[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"HdvCfuV0w7":3},"# pphi2\n\nFormal construction of the P(Φ)₂ Euclidean quantum field theory in Lean 4,\nfollowing the Glimm-Jaffe/Nelson lattice approach.\n\n> **Status at a glance.** Builds green (`lake build`), **0 sorries** — the remaining debt is a set of\n> documented project axioms. Most-developed line: the **T²_L torus** (OS0–OS2, axiom-free). The\n> **cylinder** adds OS3 (UV limit done; IR limit in progress); the full **ℝ²** OS0–OS4 target has its\n> remaining axioms concentrated in one Glimm–Jaffe Ch. 8 Nelson estimate. **Non-triviality** — that\n> the theory is genuinely interacting (`u₄ ≠ 0`) — is **proved axiom-free** on T² at weak coupling.\n> Per-route detail: [**Current status**](#current-status) below.\n>\n> **Where to look:**\n> [`planning/INDEX.md`](planning/INDEX.md) — per-axiom master status (remaining axioms, dependency\n> DAG, a discharge plan for each) · [`BRANCHES.md`](BRANCHES.md) — git branch → axiom map ·\n> [`planning/coherence-analysis.md`](planning/coherence-analysis.md) — why the pieces don't *yet*\n> compose into the single conjoined \"interacting φ⁴₂ QFT exists\" theorem ·\n> [`docs/STATUS_HISTORY.md`](docs/STATUS_HISTORY.md) — dated discharge history.\n\n## What this project proves\n\n**Main theorem** (`Pphi2/Main.lean`): For any even polynomial P of degree ≥ 4\nbounded below and any mass m > 0, there exists a probability measure μ on the\nspace of tempered distributions S'(ℝ²) satisfying all five Osterwalder-Schrader\naxioms:\n\n- **OS0 (Analyticity):** The generating functional Z[Σ zᵢJᵢ] is entire analytic\n  in z ∈ ℂⁿ.\n- **OS1 (Regularity):** ‖Z[f]‖ ≤ exp(c(‖f‖₁ + ‖f‖ₚᵖ)) for some 1 ≤ p ≤ 2.\n- **OS2 (Euclidean Invariance):** Z[g·f] = Z[f] for all g ∈ E(2) = ℝ² ⋊ O(2).\n- **OS3 (Reflection Positivity):** The RP matrix Σᵢⱼ cᵢcⱼ Re(Z[fᵢ − Θfⱼ]) ≥ 0.\n- **OS4 (Clustering):** Z[f + Tₐg] → Z[f]·Z[g] as ‖a‖ → ∞.\n\nBy the Osterwalder-Schrader reconstruction theorem, the corresponding\nmathematical theorem in the literature yields a relativistic Wightman QFT in\n1+1 Minkowski spacetime with a positive mass gap. This repository currently\nformalizes the Euclidean OS side, not the reconstruction step itself.\n\nThis is the theorem originally proved by Glimm-Jaffe (1968–1973), Nelson (1973),\nand Simon, with contributions from Guerra-Rosen-Simon and others.\n\n**Non-triviality** (that the constructed theory is genuinely interacting, not a disguised free field)\nis a separate result: the continuum φ⁴₂ measure on T² is **non-Gaussian** (`u₄ ≠ 0`) at weak coupling,\nproved **axiom-free** (`torus_pphi2_isInteractingStrict_weakCoupling`). See \"Current status →\nNon-triviality\" below.\n\n## Scope and foundations direction\n\nThis repository currently formalizes one specific Euclidean-QFT formulation:\nthe Glimm-Jaffe/Nelson construction of a positive probability measure on\n`S'(ℝ²)` for bosonic scalar `P(Φ)₂`. It does not yet formalizes gauge fields,\nfermions, chemical potential, or topological terms.\n\nA framework for defining QFTs on general spacetimes\n(compact manifolds, lattices, manifolds with boundary) with separated spacetime\ngeometry and field content is a WIP. An exploratory organizational proposal is in\n[`docs/foundational-roadmap.md`](docs/foundational-roadmap.md), with a\ntechnical reference for the current formulation-layer code in\n[`docs/formulation-layer.md`](docs/formulation-layer.md) and open design\nquestions (general manifolds, multiple fields, fermions, signed/complex\nmeasures) in\n[`docs/formulation-layer-questions.md`](docs/formulation-layer-questions.md).\n\n## Proof approach\n\nThe construction proceeds in six phases:\n\n1. **Lattice measure** — Define the Wick-ordered interaction\n   V_a(φ) = a² Σ_x :P(φ(x)):_a on the finite lattice (ℤ/Nℤ)² and construct\n   the interacting measure μ_a = (1/Z_a) exp(−V_a) dμ_{GFF,a}.\n\n2. **Transfer matrix** — Decompose the lattice action into time slices. The\n   transfer matrix T is a positive trace-class operator. This gives reflection\n   positivity (OS3).\n\n3. **Spectral gap** — Show T has a spectral gap (λ₀ > λ₁) by Perron-Frobenius.\n   This is the lattice mass gap; OS4 on the periodic torus is phrased with **cyclic**\n   Euclidean-time separation (`latticeEuclideanTimeSeparation` in `OS4_MassGap.lean`),\n   and the textbook continuum clustering picture is recovered after IR/continuum limits.\n\n4. **Continuum limit** — Embed lattice measures into S'(ℝ²), prove tightness\n   (Mitoma + Nelson's hypercontractive estimate), extract a convergent\n   subsequence by Prokhorov. OS0, OS1, OS3, OS4 transfer to the limit.\n   The free (Gaussian) case is handled separately in `GaussianContinuumLimit/`:\n   the lattice GFF measures are tight (Mitoma criterion with uniform m⁻²\n   second-moment bound from the spectral gap of −Δ_a + m²), their weak\n   limits are Gaussian (Bochner-Minlos), and the covariance converges to\n   the continuum Green's function G(f,g) = ∫ f̂(k)ĝ(k)/(|k|²+m²) dk/(2π)².\n\n5. **Euclidean invariance** — Restore full E(2) symmetry via a Ward identity\n   argument. The rotation-breaking operator has scaling dimension 4 > d = 2,\n   so the anomaly is RG-irrelevant and vanishes in the continuum limit; in the\n   super-renormalizable `P(Φ)₂` setting one allows at most polynomial\n   `|log a|` corrections, still multiplied by the vanishing `a²` factor.\n\n6. **Assembly** — Combine all axioms into the main theorem.\n\n## Four routes (spacetimes)\n\nThe construction is carried out on four spacetimes, each targeting different\nOS axioms. See [ROUTES.md](ROUTES.md) for the detailed comparison.\n\n### Route A: ℝ² (Euclidean plane) — OS0–OS4\nThe full construction targets S'(ℝ²) and proves all five OS axioms.\nThe continuum limit involves both UV (a → 0) and IR (volume → ∞) limits.\n**Status:** the full target (UV **and** IR limits). Remaining axioms are concentrated in\n**Cluster A** — the Glimm–Jaffe Ch. 8 dynamical-cutoff Nelson estimate (a multi-week deliverable,\n~6–8 wk per the Gemini estimate in [`docs/lattice-action-normalization-fix.md`](docs/lattice-action-normalization-fix.md))\n— plus the continuum-inheritance bridges; the Phase-B Glimm–Jaffe Fourier estimates are now theorems\n(`#print axioms Pphi2.rough_error_variance` ⟹ `[propext, Classical.choice, Quot.sound]`). Current\ncounts: [`docs/AXIOM_STATUS.md`](docs/AXIOM_STATUS.md). Discharge history (Stage-1 fix, Phase-2\ndischarges, the isotropic-cylinder branch): [`docs/STATUS_HISTORY.md`](docs/STATUS_HISTORY.md).\n\n### Route B: T²_L (symmetric torus) — OS0–OS2\nFinite-volume warm-up isolating the UV limit. Lattice (ℤ/Nℤ)² with\nspacing a = L/N → 0. The interacting continuum limit `torusInteractingLimit_exists`\nis proved via Mitoma-Chebyshev tightness + Nelson's exponential estimate\n(proved: physical volume a²N²=L² is constant). OS3 dropped (→ Routes B', C).\n\n**`TorusInteractingOS.lean`: 0 local axioms, 0 sorries** — OS0–OS2 complete (OS0 via a fully verified\nOsgood lemma + Gaussian integrability; OS2 via gaussian-field's proved time-reflection / translation\nidentities). See [`docs/torus-interacting-os-proof.md`](docs/torus-interacting-os-proof.md) for the\nproof overview and [`docs/STATUS_HISTORY.md`](docs/STATUS_HISTORY.md) for the discharge log.\n\n### Route B': T_{Lt,Ls} → S¹_{Ls} × ℝ (asymmetric torus → cylinder) — OS0–OS3\nExtends Route B to the cylinder by taking the time direction to infinity.\nThe construction proceeds in two limits:\n\n1. **UV limit (DONE):** On the asymmetric torus T_{Lt,Ls} = (ℝ/Lt ℤ) × (ℝ/Ls ℤ)\n   with lattice (ℤ/Nℤ)² and geometric-mean spacing √(Lt·Ls)/N, take N → ∞.\n   Route B's OS0–OS2 proofs are fully adapted to the asymmetric case.\n   `AsymTorusOS.lean`: OS0–OS2 complete; the remaining input is the Cluster-A asym Nelson estimate\n   `asymNelson_exponential_estimate` (`AsymTorusInteractingLimit.lean`).\n\n2. **IR limit (in progress):** Take Lt → ∞ with Ls fixed. The torus measures\n   μ_{P,Lt,Ls} on T_{Lt,Ls} converge weakly to a measure μ_{P,Ls} on the\n   cylinder S¹_{Ls} × ℝ. Tightness follows from uniform-in-Lt moment bounds\n   via the **method of images** (gaussian-field `Cylinder/MethodOfImages.lean`).\n   The IR limit files are in `IRLimit/` with 0 local axioms and 0 sorries,\n   plus explicit nonlocal inputs for eventual Green moments and eventual\n   pullback reflection positivity.\n   `limit_exponential_moment` (MCT + truncation) is now fully proved.\n   OS2 (time reflection) of the limit measure is **proved** via characteristic\n   functional convergence.\n\nThe cylinder S¹_{Ls} × ℝ has a natural time axis ℝ, enabling:\n- **OS3 (Reflection positivity):** Time reflection Θ: t ↦ -t is a clean\n  involution on S¹_{Ls} × ℝ. The RP matrix for positive-time test functions\n  is positive semidefinite, proved from the lattice RP (transfer matrix\n  positivity) + weak limit transfer.\n- **Transfer matrix:** The cylinder admits a Hilbert space decomposition\n  L²(S¹_{Ls}) via spatial slicing at fixed time. The transfer operator\n  T = exp(-H) where H is the P(φ)₂ Hamiltonian. Spectral gap of T\n  gives the mass gap and clustering.\n\n**Advantages over Route C:** reuses all Route B infrastructure (0 axioms for OS0–OS2); only OS3 (RP)\nand the `Lt → ∞` limit are new. **Status:** UV limit complete (`AsymTorusOS.lean`, 0 local axioms);\nIR limit in progress (`IRLimit/`, 0 local axioms, 0 sorries) — `limit_exponential_moment` and OS2\ntime-reflection proved, with the uniform cylinder exponential moment and OS3 reduced to the explicit\neventual Green-moment / pullback-RP hypotheses (`AsymTorusSequenceHasUniformGreenMomentBound`,\n`CylinderMeasureSequenceEventuallyReflectionPositive`), each with proved bridges from stronger\nstatements. Discharge detail: [`docs/STATUS_HISTORY.md`](docs/STATUS_HISTORY.md).\n\n### Route C: S¹_L × ℝ (cylinder, direct) — OS0–OS3\nDirect Nelson/Simon construction with natural time axis ℝ for OS reconstruction.\nThe field is a distribution (not a function), requiring isonormal Gaussian extension.\nOS3 uses Laplace factorization of the cylinder Green's function.\n**21 axioms + 0 sorries** (preserved in `future/`, not in active build).\n\n### Which OS axiom comes from which route?\n| OS axiom | Best route | Why |\n|----------|-----------|-----|\n| OS0 (Analyticity) | B, B' | Exponential moment bounds (proved) |\n| OS1 (Regularity) | B, B' | Clean moment bounds (proved) |\n| OS2 (Symmetry) | B' or A | B' for cylinder symmetries, A for full E(2) |\n| OS3 (RP) | B' or C | Natural time half-space on cylinder |\n| OS4 (Clustering) | B' or A | Transfer matrix spectral gap |\n\n## Construction parameters and renormalization\n\nThe construction takes two inputs:\n\n- **P** (`InteractionPolynomial`) — an even polynomial of degree ≥ 4, bounded below.\n  Examples: P(τ) = λτ⁴, P(τ) = λτ⁴ + μτ², P(τ) = (τ²−a²)⁴.\n  P may have a nonzero quadratic coefficient; the physical mass receives\n  contributions from both the Gaussian mass and the quadratic term in P.\n\n- **mass** (`mass : ℝ`, `0 \u003C mass`) — the mass parameter in the Gaussian\n  reference measure, whose covariance is (−Δ_a + mass²)⁻¹. This must be\n  strictly positive so the lattice operator is invertible (the zero mode\n  has eigenvalue mass²). This is a technical requirement for the Gaussian\n  reference measure, not a physical restriction on the theory.\n\nThe expansion is always around φ = 0, but this does not force the theory\ninto the symmetric phase. An even polynomial can have its global minima at\n±a ≠ 0 (e.g. P(τ) = (τ²−a²)⁴); the functional integral determines which\nphase the theory is in.\n\n**Renormalization:** P(Φ)₂ is super-renormalizable in d = 2. The only UV\ncounterterm is the Wick ordering constant c_a = G_a(0,0) ~ (1/2π)log(1/a),\nwhich is the lattice propagator at coinciding points. The Wick-ordered\ninteraction :P(φ(x)):_a subtracts the divergent self-contractions at each\nlattice spacing. No mass, coupling constant, or wave function\nrenormalization is needed beyond Wick ordering.\n\n## Consistency checks\n\nBeyond the OS axioms themselves, the construction should satisfy additional\nconsistency checks:\n\n- **Mass reparametrization invariance.** The physical measure depends on the\n  total action, not on how it is split between the Gaussian reference measure\n  and the interaction. Shifting the bare mass m → m' while compensating\n  P → P + m²/2 − (m')²/2 leaves the total quadratic term (−Δ + m²) + P\n  unchanged, so the resulting continuum measure must be identical.\n\n- **Wick ordering scheme independence.** The Wick ordering constant\n  c_a = G_a(0,0) depends on the bare mass m through the propagator. A mass\n  shift changes c_a, but the compensating shift in P absorbs this, so the\n  Wick-ordered interaction :P(φ):_a is scheme-independent up to the total\n  action.\n\n- **Torus–infinite volume consistency.** For test functions supported well\n  inside T²_L (away from the boundary identification), the torus and\n  infinite-volume Schwinger functions should agree in the L → ∞ limit.\n\n## Current status\n\nAll six phases are structurally complete and the full project builds\n(`lake build`).\n\nCurrent counter (`./scripts/count_axioms.sh`, 2026-06-23): pphi2 **28 raw /\n26 real axioms**, 0 sorries; gaussian-field **3 axioms**, 0 sorries. The eight\nreal Layer-B2 Route-A axioms are the six GNS bridge obligations isolated in\n`Pphi2/AsymTorus/AsymBridgeInstance.lean`, B5b single-slice stability in\n`Pphi2/AsymTorus/AsymB5bSingleSlice.lean`, and the final lattice Route-A\nassembly input `asymInteractingVariance_le_freeVariance_lattice_Lt_uniform`.\n\nDetailed axiom/sorry inventory lives in the single sources of truth:\n[`planning/INDEX.md`](planning/INDEX.md) (per-axiom master status machine for the remaining axioms),\n[`docs/AXIOM_STATUS.md`](docs/AXIOM_STATUS.md) (axiom inventory), [`BRANCHES.md`](BRANCHES.md) (which\ngit branch carries the live code for each axiom), and [`status.md`](status.md) (full inventory).\nDated discharge narrative is archived in [`docs/STATUS_HISTORY.md`](docs/STATUS_HISTORY.md).\n\n### Per-route snapshot (the OS-axiom construction)\n\n| Route (spacetime) | OS axioms | State |\n|---|---|---|\n| **B — T²_L** symmetric torus | OS0–OS2 | **Complete**, 0 local axioms (`TorusInteractingOS.lean`). The most developed route; UV-only limit (N→∞ at fixed volume `L`). |\n| **B′ — cylinder** (asym torus → S¹_{Ls}×ℝ) | OS0–OS3 | UV limit (N→∞) **done**, 0 local axioms; IR limit (Lt→∞) in progress. Remaining nonlocal inputs: the eventual Green-moment bound and eventual pullback RP for the asym family. |\n| **A — ℝ²** Euclidean plane | OS0–OS4 | Full target (UV **and** IR limits). Remaining axioms concentrated in **Cluster A** (Glimm–Jaffe Ch. 8 dynamical-cutoff Nelson) plus continuum-inheritance bridges. |\n| **C — cylinder, direct** | OS0–OS3 | Preserved in `future/` (21 axioms), **not** in the active build. |\n\nThe torus continuum limit (`TorusContinuumLimit/`) is the cleanest backbone: fixing physical volume\n`L` and taking only `N→∞` isolates the UV limit from IR issues. Prokhorov extraction on the\nconfiguration space is **proved** (not axiomatized) via gaussian-field's DM-basis\n`prokhorov_configuration`; the same pipeline handles both Gaussian and interacting (P(φ)₂) measures\nvia Cauchy–Schwarz density transfer. The torus Gaussian limit satisfies OS0–OS3 (`TorusOSAxioms.lean`,\nall proved).\n\n### Non-triviality (is the theory genuinely interacting?)\n\nSeparate from the OS axioms: is the continuum φ⁴₂ measure **non-Gaussian** (`u₄ ≠ 0`)? Both sub-routes\nlive on the **T² (Route B) spacetime** — these \"weak-coupling / dilation\" sub-routes are *not* the\nspacetime routes A/B/B′/C above.\n\n- **Weak-coupling sub-route — DONE, axiom-free.** `torus_pphi2_isInteractingStrict_weakCoupling`\n  (`TorusContinuumLimit/TorusCouplingResult.lean`): for some small coupling `g₀ ∈ (0,1]`, the\n  continuum limit of the coupling-`g₀` interacting torus measures has a *strictly negative* connected\n  four-point (`TorusIsInteractingStrict`, hence `TorusIsInteracting`). `#print axioms` ⟹\n  `[propext, Classical.choice, Quot.sound]`. Design: [`planning/route-A-weak-coupling-plan.md`].\n  (Currently on branch `route-a-weak-coupling`, PR #48.)\n- **`λ=1` / large-mass normalization — DEFERRED.** Upgrading to full coupling via the continuum\n  dilation is sound but entangled with unbuilt clustering / spectral-gap machinery — see the deferral\n  note in [`planning/continuum-rescaling-plan.md`].\n\nA **shared foundations layer** (`Common/QFT/Euclidean/{Formulations,ReconstructionInterfaces}.lean`)\nseparates concrete measure models, tensor-moment / distributional Schwinger data, explicit\nreconstruction hypotheses, and backend-independent reconstruction rules.\n\n## Nontrivial infrastructure notes\n\n- **Configuration-space Prokhorov bridge**:\n  [SobolevProkhorovPlan.lean](Pphi2/ContinuumLimit/SobolevProkhorovPlan.lean)\n  records the staged theorem API to replace\n  `prokhorov_configuration_sequential`.\n- **Transfer-matrix spectral infrastructure (Jentzsch)**:\n  [Jentzsch.lean](Pphi2/TransferMatrix/Jentzsch.lean) contains the\n  positivity-improving/Perron-Frobenius spectral input used for mass-gap-level\n  statements.\n- **Convolution operator infrastructure**:\n  [L2Convolution.lean](Pphi2/TransferMatrix/L2Convolution.lean) centralizes\n  Young-type analytic dependencies for `L²` convolution operators.\n- **Global inventory and difficulty tracking**:\n  [status.md](status.md) and [docs/axiom_proof_plans.md](docs/axiom_proof_plans.md).\n\n## File structure\n\n```\nPphi2/\n  Polynomial.lean                    -- Interaction polynomial P(τ)\n  WickOrdering/                      -- Phase 1: Wick monomials and counterterms\n  InteractingMeasure/                -- Phase 1: Lattice action and measure\n  TransferMatrix/                    -- Phase 2-3: Transfer matrix, positivity, spectral gap\n    L2Multiplication.lean            -- Multiplication operator M_w on L²\n    L2Convolution.lean               -- Convolution operator Conv_G on L² (Young's inequality)\n    L2Operator.lean                  -- Transfer operator T = M_w ∘ Conv_G ∘ M_w\n    Jentzsch.lean                    -- Jentzsch's theorem, Perron-Frobenius spectral properties\n  OSProofs/\n    OS3_RP_Lattice.lean              -- Phase 2: Reflection positivity on the lattice\n    OS3_RP_Inheritance.lean          -- Phase 2: RP closed under weak limits\n    OS4_MassGap.lean                 -- Phase 3: Clustering from spectral gap\n    OS4_Ergodicity.lean              -- Phase 3: Ergodicity from mass gap\n    OS2_WardIdentity.lean            -- Phase 5: Ward identity for rotation invariance\n  OSforGFF/                          -- Matrix positivity library (from OSforGFF)\n    PositiveDefinite.lean            -- Positive definite functions\n    FrobeniusPositivity.lean         -- Frobenius inner product positivity\n    SchurProduct.lean                -- Schur (Hadamard) product theorem\n    HadamardExp.lean                 -- Entrywise exponential preserves PSD/PD\n    TimeTranslation.lean             -- Schwartz space time translation continuity\n  ContinuumLimit/                    -- Phase 4: Embedding, tightness, convergence\n    Hypercontractivity.lean          -- Nelson's estimate (Option A: Cauchy-Schwarz density transfer)\n  GaussianContinuumLimit/            -- Phase 4G: Free GFF continuum limit (reusable infrastructure)\n    EmbeddedCovariance.lean          -- gaussianContinuumMeasure, embeddedTwoPoint, continuumGreenBilinear\n    PropagatorConvergence.lean       -- Lattice propagator → continuum Green's function; uniform bound\n    GaussianTightness.lean           -- Tightness of {ν_{GFF,a}} via Mitoma criterion\n    GaussianLimit.lean               -- Existence + Gaussianity of the limit; IsGaussianContinuumLimit\n  TorusContinuumLimit/               -- Phase 4T: Torus continuum limit (UV only, L fixed)\n    TorusEmbedding.lean              -- torusEmbedLift, torusContinuumMeasure, Green's function\n    TorusPropagatorConvergence.lean  -- Lattice → continuum eigenvalues, uniform bound, positivity\n    TorusTightness.lean              -- Tightness via Mitoma on torus (finite volume)\n    TorusConvergence.lean            -- Prokhorov extraction (PROVED, not axiomatized)\n    TorusGaussianLimit.lean          -- Gaussian identification, IsTorusGaussianContinuumLimit\n    TorusInteractingLimit.lean       -- P(φ)₂ tightness + existence (Cauchy-Schwarz transfer)\n  GeneralResults/\n    FunctionalAnalysis.lean          -- Pure Mathlib results: Cesàro, Schwartz Lp, trig identities, log-decay, CF defect control\n  ContinuumLimit/\n    CharacteristicFunctional.lean    -- Continuum CF analyticity/invariance/reality/ergodicity support\n    TimeReflection.lean              -- Continuum time reflection on Schwartz space and distributions\n  OSAxioms.lean                      -- Phase 6: OS axiom definitions (matching OSforGFF)\n  FormulationAdapter.lean            -- Exports `Pphi2` to the shared formulation interfaces\n  Main.lean                          -- Phase 6: Main theorem assembly\n  Bridge.lean                        -- Bridge between pphi2 and Phi4 approaches\nCommon/\n  QFT/Euclidean/Formulations.lean    -- Shared formulation layers: measure / Schwinger / reconstruction input\n  QFT/Euclidean/ReconstructionInterfaces.lean -- Backend-independent linear-growth / reconstruction interfaces\n```\n\n## Dependencies\n\n- [gaussian-field](https://github.com/mrdouglasny/gaussian-field) — Gaussian\n  free field on nuclear spaces, lattice field theory, FKG inequality\n- [Mathlib](https://github.com/leanprover-community/mathlib4) — Lean 4\n  mathematics library\n\n## Building\n\nRequires Lean 4. gaussian-field is a git dependency (fetched automatically).\n\n```bash\ngit clone https://github.com/mrdouglasny/pphi2.git\ncd pphi2\nlake build\n```\n\n### Continuous integration\n\nPull requests and pushes to `main` run [GitHub Actions](.github/workflows/ci.yml)\nusing [leanprover/lean-action](https://github.com/leanprover/lean-action): install\nthe toolchain from `lean-toolchain`, optionally `lake exe cache get` for Mathlib,\nthen `lake build`. Workflow YAML is checked with [actionlint](.github/workflows/actionlint.yml).\n[Dependabot](.github/dependabot.yml) proposes weekly updates for action versions.\n\n## Documentation\n\n### Expository\n- [docs/construction-overview.md](docs/construction-overview.md) —\n  End-to-end overview of the six-phase construction, from lattice to QFT\n- [docs/wick-ordering.md](docs/wick-ordering.md) — Wick ordering,\n  renormalization, and why only one counterterm is needed\n- [docs/nuclear-spaces-and-measures.md](docs/nuclear-spaces-and-measures.md) —\n  Nuclear spaces, the Gel'fand triple, and Gaussian measure construction\n- [docs/transfer-matrix-and-mass-gap.md](docs/transfer-matrix-and-mass-gap.md) —\n  Transfer matrix, Jentzsch's theorem, and the mass gap\n- [docs/reflection-positivity.md](docs/reflection-positivity.md) —\n  Reflection positivity (OS3), the perfect square argument, and OS\n  reconstruction\n- [docs/hypercontractivity.md](docs/hypercontractivity.md) — Why\n  hypercontractivity is needed and how it transfers Gaussian estimates to\n  the interacting measure\n- [docs/tightness-and-weak-convergence.md](docs/tightness-and-weak-convergence.md) —\n  Tightness, weak convergence, Prokhorov's theorem, and uniqueness of\n  the limit\n\n### Technical\n- [status.md](status.md) — Complete axiom/sorry inventory with difficulty\n  ratings and priority ordering\n- [docs/plan.md](docs/plan.md) — Development roadmap and construction outline\n- [docs/foundational-roadmap.md](docs/foundational-roadmap.md) — Why the repo\n  is being refactored around formulation layers (measure / Schwinger /\n  reconstruction)\n- [AXIOM_AUDIT.md](AXIOM_AUDIT.md) — Self-audit of all axioms\n  (pphi2 + gaussian-field + markov-semigroups + gaussian-hilbert) with\n  correctness ratings; format per `~/.claude/AXIOM_AUDIT_FORMAT.md`.\n- [docs/mathlib_candidates.md](docs/mathlib_candidates.md) — Standard results\n  suitable for Mathlib contribution (~50 across pphi2 + gaussian-field)\n- [docs/gemini_review.md](docs/gemini_review.md) — External review of axioms\n  with references and proof strategies\n- [docs/torus_continuum_limit_plan.md](docs/torus_continuum_limit_plan.md) —\n  Plan for torus OS axioms (Gaussian + interacting P(φ)₂)\n- [docs/os_axioms_lattice_plan.md](docs/os_axioms_lattice_plan.md) — Design\n  notes for OS axiom formulations\n\n## References\n\n- J. Glimm and A. Jaffe, *Quantum Physics: A Functional Integral Point of View*,\n  Springer (1987)\n- B. Simon, *The P(φ)₂ Euclidean (Quantum) Field Theory*, Princeton (1974)\n- E. Nelson, \"Construction of quantum fields from Markoff fields,\" *J. Funct. Anal.* (1973)\n- K. Osterwalder and R. Schrader, \"Axioms for Euclidean Green's functions I, II,\"\n  *Comm. Math. Phys.* 31 (1973), 42 (1975)\n\n## Imported material\n\nThe files under `Pphi2/OSforGFF/` (PositiveDefinite, FrobeniusPositivity, SchurProduct,\nHadamardExp) are imported from the\n[OSforGFF](https://github.com/mrdouglasny/OSforGFF) project, authored by\nMichael R. Douglas, Sarah Hoback, Anna Mei, and\nRon Nissim. These provide the Schur product theorem and entrywise exponential\npositivity results used in the OS3 reflection positivity proof.\n\n## Related work\n\n- Xi Yin, [Phi4](https://github.com/xiyin137/Phi4) — Formalization of φ⁴ quantum\n  field theory in Lean 4\n\n## Future projects\n\n- **Unified OS axiom framework.** Currently the infinite-volume OS axioms\n  (`OSAxioms.lean`) and torus OS axioms (`TorusOSAxioms.lean`) are defined\n  separately. These should be unified into a single parametric `SatisfiesOS`\n  structure taking a `SpaceTime` parameter that encodes the geometry:\n  whether space is compact (torus vs ℝ², controlling ergodicity/clustering),\n  whether a distinguished time direction exists (enabling reflection positivity),\n  the symmetry group (E(2) vs translations × D4), and so on. The OS axioms\n  and other consistency tests (e.g. moment bounds, support properties) would\n  then be stated once and instantiated for each spacetime. Both the Gaussian\n  and interacting measures would be verified against the same axiom bundle.\n\n## Author\n\nMichael R. Douglas and collaborators\n\n## License\n\nCopyright (c) 2026 Michael R. Douglas. Released under the Apache 2.0 license.\n",1782661975637]