{ "schema_version": 2, "cell_id": "ASTIS-SHARED-random-scan-heat-bath", "route": "shared", "title": "Uniform random-site heat-bath singleton transition law", "mode": "exploratoryProof", "source_anchor": "Discrete Sampling arXiv:2307.13826v4 section 1.1 PDF4 defines positive atom support; section 1.3 PDF5 general update starts in that support and chooses a site uniformly. MCMC2407.12751v1 section2.1.1 printed48/PDF54 motivates conditional component moves. Exact source fidelity remains separately reviewed.", "target_statement": "For a probability measure mu on Fin(n+1)->Bool and mu{x} nonzero, randomScan mu x {y} equals (n+1)^-1 times the sum over sites i of the normalized target singleton mu{y}/mu(F_i(x)) if every j!=i has y j=x j, and zero otherwise. The actual randomScan reuses finiteMixture with uniform NNReal weights and the existing coordinate heatBath.", "status": "independently_verified", "parents": [ "AutoSamplingTheory.TechnicalLemmas.Probability.KernelMixture.finiteMixture", "AutoSamplingTheory.TechnicalLemmas.Probability.KernelMixture.finiteMixture_apply", "AutoSamplingTheory.TechnicalLemmas.Probability.KernelMixture.finiteMixture_isMarkovKernel", "AutoSamplingTheory.TechnicalLemmas.Probability.CoordinateHeatBath.heatBath", "AutoSamplingTheory.TechnicalLemmas.Probability.CoordinateHeatBath.heatBath_eq_cond", "MeasureTheory.measure_mono", "ProbabilityTheory.cond_apply'", "Fin.exists_succAbove_eq", "Fin.succAbove_ne" ], "consumers": [ "mcmc", "discrete-sampling" ], "shared_floor_audit": { "searched": [ "Samplinglib TechnicalLemmas/Probability and source-present module cards: KernelMixture, CoordinateHeatBath, CoordinateHeatBathConditional, KernelInvariance; existing focused tests and compact verified kernel-cell synthesis", "Active shared Frontier Cells, Libraries/shared-foundations.yml and frontloaded-shared-spine.json; no operational uniform-scan singleton law found", "Mathlib db584cd6d46c92f209a44c0f1c829460d327499d, Apache2.0: ConditionalProbability.lean cond_apply', Measure finiteSum/smul and measure_mono; Data/Fin/SuccPred.lean exists_succAbove_eq/succAbove_ne; ENNReal/Inv.lean coe_inv (requires nonzero denominator)" ], "classification": "adapt", "decision": "new_canonical_shared", "canonical_declaration": "AutoSamplingTheory.TechnicalLemmas.Probability.RandomScanHeatBath.randomScan_apply_singleton", "canonical_shared_cell": "ASTIS-SHARED-random-scan-heat-bath", "notes": "One substantive integration node, including construction and Markov adapter; no duplicate kernel mixture, resampling, generic positive-fiber or invariant-power wrapper. KernelInvariance and invariance facts are consumer-test inputs, not new proof parents of the singleton law." }, "evidence": { "focused_checks": [ "Early direct public theorem check PASS with Lean4.33.0 and LEAN_NUM_THREADS=1; public module built16s.", "Final lake build Tests.RandomScanHeatBath PASS exit0,2982 jobs; tests built21s. Exact construction, Markov instance and singleton theorem signatures printed; only propext, Classical.choice and Quot.sound axioms.", "Compiled actual-kernel uniform two-bit row: stay1/2, each one-bit change1/4, two-bit change0. Diagonal positive-support target permits forbidden ambient states and is absorbing at a supported start.", "Native supported-atom inclusion proves nonzero finite denominators; a nonzero Dirac target at an invalid ambient start has null fibers, and actual Markov randomScan differs from the zero normalized-fiber mixture. One-site n=0 case and existing invariance/power consumer pass.", "Scoped fake-closure, trailing-whitespace and diff checks clean; Frontier Cell checker PASS9 cells." ], "independent_verification": "2026-09-08: random_scan_proof_review independently accepted frozen proof commit 90581ec87c039ae9597f76d3a434eb08e14aa579, distinct from random_scan_worker and random_scan_decoder. Public/test Git blobs and SHA256 matched before and after review. Fresh direct public and independent edge elaborations passed with explicit Lean4.33.0 and LEAN_NUM_THREADS=1; exactly one final Tests.RandomScanHeatBath focused build passed, 2982 jobs (unchanged cached test replay). Public and added theorem axioms are only propext, Classical.choice, Quot.sound; canonical scoped fake-closure scan clean. Added generic forbidden-destination and two-distinct-changes zero laws plus actual nonuniform target transition P(00,10)=3/8. Independently inspected final ASTIS-RT-20260908-RandomScanLaw source-reviewed/accepted equivalent-after-elaboration audit, verified immutable source response b62914b54efc84122813b1530f2175ea35bdf6f170861bc1bdd8c67d14410935 and same-reviewer mechanical amendments; seven slots/five original deltas/verdict unchanged. Fresh semantic gate PASS2 audits/1repair. Acceptance covers only the cited positive-support explanatory one-update target, NOT literal printed step2/full numbered algorithm, null-fiber normalized law, executable sampler, reversibility, mixing, chapters, Registry integration or stabilization. Detailed report: runs/20260908-samplewiki-resume/random-scan-independent-review.md.", "root_build": "Owned focused modules only; sole stabilization lane owns public aggregation and root gate.", "graph_regeneration": "Root owns source card, diagram and graph integration.", "pr": "" }, "blocked": { "reason": "", "children": [] }, "source_detail_audit": { "primary_edition": "Discrete2307.13826v4 SHA2563cc2f911b33bb5538157ef8a70f0c7e0f3c812ecd06dc9c1d5ea0bfdae11a52a; MCMC2407.12751v1 SHA25660ecdd243a5815771332b5461e7eae59d41242933e53a721785870b6877f008f. Worker independently byte-checked both originals and visually inspected complete discrete p4/p5 and MCMC p54.", "primary_anchor": "Discrete section1.1 support definition PDF4 and section1.3 general three-step update PDF5; MCMC section2.1.1 conditional Gibbs proposal PDF54.", "detail_status": "omitted", "gap": "Combine the already-defined coordinate conditional law with actual uniform site selection, and derive each positive retained fiber from source-supported x by singleton inclusion. No extra positive-fiber assumption is supplied. The source copy-index discrepancy has a separate exact-proposal review; it is not silently corrected by this Lean proof.", "consulted": [ { "source": "Pinned Mathlib db584cd6d46c92f209a44c0f1c829460d327499d, Apache2.0", "anchor": "ConditionalProbability.lean216-221 cond_apply/cond_apply'; Data/Fin/SuccPred.lean succAbove_ne and exists_succAbove_eq; ENNReal/Inv.lean67 coe_inv", "hypothesis_adapter": "Bool carries discrete measurable, measurable singleton, nonempty and StandardBorel instances. Probability gives finite target; source support supplies nonzero singleton and monotonicity supplies every fiber. The n+1 index makes the uniform denominator nonzero. Operational all-other-site equality is equivalent to native retained tuple equality." } ], "fidelity_boundary": "Source-facing finite Boolean probability specialization and one uniform update. Graph vertices are represented by Fin(n+1), spins by Bool, source support by mu{x}!=0 within the full cube. Canonical audit ASTIS-RT-20260908-RandomScanLaw independently accepts the explicitly cited explanatory one-update target as equivalent-after-elaboration. This does not certify literal printed step2 or the full numbered algorithm; the historical limited-context discrepancy audit is not overwritten. No null-fiber law, concrete Gibbs normalization, general-state MH, reversibility, mixing or executable cost is asserted." }, "semantic_roundtrip_required": true, "conceptual_mirror_audit": { "status": "none-found", "discovery_ids": [], "reason": "Reviewed conceptual protocol, family:proximal-energy, family:conditional-dependence and transport:mcmc-gibbs-coordinate. The new step is literal finite-mixture plus atomic-conditioning reuse, not a new cross-domain analogy. Existing block optimization candidate is not certified or duplicated." }, "mcmc_contract": { "state_space": "Finite Boolean cube Fin(n+1)->Bool with its standard discrete measurable structure.", "target": "Any probability measure on the ambient cube, zero singleton masses allowed.", "support": "Actual source state space Omega is the positive atom support; law theorem requires mu{x}!=0, not full ambient positivity.", "operator": "One actual randomScan kernel: finiteMixture of coordinate heatBath laws with fixed reciprocal(n+1) NNReal weights; action on output laws.", "time_model": "discrete-time", "clock": "Exactly one uniformly selected site per transition, not a sweep or continuous-time rate.", "invariance": "Existing finiteMixture_invariant and heatBath_invariant consumed only in tests; no new invariant wrapper.", "invariance_class": "exact-target", "ergodicity": "Not assumed or concluded; supported targets can be disconnected under single-site moves.", "initialization": "Positive target singleton at x; each conditioning denominator proved nonzero by inclusion.", "metric": "Exact pointwise singleton transition probabilities, not an error rate.", "cost": "Noncomputable measure-level interface; no algorithmic conditional sampler or work guarantee.", "objective": "Identify exact one-move random-scan transition law on source-valid starts.", "source_proof_status": "Uniform choice and conditional move are source algorithm steps. Independent ASTIS-RT-20260908-RandomScanLaw accepts only the cited explanatory one-update target as equivalent-after-elaboration; complete printed algorithm remains outside acceptance.", "periodicity": "No aperiodicity, absolute-gap or convergence claim is needed for this one-step identity." }, "discrete_state_contract": { "state_space": "Fin(n+1)->Bool, n natural, with at least one site.", "support": "Positive target singleton support in the ambient cube; forbidden ambient states permitted.", "target": "Probability normalization explicit; arbitrary law, not a constructed graph/temperature Gibbs density.", "operator": "Uniform finite mixture of actual coordinate heatBath transitions acting on laws.", "time_model": "discrete-time", "clock": "One random-site update; probabilities weighted exactly by reciprocal(n+1).", "reversibility": "Not asserted; exact singleton law is the only substantive new conclusion.", "pinning": "Retain all sites except i. Positive starting atom is contained in every retained fiber, yielding strictly positive finite denominators.", "regime": "Any finite Boolean probability target; no graph degree, temperature, uniqueness or mixing condition.", "metric": "Exact singleton transition law.", "cost": "No executable sampling or complexity claim.", "source_proof_status": "Source support and uniform selection audited on complete pinned pages; independent ASTIS-RT-20260908-RandomScanLaw accepts the expressly scoped explanatory one-update law, not the complete printed numbered algorithm.", "aperiodicity_or_absolute_gap": "Not applicable to one-step identification; no convergence theorem." } }