{"id":"e8553161-2119-49ec-a65a-84fa1dac40ae","arxiv_id":"2607.11404","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Under computable compactness and overtness of invariant measures, the max ergodic average of a computable potential is computable and maximising measures form a Π1 set; for finite-range SFT potentials an explicit max-mean-cycle algorithm computes both exactly.","lead":"The paper proves that under standard computability assumptions on a dynamical system, the zero-temperature maximum ergodic average of a computable potential is a computable real, and the set of maximising measures is a Π1-computable compact. For finite-range potentials on one-dimensional SFTs it also supplies an explicit finite-time graph algorithm (with code) that recovers both quantities exactly.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the overtness hypothesis as the sole substantive limitation and notes that the authors already flag it. The abstract bounds follow immediately once φ* is known to be computable and MT is Π1 (or computably compact/overt). The algorithmic reduction is self-contained once the existence of a finite-range cohomologous h is granted by Bousch; the period-shortening step that produces a cycle of length ≤|Ar| is elementary graph theory and does not introduce new assumptions. The shipped repository further corroborates the concrete complexity claim. Consequently the ACCEPT verdict stands; no adjustment is warranted.","tokens_in":20610,"tokens_out":379,"duration_ms":4188,"concrete_test":"Independently re-derive the equality F(w)=β ⇔ w∈L(X) of Corollary 5.5 from the cohomologous function h of Bousch without invoking the period-shortening argument of the proof of Proposition 5.4; if the equivalence fails for some finite-range φ, the exact algorithm of Section 5 would be incomplete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims of Theorem 3.2 and the Section 5 algorithm rest on standard preservation properties of computable compactness/overtness (Propositions 2.12–2.15) and on a correct reduction of finite-range maximisation to max-mean-weight cycles on the De Bruijn graph (Proposition 5.4 and Corollary 5.5). The paper itself documents the non-uniformity of hypothesis 2○ (overtness of MT(X)) and supplies working code for the algorithmic part. No internal inconsistency or hidden gap that would falsify the stated bounds under the stated hypotheses was found.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies computability properties of zero-temperature ergodic optimisation. For a computable dynamical system (X,T) and computable continuous potential φ, under the hypotheses that X is computably compact and that the set MT(X) of invariant measures is computably overt, Theorem 3.2 shows that the maximum ergodic average β(φ) is a computable real number and that the set Mmax(φ) of maximising measures is a Π1-computable compact subset of M(X). In the special case of one-dimensional subshifts of finite type with finite-range potentials taking values in a computable ordered field F, Section 5 reduces the problem to maximum-mean-weight cycles on the associated De Bruijn graph and supplies an explicit algorithm of complexity O(|A|^{3r+1}) that computes both β(φ) and a finite set of forbidden patterns defining an SFT whose invariant measures are exactly the maximising ones; a matching public code repository is provided.","tokens_in":20750,"tokens_out":768,"duration_ms":14383,"significance":"The results cleanly transfer standard tools of computable analysis (computable compactness, overtness, and preservation under continuous maps and integration) to the setting of ergodic optimisation, thereby answering the natural zero-temperature counterpart of earlier positive-temperature computability results. The explicit polynomial-time algorithm for finite-range potentials on one-dimensional SFTs, together with working code, is a concrete and immediately usable contribution that goes beyond abstract upper bounds. The paper carefully documents the limitations of the overtness hypothesis, so the scope of the claims is transparent. Overall the work strengthens the bridge between computability theory and thermodynamic formalism.","major_comments":[],"minor_comments":[{"comment":"Abstract, first sentence: “physicals systems” is a typographical error for “physical systems”.","section":null},{"comment":"Section 2.3, Definition 2.4 and Remark 2.5: the distinction between the paper’s Πk closed sets and the classical effective descriptive-set-theoretic Π0k sets is useful, but a one-sentence pointer to the literature (e.g., Weihrauch or Brattka–Presser) would help readers less familiar with the hierarchy.","section":null},{"comment":"Section 5.2, Remark 5.2: the counter-example with λ is clear, yet the subsequent claim that one can still obtain a nested sequence of outer approximations Fn is left somewhat informal; a short explicit statement of the resulting algorithm would improve readability.","section":null},{"comment":"Section 5.3: the complexity bound O(|A|^{3r+1}) is stated after the description of the adapted Karp procedure; it would be helpful to isolate the precise arithmetic-operation count (or bit-complexity when F=Q) in a displayed equation or remark.","section":null},{"comment":"References: the arXiv identifier of the companion positive-temperature paper [GST25] is missing; adding it would aid readers.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is technically solid, self-contained, and accompanied by reproducible code. It is a natural and well-executed sequel to the authors’ earlier positive-temperature work. I see no reason to delay publication."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is simple: under the two standard hypotheses (computable compactness of X and computable overtness of the invariant measures), β(φ) is computable and the maximising set is Π1, and for finite-range potentials on one-dimensional SFTs they give an explicit max-mean-cycle algorithm that recovers both quantities exactly in finite time, with working code.\n\nWhat is actually new is the clean hierarchy table (Theorem 3.2) that organises the landscape, and the reduction of the SFT problem to finding all vertices that realise the maximum mean cycle weight on the De Bruijn graph (Proposition 5.4 + Corollary 5.5). The latter is a transparent adaptation of Karp, but they correctly identify the full set of maximising vertices rather than just one cycle, and they ship the implementation. The background sections on computable metric spaces and measures are standard but well-written; the survey of which classical models satisfy the hypotheses is honest and useful.\n\nSoft spots are minor and already flagged by the authors. Overtness of MT(X) fails for some computable maps (certain logistic parameters, non-computable SRB measures) and is undecidable for general 2-d SFTs, so the abstract bounds are not uniform; that is a real limitation of the setting, not a flaw in the proofs. The general approximation procedure is non-constructive in runtime, which is why the SFT section exists. No circularity, no hidden parameters, citations look solid (Weihrauch, Bousch, Karp, their own positive-temperature paper only for motivation).\n\nThis is for people who already care about computable dynamics or zero-temperature ergodic optimisation. It will not reshape statistical mechanics, but it is the right reference for anyone who wants to know what is computable and how to compute it in the symbolic case. I would send it to a serious referee without hesitation; the math holds and the code is there. Engage with it.","headline":"Clean, usable transfer of computable-analysis tools to zero-temperature ergodic optimisation, plus a working finite-time algorithm for the SFT case that people actually code.","tokens_in":21320,"tokens_out":513,"would_cite":true,"duration_ms":10427,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A50","37B10","03D78","68Q17"],"pacs":[],"model":"grok-4.5","headline":"For computable potentials on well-behaved systems, the zero-temperature maximum average is computable and the maximising measures form a Π₁ compact set; on one-dimensional SFTs both are given by a finite-time max-mean-cycle algorithm.","keywords":["ergodic optimisation","zero temperature","computable analysis","maximising measures","subshifts of finite type","maximum mean cycle","De Bruijn graph","computable dynamical systems"],"falsifier":"Exhibit a computable dynamical system that is computably compact, a computable potential, and a proof that the set of invariant measures is computably overt, yet the maximum ergodic average is not a computable real, or the set of maximising measures is not Π₁.","tokens_in":21525,"feed_emoji":"⚙️","tokens_out":730,"duration_ms":7969,"temperature":0.7,"pith_summary":"Zero-temperature ergodic optimisation asks which invariant probability measures maximise the long-run average of a continuous potential φ. The paper shows that when the underlying dynamical system is computably compact and its set of invariant measures is computably overt, any computable φ yields a computable maximum average β(φ) and a Π₁-computable compact set of maximising measures. In the concrete setting of one-dimensional subshifts of finite type with finite-range potentials taking values in a field that permits exact arithmetic, both quantities become exactly computable: the maximising support is itself an SFT whose forbidden patterns are recovered by a maximum-mean-cycle algorithm of complexity O(|A|^{3r+1}). The result therefore turns an abstract optimisation problem into an explicit graph-theoretic computation, with a matching open-source implementation.","feed_headline":"Zero-temperature maximisers become computable on SFTs","feed_subtitle":"Finite-range potentials reduce to max-mean cycles; an explicit O(|A|^{3r+1}) algorithm recovers both the average and the support.","key_machinery":"The reduction of maximising measures for a finite-range potential to the vertices of a De Bruijn graph that realise the maximum mean cycle weight (via a cohomologous function and Karp-style dynamic programming). This reduction simultaneously computes β(φ) and produces an explicit finite set of forbidden patterns whose SFT is precisely the support of all maximising measures.","core_discovery":"Under the two standing hypotheses that the ambient space is computably compact and that the set of invariant measures is computably overt, the maximum ergodic average of any computable potential is a computable real number and the set of maximising measures is a Π₁ compact subset of the space of measures. On one-dimensional SFTs with finite-range potentials the same objects are obtained exactly, in finite time, by reducing the problem to the maximum mean weight of cycles in a De Bruijn graph.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Zero-temp maximisers form Π₁-compact set under computable hypotheses","Max ergodic average is computable for any computable potential","Finite algorithm recovers max average via max-mean cycles on SFTs","Explicit O(|A|^{3r+1}) recovery of maximisers for finite-range SFT potentials","De Bruijn graph cycles yield exact zero-temp maximising measures"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The set of invariant measures must itself be computably overt; without that density of computable measures the upper bounds on complexity no longer hold, and the paper notes that the property fails for some natural computable maps and is undecidable for general two-dimensional SFTs.","fun_headline_variants_meta":{"raw":{"variants":["Zero-temp maximisers form Π₁-compact set under computable hypotheses","Max ergodic average is computable for any computable potential","Finite algorithm recovers max average via max-mean cycles on SFTs","Explicit O(|A|^{3r+1}) recovery of maximisers for finite-range SFT potentials","De Bruijn graph cycles yield exact zero-temp maximising measures"]},"model":"grok-4.5","effort":"low","cost_usd":0.004712,"raw_usage":{"total_tokens":1250,"prompt_tokens":697,"num_sources_used":0,"completion_tokens":83,"cost_in_usd_ticks":47120000,"prompt_tokens_details":{"text_tokens":697,"audio_tokens":0,"image_tokens":0,"cached_tokens":0},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":470,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":697,"tokens_out":83,"duration_ms":4925,"temperature":1.0,"reasoning_tokens":470,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T05:48:24.708969+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a computable dynamical system that is computably compact, a computable potential, and a proof that the set of invariant measures is computably overt, yet the maximum ergodic average is not a computable real, or the set of maximising measures is not Π₁.","supporting_citations":[],"review_version":1}