{
  "schema_version": 2,
  "cell_id": "ASTIS-SHARED-canonical-fisher-transport-pairing",
  "route": "shared",
  "title": "Canonical relative score paired with an optimal coupling",
  "mode": "exploratoryProof",
  "source_anchor": "Chewi, Log-Concave Sampling, August 9 2026, SHA256 9b454ccf44fe700081e13a766ae9cabb83c3530f5fdc532d59ac335f53652597, Theorem 8.4.1 proof printed p.221/PDF p.233, Cauchy-Schwarz line. Shared analytic prerequisite only, not the numbered theorem.",
  "target_statement": "For a coupling gamma of finite-second-moment measures mu and nu, the pairing of gradient(logRatio mu pi)(x) with y-x is integrable whenever the existing smooth finite score domain holds. If gamma is quadratic-optimal, the absolute pairing integral is at most sqrt(information mu pi hscore) times wassersteinDistance mu nu toReal. Measurability, integrability and finite real cost identification are proved, not added as pairing premises.",
  "status": "proved_locally",
  "parents": [
    "Analysis.MeasurableGradient.measurable_gradient",
    "CanonicalRelativeFisher.SmoothFiniteScoreDomain",
    "CanonicalRelativeFisher.information_eq_integral_scoreSq",
    "CouplingQuadraticIntegrability.measurePreserving_fst_of_isCoupling",
    "CouplingQuadraticIntegrability.integrable_norm_sub_sq_of_isCoupling",
    "DisplacementRealQuadraticCost.integral_norm_sq_eq_wassersteinDistance_sq_toReal_of_optimal",
    "MeasureTheory.memLp_two_iff_integrable_sq_norm",
    "MeasureTheory.integral_mul_norm_le_Lp_mul_Lq"
  ],
  "consumers": [
    "samplewiki-route",
    "statistical-optimal-transport"
  ],
  "shared_floor_audit": {
    "searched": [
      "Samplinglib Probability/SDE and TechnicalLemmas InformationTheory/Measure/Analysis, module cards and active shared Frontier Cells",
      "Mathlib db584cd6d46c92f209a44c0f1c829460d327499d MeasureTheory Function L2Space/L1Space and Bochner Basic",
      "ATLAS pinned memory: separate fisher and wasserstein samplewiki-route queries returned empty JSON lists",
      "Existing external-lean-libraries/log-concave-sampling-notes.md is roadmap only; outer_repos and outer_papers are absent, not checkout-searched"
    ],
    "classification": "missing",
    "decision": "new_canonical_shared",
    "canonical_declaration": "AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalFisherTransportPairing.abs_integral_pairing_le_sqrt_information_mul_wasserstein",
    "canonical_shared_cell": "ASTIS-SHARED-canonical-fisher-transport-pairing",
    "notes": "Existing Holder is reused inside the canonical RN score / coupling marginals / optimal W2 cost join. No new Fisher hierarchy, Wasserstein definition, generic Holder alias or assumed first-variation structure."
  },
  "evidence": {
    "focused_checks": [
      "ELAN_TOOLCHAIN=leanprover/lean4:v4.33.0 LEAN_NUM_THREADS=1 lake build AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalFisherTransportPairing AutoSamplingTheory.TechnicalLemmas.InformationTheory.GeodesicFisherTransport: PASS exit 0, 3519 jobs; new public module built on first attempt in 19s",
      "Same pinned environment, lake build Tests.CanonicalFisherTransportPairing: PASS exit 0, 3520 jobs; final tests built in 19s after adding ENNReal notation scope",
      "Test separately checks arbitrary-coupling integrability, finite optimal W2 without probability/SFinite, and a canonical-KL/geodesic consumer eliminating hcs with first variation and metric representation still explicit inputs",
      "Both public declarations depend only on propext, Classical.choice, Quot.sound",
      "Scoped fake-closure/trailing-whitespace scan has no matches; scoped git diff --check clean"
    ],
    "independent_verification": "Independent PROOF GATE PASS at 74243f263bf4ed5fd632f4f3a0950b58ed77b4d7, exact frozen hashes, fresh-artifact integration and four edge checks; see runs/20260908-samplewiki-resume/fisher-transport-independent-review.md. Source-specific certification WITHHELD: independent source result fisher-transport.source-review-result.json has domain-mismatch / needs-revision, immutable SHA256 d32e17f0b765cbfe12c3c3155c2c11ab3acb3c18aa05864a973bd8d812a0a1ae. Advance remains PROVED_LOCAL and cell proved_locally; no VERIFIED or assimilation.",
    "root_build": "Sole stabilization owner",
    "graph_regeneration": "Sole stabilization owner",
    "pr": ""
  },
  "blocked": {
    "reason": "",
    "children": []
  },
  "source_detail_audit": {
    "primary_edition": "Chewi Log-Concave Sampling August 9 2026; recovered author PDF exactly matches canonical edition SHA256 9b454ccf44fe700081e13a766ae9cabb83c3530f5fdc532d59ac335f53652597",
    "primary_anchor": "Theorem 8.4.1 proof printed p.221/PDF p.233: first-order KL inequality followed by rearrangement and Cauchy-Schwarz. PDF233 and234 visually reviewed in recovered canonical edition; current August20 edition is not the source pin.",
    "detail_status": "omitted",
    "gap": "The source writes an optimal-map expectation and finite Fisher formula without the formal score-measurability, L2 pairing-integrability and real/extended transport-cost adapters. The independent KL first variation is not proved by Cauchy-Schwarz.",
    "consulted": [
      {
        "source": "Pinned Mathlib db584cd6d46c92f209a44c0f1c829460d327499d, Apache 2.0",
        "anchor": "L2Space.lean memLp_two_iff_integrable_sq_norm; L1Space/Integrable.lean MemLp.integrable_mul; Bochner/Basic.lean integral_mul_norm_le_Lp_mul_Lq and integral_map",
        "hypothesis_adapter": "Pull canonical squared score through the first marginal, use finite second moments for displacement L2, derive product/pairing integrability, and identify optimal cost with the existing W2 definition. Generic coupling statement includes graph couplings but does not construct an optimal map."
      }
    ],
    "fidelity_boundary": "One supplementary analytic coupling-pairing edge only, with independent proof PASS but source-specific certification withheld by domain-mismatch / needs-revision; see runs/20260908-samplewiki-resume/fisher-transport-independent-review.md and immutable fisher-transport.source-review-result.json. The source probability/P2 laws, nu=pi optimal graph-coupling/expectation specialization, and selected RN score/Fisher bridge are not supplied. Uses the existing chosen RN total-gradient smooth finite branch, not a Sobolev or infinite-energy extension. No source repair or domain equivalence is accepted. Route priority remains representative/score-Fisher identification, finite-entropy source-law domain, then endpoint one-sided versus two-sided first variation; the actual klDiv.toReal consumer is only a conditional integration test. KL convexity, metric/law representation, optimal coupling existence, actual heat-flow domain and Chewi 8.4.1 remain unproved here. Advance PROVED_LOCAL and cell proved_locally are unchanged; no VERIFIED or assimilation."
  },
  "conceptual_mirror_audit": {
    "status": "none-found",
    "discovery_ids": [],
    "reason": "Existing family:gap-gradient and transport:pl-lsi already retain the conceptual metric-gradient pattern. This edge is literal L2 Holder plus marginal/cost transport, not a new cross-domain correspondence."
  }
}