{"id":"89ccf5c6-e5d6-4ef1-b9ff-dacae13b6c37","arxiv_id":"1908.09056","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Gradient fields of low-temperature spin models (Ising, Potts, beach, six-vertex) are finitary factors of i.i.d. processes even when the models themselves are not.","lead":"The paper proves that the gradient of the low-temperature Ising model, the Potts model, and the beach model is a finitary factor of an i.i.d. process, meaning only the global spin value blocks finitary coding. It also introduces a new percolation representation of the six-vertex model and shows the absolute diagonal gradient and Laplacian of its height function are finitarily codable.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.4's stated threshold misses a factor 3: the proof requires c > 2 + 3p_c/(1-p_c), not c > 2 + p_c/(1-p_c).","rationale":"The reader's weakest assumption concerned dependence on Harel--Spinka's monotone-coding theorem and its adaptation to a reversed partial order. That external dependence is real, but in good faith I do not treat reliance on a stated theorem from the same research group as the primary correctness risk without evidence against it; moreover, the paper actually maps the superimposed model to the usual order before applying the theorem, so the adaptation is less severe than the reader's phrasing suggests. The sharper, internally verifiable problem is the factor-3 mismatch in Theorem 1.4: the theorem's stated range c > 2 + p_c/(1-p_c) is not the range used anywhere in the proof, which requires c > 2 + 3p_c/(1-p_c). This is not a matter of external convention: it is an arithmetic consequence of the disagreement-percolation bound in Lemma 4.10 and Theorem 4.7, and it directly narrows the parameter region for which the central six-vertex ffiid claims are established. The abstract's numerical value c ≳ 6.4 supports the reading that a factor was dropped from the statement. The fix is straightforward—either weaken the theorem statement to the proven threshold or extend the argument to the weaker range—so the result remains credible, but the paper should not be accepted without this correction. My verdict is therefore CONDITIONAL, in agreement with the reader's overall judgment even though the load-bearing concern I identify is internal rather than the external theorem dependency.","tokens_in":38371,"tokens_out":8835,"duration_ms":92792,"concrete_test":"Recompute the domination threshold in Lemma 4.10 and Theorem 4.7 with q = 2: substitute max{2, q+1} = 3 into p = α/(max{2,q+1}+α) and solve p > p_c, obtaining α > 3p_c/(1-p_c). Then scan every statement in Section 5 that invokes Proposition 4.11 or Theorem 4.7, and check whether any argument covers α in (p_c/(1-p_c), 3p_c/(1-p_c)]; if none does, correct Theorem 1.4's threshold to c > 2 + 3p_c/(1-p_c) or supply a new proof for the weaker range.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The internal threshold for the six-vertex result is inconsistent. In Theorem 1.4 the condition is stated as c > 2 + p_c/(1-p_c), and Section 5 repeats 'fix c > 2 + p_c/(1-p_c) and α = c - 2'. But the proof actually needs α > 3p_c/(1-p_c) for q = 2. Indeed, Lemma 4.10 uses a dominated Bernoulli site percolation with parameter p = α / (max{2, q+1} + α), and Theorem 4.7 requires p > p_c, so for q = 2 one needs p = α/(3+α) > p_c, i.e. α > 3p_c/(1-p_c). Proposition 4.11, Corollary 4.12, and the proof of Proposition 5.2 all invoke exactly this stronger condition (with c = 2 + α), so the full proof only establishes Theorem 1.4 for c > 2 + 3p_c/(1-p_c), about 6.365, not for c > 2 + p_c/(1-p_c), about 3.455. The abstract's 'c ≳ 6.4' matches the proof, so the theorem statement appears to have lost a factor 3. This is a concrete, checkable gap in a central claim: as written, the theorem asserts ffiid of |∇_d h| and |Δh| on a parameter interval for which the supplied arguments are not valid.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies finitary factors of i.i.d. fields for gradient-type observables of spin systems with multiple Gibbs states. The main theorems state: (Theorem 1.1) the edge percolation of unequal spins in the low-temperature Ising model is ffiid with exponential coding radius for all β>β_c(d); (Theorem 1.3) the oriented-edge gradient of the q-state Potts model is ffiid if and only if the free and wired random-cluster measures coincide; (Theorem 1.5) the gradient of the beach model is ffiid under the analogous uniqueness condition; and (Theorem 1.4) for the six-vertex/F-model height function with flat boundary conditions and c>2+p_c/(1−p_c), the fields h, ∇h, ∇_d h, and Δh are not (Z2)even-ffiid, while |∇_d h| and |Δh| are ffiid. The proofs introduce a universal cluster-tree construction on percolation clusters (Section 2) and a new 'superimposed random-cluster model' coupled to the six-vertex model (Section 4). The six-vertex positive results are proved only under the stronger threshold c>2+3p_c/(1−p_c), as detailed in major comment 1.","tokens_in":38667,"tokens_out":15061,"duration_ms":147760,"significance":"The overarching message—that global spin-flip or spin-permutation information is the only obstruction to ffiid, and that suitable gradients are ffiid—is natural and, if fully established, would be a substantial contribution to the finitary-coding literature. The paper's concrete assets include a deterministic cluster-tree factor map that is universal across percolation processes with a unique infinite cluster, and a new graphical representation of the six-vertex model that is likely to be useful independently. The results are stated as theorems proved from explicit assumptions rather than inferred from fitted parameters, and the main proof strategy is coherent. However, the factor-3 threshold gap in Theorem 1.4 means that one of the central positive claims is currently established on a smaller parameter region than the theorem statement asserts. The reversed-order application of the Harel–Spinka theorem is also only sketched. With these points addressed, the paper would be a strong addition to the field.","major_comments":[{"comment":"The parameter threshold in Theorem 1.4 is not the threshold used in the proof. The theorem states c > 2 + p_c/(1−p_c), and Section 5 repeats 'fix c > 2 + p_c/(1−p_c) and α = c−2'. However, Theorem 4.7 requires α > (p_c/(1−p_c)) max{q+1,2}, which for q=2 is α > 3p_c/(1−p_c). Equivalently, Lemma 4.10 uses a dominated Bernoulli site percolation with parameter p = α/(max{2,q+1}+α), and the condition p > p_c is exactly α > 3p_c/(1−p_c) when q=2. Proposition 4.11 and Corollary 4.12 likewise fix α > 3p_c/(1−p_c). Since α = c−2, the supplied arguments establish Theorem 1.4 only for c > 2 + 3p_c/(1−p_c) (approximately 6.365), not for c > 2 + p_c/(1−p_c) (approximately 3.455). The abstract's 'c ≳ 6.4' is consistent with the proof, so the theorem statement should be corrected to the stronger threshold or the argument must be extended to the claimed range.","section":"Theorem 1.4; opening line of Section 5; Lemma 4.10; Theorem 4.7"},{"comment":"The ffiid property of the superimposed model is obtained by applying [29, Theorem 7] to a partial order that is reversed on one sublattice, with only a two-sentence remark that the proof of that theorem extends to this setting. Because the positive directions of Theorems 1.1, 1.3, 1.4, and 1.5 depend on monotone-coding inputs, this adaptation is load-bearing. I recommend that the authors state and prove the reversed-order version as a lemma, or provide a complete verification that [29, Theorem 7] applies verbatim to the partial order in (4.8). As written, the argument relies on an unproved adaptation of an external theorem.","section":"Section 5, proof of Proposition 5.2"}],"minor_comments":[{"comment":"The sentence 'the six-vertex model can be coupled with the six-vertex model' appears to contain a typo; it should refer to the superimposed random-cluster model.","section":"Section 1.5, six-vertex outline"},{"comment":"Reference [21] is listed as 'In preparation, 2019'; the published or updated version should be cited if available.","section":"References"},{"comment":"The definition of kΔ(η) as 'the sum kΔ(η0)+kΔ(η1) of the number of open vertex-clusters' is somewhat redundant; consider simplifying to 'the number of open clusters in η0 plus the number in η1 that meet Δ'.","section":"Section 4.1, equation (4.1)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript depends on several results from the authors' own preprints ([29], [41]) and on [21], which is listed as 'In preparation'. The editor may wish to confirm the availability and status of these references. The discrepancy between the statement of Theorem 1.4 and the proof threshold should be resolved before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves something genuinely new: the gradient of the low-temperature Ising model is ffiid at every beta, with exponential tails, and similarly for Potts and beach gradients under the right random-cluster uniqueness condition. The new superimposed random-cluster model for the six-vertex model is a real contribution in its own right, and the cluster-tree construction in Section 2 is an elegant general tool. The main arc of the argument is credible and detailed.\n\nThe soft spots are real but mostly minor. The most concrete issue is the threshold in Theorem 1.4. The statement says c > 2 + pc/(1-pc), but the proof in Sections 4 and 5 requires α > 3pc/(1-pc), i.e. c > 2 + 3pc/(1-pc). The abstract's c ≳ 6.4 matches the proof, so the theorem statement appears to have lost a factor 3. This is a checkable inconsistency in a central claim, though it does not undermine the qualitative result—the paper still proves ffiid for all c above about 6.4. It should be fixed before publication, either by restating the theorem at the stronger threshold or by finding an argument that actually works at the weaker one.\n\nThe other soft spot is the use of the Harel–Spinka monotone-coding theorem in the reversed partial order. That is a legitimate external input, not circular, but the adaptation is only sketched in two sentences and deserves a fuller proof. The disagreement-percolation and exploration arguments are mostly standard, though compressed.\n\nOverall, the paper is a solid piece of work with substantial new theorems and a new graphical representation. It deserves a serious referee. I would send it to peer review, ask the authors to correct the threshold and expand the reverse-order coding argument, and then accept after those revisions.","headline":"Strong new results on finitary codings of gradients, but Theorem 1.4's six-vertex threshold has a real factor-3 inconsistency that the authors should fix before publication.","tokens_in":39196,"tokens_out":1715,"would_cite":true,"duration_ms":18881,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82B20","82B43"],"pacs":[],"model":"deepseek-v4-flash","headline":"Global spin information, not local disorder, is the only barrier to coding low-temperature spin models finitarily from i.i.d. randomness.","keywords":["finitary factor of i.i.d.","Ising model","Potts model","six-vertex model","random-cluster model","superimposed random-cluster model","gradient model","coding radius"],"falsifier":"Find a percolation process on a transitive graph with a unique infinite cluster and exponentially decaying cluster-size, distance-to-infinite-cluster, and connectivity tails for which the gradient of independently colored clusters is not a finitary factor, or show that the monotone-coding theorem fails under a reversed partial order. Either would break the chain behind Theorem 1.1, 1.3, 1.4, or 1.5.","tokens_in":1901,"feed_emoji":"🧊","tokens_out":4235,"duration_ms":100269,"temperature":0.7,"pith_summary":"The paper shows that the only obstruction to coding low-temperature spin models finitarily from i.i.d. randomness is their global spin information. For the Ising model below critical temperature, the gradient—the edges whose endpoints have opposite spins—is a finitary factor of i.i.d. with exponentially decaying coding radius at every inverse temperature, and similarly for Potts and beach model gradients once the underlying random-cluster measures have a unique Gibbs state. For the six-vertex F-model at large parameter $c$, the height function's gradient itself remains an obstruction, but the absolute value of its diagonal gradient and of its Laplacian are finitary factors. The proof supplies a general cluster-tree mechanism and a new graphical representation of the six-vertex model.","feed_headline":"Drop the global spin flip; the Ising gradient is finitarily codable","feed_subtitle":"Sampling only the differences between neighboring spins, not the spins themselves, removes the global obstruction.","key_machinery":"The load-bearing object is a deterministic cluster-tree factor map: for any percolation configuration with a unique infinite cluster, each finite cluster is assigned a parent (the largest-diameter cluster at a large enough scale), so the collection of clusters forms a tree rooted at the infinite cluster, and the tree-path between any two vertices' clusters can be recovered by a finite exploration. This turns independently colored clusters into a gradient by reading spin differences along tree paths, and it is what upgrades 'gradient of independently colored clusters' to a finitary factor. The paper's second mechanism is the superimposed random-cluster model—two random-cluster configurations, one on the even sublattice and one on its dual, with no closed primal-dual cross—which is monotone under an order reversed on one sublattice, has a unique Gibbs state for large $\\alpha$, and couples to the six-vertex spin representation with $c=2+\\alpha$.","core_discovery":"The central discovery is that global transformations—a global spin flip in Ising, a global permutation in Potts and beach models—are exactly what prevents finitary codability, and that discarding this global information through a local gradient leaves a field that is finitarily codable from i.i.d. input. For the low-temperature Ising model, the paper proves the gradient percolation configuration is ffiid with exponential tails on the coding radius for every $\\beta>\\beta_c(d)$, and derives a volume-order large deviation estimate for the energy. For the six-vertex F-model with $c>2+p_c/(1-p_c)$, it proves the height function's diagonal-gradient absolute value and Laplacian absolute value are ffiid, while the gradient and Laplacian themselves are not even $(\\mathbb{Z}^2)_{\\mathrm{even}}$-ffiid; the mechanism is a new Edwards-Sokal-type coupling between the six-vertex model and a superimposed pair of random-cluster models on primal and dual lattices.","pith_inferences":["The same cluster-tree construction should apply to any percolation process with a unique infinite cluster and exponential cluster-size, distance, and connectivity tails, so the method likely extends to other models with global symmetries, such as hard-core even/odd states, once a suitable gradient is defined.","If the superimposed model's uniqueness could be pushed below the threshold $\\alpha>3p_c/(1-p_c)$, the six-vertex ffiid result would extend to smaller $c$; the paper identifies this as open, and the coupling suggests a natural route through subcritical disagreement percolation.","Because the superimposed model at $\\alpha=0$ coincides with the critical random-cluster model with $q=4$, a deeper coupling between the two representations might transfer known random-cluster results to six-vertex problems."],"forward_implications":["The Ising energy per edge satisfies an exponential large-deviation bound in every dimension $d\\ge 2$ and at every positive temperature, including criticality, because the gradient is ffiid and ffiid fields obey the ergodic theorem at exponential rate.","The gradient of the $q$-state Potts model is ffiid exactly when the free and wired random-cluster measures coincide, giving a concrete condition under which low-temperature Potts behavior is finitarily codeable.","The beach-model gradient inherits ffiid from uniqueness of the beach-random-cluster Gibbs state.","The six-vertex absolute diagonal gradient and absolute Laplacian are ffiid for $c>2+p_c/(1-p_c)$, while the height gradient itself is not $(\\mathbb{Z}^2)_{\\mathrm{even}}$-ffiid.","The superimposed random-cluster model provides a new graphical representation of the six-vertex model with $c\\ge 2$, coupling six-vertex correlations to the connectivity of two random-cluster layers."],"supporting_citations":[{"why":"Supplies the theorem that monotone models with coinciding extremal measures are finitary factors of i.i.d., the step that makes the random-cluster and superimposed percolation processes ffiid.","marker":"[29]"},{"why":"Established that the Ising model is ffiid if and only if the Gibbs measure is unique, providing the baseline obstruction the paper removes.","marker":"[6]"},{"why":"Provides the random-cluster model definitions, the Edwards-Sokal coupling, and the free/wired measure framework used throughout.","marker":"[23]"},{"why":"Gives the exponential decay of truncated correlations for the FK-Ising measure needed for exponential tails on the coding radius.","marker":"[16]"},{"why":"Supplies Pisztora's coarse-graining good-box estimates used to verify the exponential tails condition (2.1).","marker":"[38]"},{"why":"Identifies the slab-percolation threshold with the Ising critical point, completing the good-box argument for all $\\beta>\\beta_c(d)$.","marker":"[7]"},{"why":"Supplies the exponential ergodic theorem for ffiid fields, used both for the energy large-deviation estimate and for the non-ffiid contradiction in the six-vertex Laplacian argument.","marker":"[9]"},{"why":"Provides the spatial-mixing and monotone-model ffiid toolkit and the non-ffiid argument for Markov random fields adapted to the six-vertex obstruction.","marker":"[41]"},{"why":"Gives the convergence of six-vertex height measures for $c>2$, which the paper reproves for larger $c$ and uses as background for the $c=2$ case.","marker":"[21]"}],"fun_headline_variants":["Global flip is the only barrier; Ising gradient is finitarily codable","Finitary coding works for gradients, not spins: new six-vertex insight","New six-vertex representation enables finitary coding of Laplacian","Ising and six-vertex: gradients are codable, global symmetries are not"],"cache_read_input_tokens":41344,"weakest_assumption_plain":"The proof relies on a theorem stating that a monotone model whose extremal measures coincide is a finitary factor of i.i.d.; in the six-vertex case the paper uses a version of this theorem for a partial order reversed on one sublattice, with only a two-sentence sketch, so if that theorem or its reversed-order extension fails, the positive ffiid results do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Global flip is the only barrier; Ising gradient is finitarily codable","Finitary coding works for gradients, not spins: new six-vertex insight","New six-vertex representation enables finitary coding of Laplacian","Ising and six-vertex: gradients are codable, global symmetries are not"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000381,"raw_usage":{"total_tokens":2100,"prompt_tokens":1104,"completion_tokens":996,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":720,"completion_tokens_details":{"reasoning_tokens":912}},"tokens_in":720,"tokens_out":996,"duration_ms":8662,"temperature":1.0,"reasoning_tokens":912,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:23:55.761164+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a percolation process on a transitive graph with a unique infinite cluster and exponentially decaying cluster-size, distance-to-infinite-cluster, and connectivity tails for which the gradient of independently colored clusters is not a finitary factor, or show that the monotone-coding theorem fails under a reversed partial order. Either would break the chain behind Theorem 1.1, 1.3, 1.4, or 1.5.","supporting_citations":[{"cited_title":"Finitary codings for the random-cluster model and other infinite-range monotone models","cited_arxiv_id":"1808.02333","evidence_quote":"Supplies the theorem that monotone models with coinciding extremal measures are finitary factors of i.i.d., the step that makes the random-cluster and superimposed percolation processes ffiid."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Established that the Ising model is ffiid if and only if the Gibbs measure is unique, providing the baseline obstruction the paper removes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the random-cluster model definitions, the Edwards-Sokal coupling, and the free/wired measure framework used throughout."},{"cited_title":"Exponential decay of truncated correlations for the Ising model in any dimension for all but the critical temperature","cited_arxiv_id":"1808.00439","evidence_quote":"Gives the exponential decay of truncated correlations for the FK-Ising measure needed for exponential tails on the coding radius."},{"cited_title":"Pisztora","cited_arxiv_id":null,"evidence_quote":"Supplies Pisztora's coarse-graining good-box estimates used to verify the exponential tails condition (2.1)."},{"cited_title":"Bodineau","cited_arxiv_id":null,"evidence_quote":"Identifies the slab-percolation threshold with the Ising critical point, completing the good-box argument for all $\\beta>\\beta_c(d)$."},{"cited_title":"Bosco, F","cited_arxiv_id":null,"evidence_quote":"Supplies the exponential ergodic theorem for ffiid fields, used both for the energy large-deviation estimate and for the non-ffiid contradiction in the six-vertex Laplacian argument."},{"cited_title":"Finitary codings for spatial mixing Markov random fields","cited_arxiv_id":"1803.10578","evidence_quote":"Provides the spatial-mixing and monotone-model ffiid toolkit and the non-ffiid argument for Markov random fields adapted to the six-vertex obstruction."},{"cited_title":"Glazman and R","cited_arxiv_id":null,"evidence_quote":"Gives the convergence of six-vertex height measures for $c>2$, which the paper reproves for larger $c$ and uses as background for the $c=2$ case."}],"review_version":1}