{
  "schema_version": 2,
  "cell_id": "ASTIS-SHARED-random-scan-reversibility",
  "route": "shared",
  "title": "Actual random-scan heat-bath reversibility through atomic balance",
  "mode": "exploratoryProof",
  "source_anchor": "Discrete2307.13826v4 section1.3 PDF5 general update; MCMC2407.12751v1 section2.1.1 printed48/PDF54 component Gibbs balance; Levin-Peres-Wilmer second edition section3.3.2 printed42-43/PDF58-59, equations3.6-3.7 and Exercise3.2 printed45/PDF61.",
  "target_statement": "For every probability mu on Fin(n+1)->Bool, Kernel.IsReversible (RandomScanHeatBath.randomScan mu) mu. Integrate actual supported-start law with zero-atom flux cases and a reusable countable measurable-singleton balance-to-set-lintegral bridge; no full support premise.",
  "status": "independently_verified",
  "parents": [
    "AutoSamplingTheory.TechnicalLemmas.Probability.RandomScanHeatBath.randomScan_apply_singleton",
    "MeasureTheory.lintegral_countable",
    "ENNReal.tsum_mul_right",
    "ENNReal.tsum_comm",
    "Fin.succAbove_ne"
  ],
  "consumers": [
    "mcmc",
    "discrete-sampling"
  ],
  "shared_floor_audit": {
    "searched": [
      "Samplinglib/ASTIS TechnicalLemmas/Probability, existing shared Frontier Cells and current RandomScanHeatBath card/next packet",
      "Libraries/shared-foundations.yml, frontloaded-shared-spine.json, DiscreteSampling/reuse-audit.json, external-lean-libraries and ATLAS reversibility query (no matching item)",
      "Pinned Mathlib db584cd6d46c92f209a44c0f1c829460d327499d Probability/Kernel/Invariance.lean and countable integration, Dirac/ext and ENNReal infinite sums"
    ],
    "classification": "adapt",
    "decision": "new_canonical_shared",
    "canonical_declaration": "AutoSamplingTheory.TechnicalLemmas.Probability.RandomScanHeatBath.randomScan_isReversible",
    "canonical_shared_cell": "ASTIS-SHARED-random-scan-reversibility",
    "notes": "No existing atomic-flux bridge found. Root approved one minimal generic KernelReversibility helper in this same integration packet. No external checkout search/clone is claimed: outer_repos and outer_papers are absent. Mathlib Apache2.0; no third-party Lean code port."
  },
  "evidence": {
    "focused_checks": [
      "KernelReversibility two-import generic bridge build PASS exit0,1883 jobs; no Markov/finite/probability assumptions.",
      "Actual RandomScanHeatBathReversibility direct first complete proof check PASS exit0; final public build15s.",
      "Final Tests.RandomScanHeatBathReversibility PASS exit0,2984 jobs,tests18s; Lean4.33.0 and LEAN_NUM_THREADS=1 explicitly set at every Lean invocation.",
      "Raw headerless #check output PASS exit0; both public axioms only propext, Classical.choice, Quot.sound.",
      "Infinite Nat counting kernel has total mass infinity and is reversible under infinite counting target; arbitrary kernel with zero target passes. Actual nonuniform transitions3/8 versus1/8, stationary weighted flux equality, arbitrary measurable sets, forbidden/null-fiber boundary, disconnected absorbing diagonal, n=0 and existing invariant-powers consumers compile.",
      "Scoped canonical fake-closure scan and trailing-whitespace checks clean; Frontier gate PASS10 cells."
    ],
    "independent_verification": "2026-09-08: reversibility_proof_review independently accepts frozen commit 865871bfe058e073421a478103bd5fc9f9b1a3eb as one actual-kernel integration node. Fresh direct two-module/full-test/independent-scratch elaborations PASS with explicit Lean4.33.0 and LEAN_NUM_THREADS=1; exact signatures, unrestricted generic bridge, null-atom branches, infinite-atom/zero-kernel edge cases and standard-only axioms checked. Canonical fake-closure scan clean; all three public/test bytes match commit and SHA256. Rational corroboration:1029 target/null-version cases,52896 exact flux pairs plus excluded-scan negative controls. Fresh independent semantic gate PASS3 audits/1repair; ASTIS-RT-20260908-RandomScanReversibility source-reviewed/accepted equivalent-after-elaboration, seven slots/six informational deltas exactly match immutable response5d2cfd3266f530d054cdbcefe087578ffe5dea8b6a9163ef44f3a9d3d6592528. Acceptance limited to LPW Boolean reversibility clause, not all finite alphabets, null-start pointwise normalization, mixing, Registry integration or stabilization. Report:runs/20260908-samplewiki-resume/reversibility-independent-review.md.",
    "root_build": "Sole stabilization lane only",
    "graph_regeneration": "Sole stabilization lane only",
    "pr": ""
  },
  "blocked": {
    "reason": "",
    "children": []
  },
  "source_detail_audit": {
    "primary_edition": "Discrete2307.13826v4 SHA2563cc2f911b33bb5538157ef8a70f0c7e0f3c812ecd06dc9c1d5ea0bfdae11a52a; MCMC2407.12751v1 SHA25660ecdd243a5815771332b5461e7eae59d41242933e53a721785870b6877f008f",
    "primary_anchor": "Discrete section1.3 PDF5; MCMC section2.1.1 printed48/PDF54, continuation printed49/PDF55",
    "detail_status": "omitted",
    "gap": "Discrete p5 explicitly asserts detailed balance in hard-core discussion, then defines general update. Its hard-core irreducibility/aperiodicity is not general-target evidence. MCMC proves fixed-block balance and distinguishes mixtures from compositions. Finite general random-scan reversibility is the standard derivation being integrated, with target-null ambient extension and set-integral bridge made explicit.",
    "consulted": [
      {
        "source": "Levin-Peres with Wilmer, Markov Chains and Mixing Times, second edition, author PDF https://pages.uoregon.edu/dlevin/MARKOV/mcmt2e.pdf, SHA2569ef39f9467d9647ff3f5e8747b9ce24b7a90d13be2f8156fbd827b95b661a772",
        "anchor": "section3.3.2 printed42-43/PDF58-59, equations3.6-3.7, Exercise3.2 printed45/PDF61",
        "hypothesis_adapter": "Finite nonempty site set represented by Fin(n+1), finite alphabet specialized to Bool; source X is positive support of target, represented within ambient cube. Matched fibers have equal mass; positive-start law only at nonzero atoms, arbitrary null versions disappear under zero flux. General finite alphabet and supported subtype kernel not claimed."
      },
      {
        "source": "Pinned Mathlib, Apache2.0",
        "anchor": "Kernel/Invariance.lean IsReversible and IsReversible.invariant; Lebesgue/Countable.lean lintegral_countable; InfiniteSum/ENNReal.lean tsum_mul_right and tsum_comm",
        "hypothesis_adapter": "Countable space with measurable singletons suffices for atomic bridge, without Markov or finite-mass assumptions. Actual Boolean probability specialization supplies required instances."
      }
    ],
    "fidelity_boundary": "One general-update standard derivation and its Boolean specialization, not exact arbitrary-target quotation from Discrete hard-core paragraph. Source copy-index overlay remains separate. Fresh independent audit ASTIS-RT-20260908-RandomScanReversibility is source-reviewed/accepted equivalent-after-elaboration for the expressly selected LPW Boolean reversibility clause only. Positive-support versus ambient-null and matrix versus set-integral presentations are explicit elaborations; arbitrary finite alphabets, all source conclusions, null-start pointwise conditional formulas, the CSV literal copy-index algorithm, irreducibility, aperiodicity, mixing, sweeps, continuous-time and cost remain excluded."
  },
  "semantic_roundtrip_required": true,
  "conceptual_mirror_audit": {
    "status": "none-found",
    "discovery_ids": [],
    "reason": "Reviewed conceptual policy and families, especially transport:mcmc-gibbs-coordinate, random-mixture versus composition failure boundary, finite Dirichlet and L2 coercivity. Current node is exact atomic algebra and actual-kernel integration, not a new cross-domain analogy or transport certificate."
  },
  "discrete_state_contract": {
    "state_space": "Finite Boolean cube Fin(n+1)->Bool, at least one site.",
    "support": "All probability targets, with arbitrary forbidden ambient zero atoms.",
    "target": "Probability normalization; no concrete Gibbs weight/graph/temperature construction.",
    "operator": "Existing randomScan kernel on laws, not a new kernel or matrix.",
    "time_model": "discrete-time",
    "clock": "One uniformly selected site per move, factor1/(n+1); not a sweep.",
    "reversibility": "Exact Mathlib set-lintegral IsReversible, obtained from all-ambient atomic flux symmetry.",
    "pinning": "Retained-coordinate fibers. Their positivity follows from positive start via existing law; no conditional formula asserted at null starts.",
    "regime": "Arbitrary finite Boolean probability target; no degree, temperature or uniqueness condition.",
    "metric": "Exact balance, not approximation or convergence rate.",
    "cost": "Noncomputable measure interface; no oracle/work bound.",
    "source_proof_status": "LPW section3.3.2/Exercise3.2 Boolean reversibility clause independently accepted via ASTIS-RT-20260908-RandomScanReversibility, equivalent-after-elaboration. General finite-alphabet breadth and source support versus ambient-null presentation remain explicit; Discrete hard-core scope and copy-index issue are separate, with no mixing claim.",
    "aperiodicity_or_absolute_gap": "Not assumed or proved; no convergence theorem. Disconnected support permitted."
  },
  "mcmc_contract": {
    "state_space": "Finite Boolean cube with measurable singletons.",
    "target": "Any probability measure, including zero target atoms.",
    "support": "Positive atoms define source support within ambient cube; no full support assumption.",
    "operator": "The existing uniform random-site conditional-resampling kernel on laws.",
    "time_model": "discrete-time",
    "clock": "One random site per step; no composition sweep or rate conversion.",
    "invariance": "Consequence through existing Mathlib IsReversible.invariant tested only, not a new wrapper or proof parent.",
    "invariance_class": "exact-target",
    "ergodicity": "Not assumed or concluded. Disconnected support counterexample remains valid.",
    "initialization": "No starting-state condition in weighted/set flux theorem; positive law invoked only in supported branches.",
    "metric": "Exact target flux, not TV/KL or estimator error.",
    "cost": "No executable sampling or work claim.",
    "objective": "Integrate detailed balance of actual kernel, not estimator analysis.",
    "source_proof_status": "MCMC fixed Gibbs block balance remains context. LPW section3.3.2/Exercise3.2 Boolean random-scan reversibility clause independently source-reviewed/accepted via ASTIS-RT-20260908-RandomScanReversibility, equivalent-after-elaboration. No all-alphabet, all-source-conclusion, null-start pointwise normalization or convergence claim.",
    "periodicity": "Not needed for this balance theorem; no aperiodicity/gap guarantee."
  }
}