{ "id": "ASTIS-RT-20260908-RandomScanReversibility", "state": "source-reviewed", "graph_node": "ASTIS-SHARED-random-scan-reversibility", "source": { "source_id": "lpw:mcmt2e-boolean-glauber", "anchor": "Levin–Peres, Markov Chains and Mixing Times, second edition, Section 3.3.2, printed pp.42–43 / PDF pp.58–59, equations (3.6)–(3.7), and Exercise 3.2, printed p.45 / PDF p.61. Boolean specialization of the reversibility clause only; https://pages.uoregon.edu/dlevin/MARKOV/mcmt2e.pdf#page=58 .", "original_text": "ASTIS faithful mathematical paraphrase of the Boolean specialization, not a quotation: Let V be a finite nonempty set of sites, let μ be a probability distribution on Boolean configurations, and let Ω be its positive-probability support. In one update from x in Ω, select a site uniformly, retain the other sites at their own old values, and draw the selected site's value from μ conditional on those retained values. This Glauber chain is reversible for μ: target-weighted transition flow from any subset A of Ω to any subset B equals the reverse flow. The finite site labels can be represented by Fin(n+1). This target records the reversibility clause, not the source's arbitrary finite-spin-alphabet generality, its separate stationary-law consequence, or any convergence claim.", "text_sha256": "e30f278b420b6fd8066900cd42ab73dd8dabedcc31ab146c9aa9f822f2a2301f", "wording_status": "faithful ASTIS paraphrase of explicitly selected Boolean reversibility clause; not original source text", "pdf_sha256": "9ef39f9467d9647ff3f5e8747b9ce24b7a90d13be2f8156fbd827b95b661a772", "pdf_url": "https://pages.uoregon.edu/dlevin/MARKOV/mcmt2e.pdf" }, "lean": { "declaration": "AutoSamplingTheory.TechnicalLemmas.Probability.RandomScanHeatBath.randomScan_isReversible", "file": "AutoSamplingTheory/TechnicalLemmas/Probability/RandomScanHeatBathReversibility.lean", "statement": "∀ {n : ℕ}\n (μ : MeasureTheory.Measure (Fin (n + 1) → Bool)) [inst : MeasureTheory.IsProbabilityMeasure μ],\n (AutoSamplingTheory.TechnicalLemmas.Probability.RandomScanHeatBath.randomScan μ).IsReversible μ", "statement_sha256": "0fc768fc777e10db15f906ad2952d4b78141c33d86379d2da1bbdcd79453eee6", "compiled": true, "formalizer": "random_scan_reversibility_worker", "decoder_context": [ "Fin(n+1) has n+1 sites and Bool has two values, with the product of discrete measurable structures. μ is a measure on this finite configuration space; IsProbabilityMeasure μ means μ(univ)=1.", "For i:Fin(n+1), let r_i(x) be the retained tuple (x(i.succAbove j)) indexed by j:Fin n. Let e_i(x)=(r_i(x),x i); this is the measurable coordinate-splitting equivalence. Its inverse inserts the selected value into the retained tuple.", "Write ν_i=(e_i)_#μ, with # meaning pushforward. Let C_i be the library-selected regular conditional Markov distribution of the second component given the first under ν_i. Define H_i(x)=(e_i⁻¹)_#(δ_{r_i(x)}⊗C_i(r_i(x))). Here δ is a Dirac probability and ⊗ is the product measure. The conditional-distribution characterization is only first-marginal-almost-everywhere; no normalized-fiber value is stipulated at null retained inputs.", "The kernel named randomScan μ in the statement is the finite sum of H_i with the constant, state-independent weight 1/(n+1) for every i. The construction uses the corresponding NNReal constant densities on component kernels.", "Kernel.IsReversible K μ is defined as follows: for every pair of measurable sets A and B, the nonnegative set integral of x↦K(x,B) over A under μ equals the nonnegative set integral of y↦K(y,A) over B under μ. Both integrals take values in the extended nonnegative reals." ] }, "reconstruction": { "text": "For every natural number n and every probability measure μ on the finite binary configuration space X = {x : Fin(n+1) → Bool}, equipped with the product discrete measurable structure, the following uniformly weighted random-scan kernel K is reversible with respect to μ. For each site i, let r_i(x) retain all coordinates other than i, let e_i(x) = (r_i(x), x(i)) be the coordinate-splitting measurable equivalence, and let C_i be the library-selected regular conditional Markov distribution of the second coordinate given the first under the pushforward (e_i)_#μ. Define H_i(x) = (e_i⁻¹)_#(δ_{r_i(x)} ⊗ C_i(r_i(x))) and K(x) = Σ_{i∈Fin(n+1)} [1/(n+1)] H_i(x). The conditional-distribution characterization of C_i holds only almost everywhere with respect to the retained-coordinate marginal; no particular normalized-fiber value is specified at null retained inputs. Then for every pair of measurable sets A,B ⊆ X, the extended-nonnegative set integrals satisfy ∫⁻_{x∈A} K(x,B) dμ(x) = ∫⁻_{y∈B} K(y,A) dμ(y).", "text_sha256": "b246529138f2137cffaf162b57aa08a81a7b70c160731b42ba5cd817f5e43080", "decoder": "/root/reversibility_blind_decoder; independent blind reconstruction using only the supplied decoder packet", "decoder_run_sha256": "762441256054127d9c046b027d4ee68c9efee454280fff0ba5f4f06e896f8c3d", "decoder_packet_sha256": "ac2608ec5077c6e91f5748f3d8cd3c5e09e54f7be05f1e42258d13a1b89c692d", "source_text_visible": false, "lean_statement_sha256": "0fc768fc777e10db15f906ad2952d4b78141c33d86379d2da1bbdcd79453eee6", "input_artifacts": [ "lean-statement", "approved-definition-context" ], "run_artifact": "runs/20260908-samplewiki-resume/reversibility.decoder-result.json", "publication_note": "Before persistence, the same decoder explicitly approved replacing only the workstation-specific input_packet_path prefix with this repository-relative path. All semantic text and hashes were unchanged." }, "semantic_slots": { "objects": { "original": "Section 3.3.2 defines a single-site Glauber chain: choose a site uniformly, preserve all other coordinates, and sample from the target distribution conditioned on that retained configuration. The selected target specializes the finite spin alphabet to Bool.", "reconstructed": "The kernel splits off one Boolean coordinate, preserves the retained coordinates through a Dirac measure, resamples the selected coordinate using a library-selected conditional Markov distribution, reconstructs the configuration, and averages these kernels uniformly.", "relation": "explicit-elaboration", "evidence": "PDF p.58 / printed p.42, equation (3.6), fixes every coordinate other than the selected site; PDF p.59 / printed p.43, equation (3.7), gives the normalized conditional distribution. The packet's blind_reconstruction describes the same retain/resample/reassemble operation. On every positive-mass retained configuration, finite-space conditional distributions satisfy that normalized formula pointwise." }, "domains": { "original": "The source chain has state space equal to the positive-probability support of a distribution on S^V. Within the selected Boolean specialization, this is Ω ⊆ Bool^V.", "reconstructed": "The kernel is defined on the entire finite discrete cube X = Fin(n+1) → Bool, with an arbitrary probability measure μ that need not give positive mass to every configuration.", "relation": "explicit-elaboration", "evidence": "PDF p.58 / printed p.42 explicitly makes the source state space the support and restricts equation (3.6) to states in that support. The packet reconstruction instead uses the ambient cube. If μ({x}) > 0, every retained-coordinate fiber through x has positive mass; equation (3.7) then identifies the reconstructed update and shows it remains in Ω. Consequently its restriction to Ω is the source chain, while μ-weighted transition integrals ignore ambient starting states outside Ω." }, "quantifiers": { "original": "For every finite nonempty site set and every Boolean-configuration probability distribution, the resulting Glauber chain is reversible. The selected paraphrase expresses reversibility for every pair of subsets of its support.", "reconstructed": "For every natural n and every probability measure μ on Fin(n+1) → Bool, the selected random-scan kernel satisfies the equality for every pair of measurable subsets of the ambient cube.", "relation": "equivalent", "evidence": "The source's general construction on PDF pp.58–59 ranges over finite site sets and probability distributions. Every finite nonempty site set admits a relabeling by Fin(n+1). Every subset of a finite discrete cube is measurable. The ambient-set equality reduces to the equality for intersections with Ω, because the measure is concentrated on Ω and updates starting in Ω stay there." }, "assumptions": { "original": "Finite site and spin sets, a probability distribution, and use of its support as the chain's state space. Nonemptiness of the site set is implicit in selecting a site uniformly. Positive support ensures that each conditioning denominator used from a chain state is positive.", "reconstructed": "Finite nonempty sites and Boolean spins are encoded in the type; μ is assumed to be a probability measure. No full-support assumption on the ambient cube is imposed. Conditional characterization is only almost everywhere under each retained-coordinate marginal.", "relation": "explicit-elaboration", "evidence": "PDF p.58 states finiteness and the support condition and prescribes uniform site selection. Equations (3.6)–(3.7) imply π(X(x,v)) ≥ π(x) > 0 for a source state x. In the finite reconstruction, almost-everywhere conditional characterization therefore determines the conditional law at every retained input arising from positive-support states. No irreducibility, graph structure, independence, or ambient strict-positivity assumption is needed for the selected clause." }, "conclusion": { "original": "The selected clause states reversibility with respect to the target distribution for the support-state Glauber chain.", "reconstructed": "For every measurable A,B in the ambient cube, the extended-nonnegative integrals of K(x,B) over A and K(y,A) over B against μ are equal.", "relation": "equivalent", "evidence": "PDF p.59 / printed p.43 states reversibility, and Exercise 3.2 on PDF p.61 / printed p.45 asks for its verification. Independently, equations (3.6)–(3.7) give the symmetric single-site weighted flow π(x)π(y)/π(X(x,v)) when x and y agree off v, and zero otherwise. Uniform averaging preserves symmetry. Finite summation gives the set-integral equality, and singleton sets recover detailed balance. Both flow integrals are bounded by one." }, "scopes": { "original": "The inspected source covers arbitrary finite spin alphabets and states both stationarity and reversibility. The packet explicitly pins only the Boolean specialization of the reversibility clause.", "reconstructed": "One-step Boolean random-scan reversibility, expressed for a selected conditional-kernel construction. The displayed declaration does not separately assert stationarity, convergence, mixing, or general finite-alphabet coverage.", "relation": "equivalent", "evidence": "PDF p.58 introduces arbitrary finite S; PDF p.59 and Exercise 3.2 state both stationarity and reversibility. The packet source.anchor and source.original_text explicitly exclude arbitrary-spin-alphabet generality and the separate stationary-law consequence. The supplied declaration and reconstruction match that selected scope, not the complete breadth of the source statement." }, "constant_dependencies": { "original": "Each discrete update selects exactly one site uniformly, with weight 1/|V|, and uses a conditional law determined by the target distribution and retained coordinates.", "reconstructed": "There are n+1 sites, each receiving mixture weight 1/(n+1). Conditional kernels depend on μ and the selected site; the scan weight does not depend on μ or the current configuration.", "relation": "equivalent", "evidence": "Uniform site selection is prescribed on PDF pp.58–59. The packet reconstruction explicitly gives K as the sum of n+1 single-site kernels with coefficient 1/(n+1). Thus one kernel application corresponds to one source update, not an entire sweep or a rescaled continuous-time clock. No additional rate or quantitative bound occurs in the selected conclusion." } }, "deltas": [ { "id": "D1", "slot": "scopes", "severity": "informational", "description": "The formal target is a Boolean specialization, not a formalization of the source's arbitrary finite-spin-alphabet generality. This restriction is already explicit in the pinned target and is acceptable within that scope.", "evidence": "PDF p.58 / printed p.42, Section 3.3.2, allows an arbitrary finite set S. The packet source.anchor identifies a Boolean specialization; lean.statement and blind_reconstruction use Bool." }, { "id": "D2", "slot": "domains", "severity": "informational", "description": "The source chain lives on positive support, whereas the reconstructed kernel lives on the full Boolean cube. These are not literally identical state-space presentations. Restriction to positive support identifies the source chain, and the ambient extension leaves the selected μ-weighted reversibility statement unchanged.", "evidence": "PDF p.58 explicitly sets the source state space equal to the support. Equations (3.6)–(3.7) restrict its updates to support configurations. The packet reconstruction admits arbitrary μ on the full cube. For positive-support x, the retained fiber has mass at least μ({x}) > 0, so the reconstructed conditional update assigns zero mass outside support; ambient source states outside support have zero μ-weight." }, { "id": "D3", "slot": "objects", "severity": "informational", "description": "The reconstruction uses a selected regular conditional version, whose values at null retained inputs are not specified by normalized fibers. The source never needs those values. This does not permit a claim of pointwise equality with a source transition rule at every ambient null state. Null current states with positive-mass retained contexts still have their corresponding single-site conditional laws determined.", "evidence": "Equation (3.7) on PDF p.59 defines a normalized fiber law only for source states introduced in equation (3.6), whose fibers have positive mass. The packet blind_reconstruction explicitly limits its conditional characterization to almost everywhere under the retained marginal. On finite positive-mass retained atoms this determines the law exactly; a null retained fiber contains no positive-support starting state and therefore contributes no μ-weighted starting flow." }, { "id": "D4", "slot": "assumptions", "severity": "informational", "description": "Nonempty sites are made explicit through Fin(n+1). The source's prose says finite sites without separately spelling out nonemptiness, but its instruction to choose a site uniformly presupposes it. No empty-site interpretation is certified here.", "evidence": "PDF p.58 / printed p.42 first introduces finite V and then instructs the chain to select a vertex uniformly from V. The packet's pinned paraphrase explicitly says nonempty, and lean.statement uses Fin(n+1), whose cardinality is always positive." }, { "id": "D5", "slot": "conclusion", "severity": "informational", "description": "Finite-state reversibility is elaborated into an equality of extended-nonnegative set integrals over arbitrary ambient measurable sets. This is equivalent for the selected target, rather than an additional convergence or trajectory-level conclusion.", "evidence": "PDF p.59 and Exercise 3.2 assert reversibility. Equations (3.6)–(3.7) produce symmetric target-weighted single-site transition probabilities; summing over sites and then over pairs of states yields the reconstructed integral equality. Conversely, singleton sets recover the pairwise equality. Finite discrete measurability and the support reduction described in D2 justify the ambient-set formulation." }, { "id": "D6", "slot": "scopes", "severity": "informational", "description": "The source also states stationarity, but the supplied declaration records only the explicitly selected reversibility clause. Acceptance therefore does not certify that this declaration separately formalizes every assertion in Section 3.3.2 or Exercise 3.2.", "evidence": "PDF p.59 / printed p.43 and Exercise 3.2 on PDF p.61 / printed p.45 both mention stationarity together with reversibility. The packet source.original_text expressly excludes the separate stationary-law consequence, and lean.statement concludes only IsReversible μ." } ], "verdict": "equivalent-after-elaboration", "source_review": { "state": "accepted", "reviewer": "/root/reversibility_source_review; independent source-facing semantic reviewer", "independent_from_formalizer": true, "independent_from_decoder": true, "evidence": "Reviewed only the anti-anchored input packet, the mandatory PDF skill, and the specified primary PDF. Source: :codex-file-citation{path=\".astis/source-cache/levin-peres-mcmt2e.pdf\" purpose=\"source\"}. Independently hashed the local PDF bytes: SHA256 9ef39f9467d9647ff3f5e8747b9ce24b7a90d13be2f8156fbd827b95b661a772, matching the separately supplied neutral provenance digest; that digest was not treated as a generated-packet field. Rendered and visually inspected complete PDF pages 58, 59, and 61, corresponding to printed pages 42, 43, and 45, and corroborated their text with pypdf extraction. Findings concern Section 3.3.2, equations (3.6)–(3.7), and Exercise 3.2 within the selected Boolean reversibility scope. Recomputed the packet hash after removing packet_sha256 and serializing with sorted keys, compact separators, ensure_ascii=false, and UTF-8; it exactly matched the claimed canonical hash. No implementation, proof, source map, previous audit, previous verdict, or repair proposal was accessed. The packet's compiled flag was not independently verified. This result is semantic comparison of the supplied statement and reconstruction with the selected source, not proof certification or an audit of hidden definitions.", "review_run_sha256": "5d2cfd3266f530d054cdbcefe087578ffe5dea8b6a9163ef44f3a9d3d6592528", "reviewer_packet_sha256": "eb5e08c10f3061c782b7adedad1cc35d183b0d0776c92482cecc385809f098b6", "run_artifact": "runs/20260908-samplewiki-resume/reversibility.source-review-result.json", "interpretation": "Accepted only for the explicitly selected Boolean specialization of the reversibility clause. The arbitrary finite-spin-alphabet statement and separate source conclusions are not fully formalized by this declaration." }, "repairs": [] }