`), `cos_variance_decay` (`Var_γ(P_t cos) ≤ e^{-2t} Var_γ(cos)`) | the 3 atomic Bakry-Émery building-block axioms in `General/OUEntropyDecomposition.lean` (Fisher info decay + de Bruijn identity); the previous broad `ouSemigroup_entropy_sq_decay_bound` was discharged from these via FTC + ε-regularization. The full BGL Proposition 4.7.1 chain is now proved end-to-end (`ouSemigroup_contDiff` via Hermite IBP in `EuclideanHermite.lean`, the entropy decay via the decomposition above in `EuclideanEntropyDecay.lean`). |\n| **C. Conditional BL → Gaussian Poincaré** | `brascampLieb_recovers_gaussian_poincare` (conditional on a `LogConcaveMeasure ℝ` with `μ = γ`) | none |\n| **C. Gaussian `LogConcaveMeasure` — structural** | `V_gauss = x²/2`, `contDiff_V_gauss`, `hessianBilin_V_gauss` (= `v · w`), `hV_gauss_convex`, `hV_gauss_curvature` (`‖v‖² ≤ Hess V(v,v)`) | none |\n| **C. Gaussian `LogConcaveMeasure` — resolvent fields** | `gaussianLogConcaveMeasure : LogConcaveMeasure ℝ` (built directly, no `Classical.choose`) | four named axioms: `gaussianResolvent`, `gaussianResolvent_ibp`, `gaussianResolvent_ibp_integrable`, `gaussianBochner_identity` (BGL §1.15-1.16: OU resolvent via Lax-Milgram + Stein IBP + Bochner-Weitzenböck) |\n| **C. Unconditional BL → Gaussian Poincaré** | `gaussian_brascampLieb_poincare` (`Var_γ(f) ≤ ∫‖f'‖² dγ` for `C¹` `f`) | the four resolvent axioms above |\n\n**Verification summary.** Set A confirms that the Mehler integral\n`ouSemigroup` evaluates to the textbook eigenvalues on the first four\nHermite polynomials — operationally verifying it really is the Mehler\nkernel, without invoking the remaining atomic OU axiom. Set B\nconfirms the abstract `satisfiesPoincare` / `variance_decay` theorems\nplug correctly into the concrete instance with `ρ = 1`. Set C confirms\nthat the Brascamp-Lieb route to Gaussian Poincaré with `ρ = 1` agrees\nwith the Bakry-Émery route on the same constant. The Gaussian\n`LogConcaveMeasure ℝ` is now built constructively in its structural\nfields (`V = x²/2`, computed 1D Hessian, convexity, curvature ρ = 1);\nonly the OU-resolvent ingredients (Lax-Milgram + Stein IBP +\nBochner-Weitzenböck) remain as four named textbook axioms with\nBGL §1.15-1.16 citations.\n\n### TV coupling (Coupling/)\n\n- `tvDist_le_coupling`: TV ≤ P(σ ≠ τ) for any coupling — **proved**\n- `maximal_coupling`: optimal coupling construction — **proved**\n- `tvDist_eq_inf_coupling`: coupling characterization — **proved**\n- `canonicalMaximalCoupling`: constructive pointwise-min coupling — **proved**\n- Giry measurability of canonical coupling — **proved** (for countable S)\n- `dobrushin_iterated_coupling_fintype`: min-disagreement coupling (finite S) — **proved**\n- `prokhorov_coupling_theorem`: min-disagreement coupling (compact S) — **proved**\n  via Prokhorov compactness + Portmanteau lsc + kernel Radon-Nikodym\n\n### Dobrushin uniqueness (Dobrushin/)\n\n- `GibbsSpec`, `IsGibbsMeasure`: Gibbs specifications — **defined**\n- `DobrushinCondition`: influence matrix condition — **defined**\n- `dobrushin_uniqueness`: Dobrushin uniqueness theorem — **proved**\n- `influenceCoeff`, `influenceCoeff_le_of_cylinder_ratio_bound` — **proved**\n- Single-site disintegration (`condSingleSiteMeasure`) — **proved**\n- Multi-site covariance bounds via `condKernel` — **proved**\n- `condKernel_ae_bound`: condKernel fiber inherits DLR — **proved**\n- Neumann series exponential decay — **proved**\n\n### Matrix semigroup theory (Matrix/)\n\n- **Heat kernel positivity** for Z-matrices: exp(-tM) ≥ 0 entrywise —\n  proved via Metzler shift decomposition\n- **Euler factor nonnegativity**: I - (t/n)M ≥ 0 for large n — proved\n- **Entrywise-nonneg matrix algebra**: pow, mul, add, smul — proved\n\n## Application to P(Phi)_2\n\nThe Gaussian results (Phases 1-3) provide the free-field infrastructure\nfor stochastic quantization. For the interacting theory, the simple\nHolley-Stroock perturbation does not apply (the interaction V_a is\nunbounded above, so osc(V) = infinity). See [docs/plan.md](docs/plan.md)\nfor the Gemini review and three alternative approaches for the\ninteracting spectral gap.\n\n## Development plan\n\nSee [docs/plan.md](docs/plan.md) for the full development plan, timeline,\nrisk assessment, and Gemini review.\n\n### Discharging the Gross axiom (Route A)\n\nThe Gross LSI ⇒ hypercontractivity axiom\n(`gross_lsi_implies_hypercontractive`) is being discharged as a\nmulti-stage program. Workstream-level roadmaps and design decisions:\n\n- [`plans/gross-discharge.md`](plans/gross-discharge.md) — overall\n  Route A roadmap (H0/G1/G2/P2/P3/W workstreams). H0, G1, G2 ✅;\n  P2 Leibniz kernel ✅ (2026-05-20).\n- [`plans/archive/p2-strongL2-leibniz-discharge.md`](plans/archive/p2-strongL2-leibniz-discharge.md)\n  — discharge plan for the P2 Bochner-Leibniz kernel.\n- [`plans/archive/grosspow-hasderivwithinAt-structural-blockers.md`](plans/archive/grosspow-hasderivwithinAt-structural-blockers.md)\n  — analysis of the 4 structural blockers encountered while\n  composing `grossPow_hasDerivWithinAt`; 3 of 4 discharged from\n  existing structure fields (2026-05-20).\n- [`plans/archive/gross-design-strictly-positive-escape.md`](plans/archive/gross-design-strictly-positive-escape.md)\n  — design decision (Path A vs Path B) for the remaining Blocker 3.\n  Gemini-vetted; Path A (strictly-positive escape, matching Gross\n  1975's original proof structure) chosen.\n\n## Dependencies\n\n- [hille-yosida](https://github.com/mrdouglasny/hille-yosida) — C₀-semigroup\n  theory, generators, resolvents, Hille-Yosida bound\n- [gaussian-field](https://github.com/mrdouglasny/gaussian-field) — Gaussian\n  measures on nuclear spaces, hypercontractivity, tightness\n- [Mathlib](https://github.com/leanprover-community/mathlib4) — Lean 4\n  mathematics library\n\n## Building\n\n```bash\nlake update\nlake build\n```\n\n## References\n\n- Bakry, Gentil, and Ledoux, *Analysis and Geometry of Markov Diffusion\n  Operators*, Springer, 2014.\n- Gross, \"Logarithmic Sobolev inequalities,\" *Amer. J. Math.* 97 (1975),\n  1061-1083.\n- Holley and Stroock, \"Logarithmic Sobolev inequalities and stochastic\n  Ising models,\" *J. Stat. Phys.* 46 (1987), 1159-1194.\n- Da Prato and Zabczyk, *Stochastic Equations in Infinite Dimensions*,\n  Cambridge, 2014.\n- Nelson, \"The free Markoff field,\" *J. Funct. Anal.* 12 (1973), 211-227.\n- Fukushima, Oshima, and Takeda, *Dirichlet Forms and Symmetric Markov\n  Processes*, de Gruyter, 2011.\n- Simon, *Functional Integration and Quantum Physics*, Academic Press,\n  1979 (Ch. 22: diamagnetic inequality).\n- Dobrushin, \"The description of a random field by means of conditional\n  probabilities,\" *Theor. Prob. Appl.* 13 (1968), 197–224.\n- Lindvall, *Lectures on the Coupling Method*, Wiley, 1992.\n- Chatterjee, *Gauge Theory Lecture Notes*, 2026.\n\n## Author\n\nMichael R. Douglas\n\n## License\n\nCopyright (c) 2026 Michael R. Douglas. Released under the Apache 2.0 license.\n",1784670034541]