{"id":"9732fc51-fa71-4435-8375-19c4c2cb077c","arxiv_id":"2507.07753","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Exact renormalisation equations yield the relative frequency of any patch in a primitive substitution tiling, with transfer to symbolic and other suspension systems.","lead":"This math paper gives a recipe for computing the exact relative frequency of any finite pattern (patch) in tilings built from substitution rules, like the Fibonacci tiling. It uses scaling symmetries to turn the problem into solving a finite system of linear equations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4's uniqueness proof is internally inconsistent: the self-consistent vector contains zero entries for illegal patches, so the asserted block normal form cannot satisfy Lemma 3's strict-positivity conclusion.","rationale":"The central claim is that exact renormalisation determines patch frequencies uniquely, and this transfers to symbolic systems. That claim stands or falls with Theorem 4: if the self-consistent system has several nonnegative solutions, the word 'exact' is unjustified. The reader correctly identifies the block-normal-form step as the weakest point. My pass sharpens this: the step is not merely unproved; as stated it is incompatible with the definition of the vector. Since ν^(m) includes zero entries for illegal patches such as Fibonacci's aaa at (τ,τ), Lemma 3 cannot deliver the strictly positive eigenvector that the proof invokes. Therefore the proof as written cannot be repaired by a routine check of Eq. (5); it needs a support-restricted formulation. This does not show that the main method is wrong, and the Fibonacci pair-correlation example is consistent, so conditional acceptance remains appropriate. I do not recommend rejection because the flaw is localised to the uniqueness proof and is likely fixable by working on the support of the frequency vector. The unproved Theorem 12 is a secondary concern; Theorem 4 is the gate for the entire exactness claim.","tokens_in":13132,"tokens_out":18725,"duration_ms":226576,"concrete_test":"For the Fibonacci substitution, build the finite self-consistent matrix M^(3) from Eq. (2) over all triples (a1,a2,a3) and distances x1∈Λ_{a2}−Λ_{a1}, x2∈Λ_{a3}−Λ_{a2} satisfying (3). Then compute the normal form of M^(3) and check whether any diagonal block in the normal form has eigenvalue τ. Independently, use the model-set formula of Proposition 5 to verify that ν_{aaa}(τ,τ)=0 and that this zero vector is part of the nonnegative eigenvector of M^(3) for eigenvalue τ. If the normal form contains a lower block with eigenvalue τ, or if the matrix admits no strictly positive eigenvector, the proof's assertion (5) fails; if it does admit a strictly positive eigenvector, this contradicts ν_{aaa}(τ,τ)=0 and reveals the vector was not defined on the claimed domain.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Theorem 4. The vector ν^(m) is defined to include all entries ν_{a1...am}(x1,...,x_{m-1}) with xi ∈ Λ_{a_{i+1}}−Λ_{a_i} satisfying (3). For m ≥ 3 this includes illegal patches, whose frequency is 0. For example, in the Fibonacci substitution the triple (a,a,a) with distances (τ,τ) lies in the self-consistent set, since τ ≤ τ^2 = (max displacement)/(λ−1), yet aaa is not a legal word, so ν_{aaa}(τ,τ)=0. The proof then asserts that M^(m) can be permuted into the block normal form (5) with M^(1)=M_ϱ and no lower block having eigenvalue λ, and it invokes Lemma 3. Lemma 3 would imply a strictly positive eigenvector for λ. But the actual frequency vector is a nonnegative eigenvector with zero entries; if the asserted normal form held, the λ-eigenspace would be one-dimensional and positive, forcing ν_{aaa}(τ,τ)>0, a contradiction. Hence either (5)–(6) is not the normal form of the matrix as defined, or 'positive solution' is meant on a restricted support that is never specified. The text's 'it is clear' and the final coupling paragraph do not resolve the issue; a correct proof must define the support of legal patches and prove uniqueness on that support. Until this is fixed, the exact computation is not well-defined.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an explicit method for computing patch frequencies in geometric realisations of primitive substitutions, using exact renormalisation relations. It introduces a self-consistent finite subsystem of renormalisation equations for m-tile patch frequencies, claims a unique positive solution (Theorem 4), gives a model-set formula for the frequencies (Proposition 5), and derives transfer formulas from the self-similar geometric setting to the symbolic and general suspension settings (Theorems 10 and 12), with a worked Fibonacci example in §4.1. The central claim is that the renormalisation system yields exact frequencies and a legality test for arbitrary patches.","tokens_in":13408,"tokens_out":7600,"duration_ms":84996,"significance":"If the proof gaps are successfully repaired, the paper would provide a natural and useful generalisation of the pair-correlation renormalisation of Baake–Gähler–Mañibo [2] to arbitrary m-tile patches, together with explicit transfer formulas to symbolic and suspension systems. The Fibonacci example is concrete and the transfer formulas in §4 are explicit and falsifiable. The exposition is generally clear and the literature is adequately cited. However, the main theorem behind the uniqueness of the renormalisation solution is currently asserted rather than proved, and the master equation is misprinted, so the contribution is not yet fully substantiated.","major_comments":[{"comment":"The summation structure in Eq. (2) is not well defined. The indices k and ℓ are both summed from 1 to m, and for each pair (k,ℓ) one sums over α_k and r(k)∈t_{a_ℓ α_k}, but no summation over the remaining supertile letters α_i (i≠k) appears, and the role of ℓ is unexplained. The proof in the following paragraph indicates that the intended sum should be over all choices α_1,...,α_m∈A and over r(k)∈t_{a_k α_k} for each k. As printed, the formula cannot be evaluated, so the statement of Theorem 2 is incomplete. Please correct the summation and verify the displayed equation.","section":"Section 3, Eq. (2)"},{"comment":"The matrix M^(m) is never explicitly defined, and the claim that it can be permuted into the block normal form (5) with M^(1)=M_ϱ and lower blocks \\widetilde M^(ℓ) given by (6) is asserted with the phrase “it is clear.” The uniqueness conclusion depends entirely on this normal-form structure and on the absence of the Perron eigenvalue λ from the B^(ℓ) blocks, so the proof needs a precise construction of M^(m), a proof of the normal form, and a verification of the conditions of Lemma 3. As it stands, the argument is not checkable.","section":"Section 3, proof of Theorem 4"},{"comment":"The application of Lemma 3 is inconsistent with the definition of ν^(m). The vector is defined on all combinations satisfying (3), including illegal patches whose frequency is zero. In the Fibonacci example, the triple (a,a,a) with distances (τ,τ) satisfies (3) but is not a legal word, so ν_{aaa}(τ,τ)=0. Lemma 3, however, would imply that the Perron eigenvector is strictly positive, forcing ν_{aaa}(τ,τ)>0. Thus either the asserted block normal form is not the normal form of M^(m), or uniqueness is claimed on an unspecified support. The proof must define the support of legal patches and prove uniqueness on that support.","section":"Section 3, proof of Theorem 4"},{"comment":"Theorem 12 is stated with no proof; the displayed statement is followed immediately by □. Since the theorem is one of the main transfer results, and since part (ii), the rationally dependent case, does not follow trivially from Theorem 10 or from the preceding text, a proof or a precise reduction to the earlier arguments is needed.","section":"Section 4.2, Theorem 12"}],"minor_comments":[{"comment":"The notation ν^s_{αβ}(m) is used for m∈Z in the final sentence, but the definition and the summation formulas only make sense for non-negative integers; please state the intended range of m.","section":"Section 4.1, Proposition 7"},{"comment":"The typeset floor and ceiling notation in the proof is garbled, and the range of n should be written explicitly as n ∈ {⌊m/τ⌋, ⌈m/τ⌉} instead of relying on the surrounding prose.","section":"Section 4.1, Lemma 8"},{"comment":"The paper would be much easier to check if, after Eq. (2), a concrete small-m example (for instance m=3 for the Fibonacci substitution) were given showing the indexing of the vector ν^(m) and the corresponding matrix M^(m).","section":"Section 2 and Section 3"},{"comment":"The paper cites the author's thesis [23] for the Kolmogorov-consistency-type relation between three- and four-tile frequencies; since [23] is listed as “in preparation,” please provide the specific statement or a proof in the current text.","section":"Section 3, after Proposition 1"}],"recommendation":"major_revision","confidential_remarks":"The core idea is plausible and the Fibonacci example is convincing, but the proof of Theorem 4 needs substantial repair: the matrix is not defined, the normal form is asserted, and the positivity argument conflicts with zero entries for illegal patches. These are load-bearing issues for the central claim. I recommend major revision rather than rejection because the problems are localisable and likely fixable within the paper's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing you should know: this is a useful extension of Baake–Gähler–Mañibo to arbitrary m-tile patch frequencies, with clean transfer theorems to symbolic and suspension settings under an irreducibility assumption. The Fibonacci example is concrete and checks out. But the uniqueness proof of Theorem 4 has a load-bearing gap: the vector ν^(m) is defined on all distance combinations in the difference sets, including illegal patches whose frequency is zero. Lemma 3, invoked for uniqueness, guarantees a strictly positive eigenvector when the matrix is in the asserted normal form. That cannot hold if the actual frequency vector has zero entries—e.g., in Fibonacci, ν_{aaa}(τ,τ)=0 while the position lies in the self-consistent set. So either the normal form (5)–(6) is not the normal form of the matrix as defined, or 'positive' is meant on a restricted support that is never specified. The proof's 'it is clear' covers exactly the step that needs an argument.\n\nWhat is good: Theorem 2 is a natural and plausible extension of the pair-correlation renormalisation equations, and the paper is honest about building on [2]. The transfer theorem (Theorem 10) and the suspension variant (Theorem 12) are genuinely useful, and Lemma 9 on rational independence of PF eigenvector entries is clean. The writing is mostly clear.\n\nOther soft spots are minor in comparison: Theorem 12 is stated with no proof—just a box—and Eq. (2) has mangled summation indices (k,ℓ and α_k) that make it hard to parse. None of these are fatal by themselves.\n\nMy take: the core method is very likely correct, but as written the central claim of well-definedness is not fully supported. A referee should ask for a proof of Theorem 4 that either defines the support of legal patches and proves uniqueness there, or explains why the alleged normal form is compatible with zero frequencies. After that, this would be a solid contribution for people working on aperiodic order and diffraction. Yes, send it to peer review.","headline":"Useful extension of pair-correlation renormalisation to higher patches, but the uniqueness proof of Theorem 4 has a real gap over zero frequencies of illegal patches.","tokens_in":13941,"tokens_out":3243,"would_cite":false,"duration_ms":37050,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37B10","37A30","52C23"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper derives exact renormalisation equations for patch frequencies in self-similar inflation tilings, proves the self-consistent part has a unique positive solution, and transfers the frequencies to symbolic and suspended systems…","keywords":["substitution dynamical systems","inflation tilings","patch frequency","renormalisation equations","symbolic dynamics","suspension flows","Fibonacci substitution","Perron-Frobenius eigenvector"],"falsifier":"For a concrete substitution such as Fibonacci, write out the matrix $M^{(3)}$ for the three-tile self-consistent subsystem and test whether it can be permuted into the asserted block triangular normal form with no lower block carrying the largest eigenvalue; a lower block that carries that eigenvalue, or a second positive eigenvector, would refute the uniqueness claim.","tokens_in":12899,"feed_emoji":"📐","tokens_out":13041,"duration_ms":124679,"temperature":0.7,"pith_summary":"This paper aims to turn the long-known existence of patch-frequency measures for substitutions into an exact computational tool. For self-similar tilings built from an inflation rule it derives renormalisation equations expressing the frequency of any m-tile patch as a sum of smaller, rescaled patch frequencies, and it proves that the finite self-consistent part of these equations has a unique positive solution. Under the condition that the substitution matrix has irreducible characteristic polynomial, the same geometric frequencies transfer to the symbolic hull and to suspension flows with arbitrary tile lengths. The payoff is that one can in principle decide whether any given finite patch is legal and compute its exact relative frequency, something the standard induced-substitution method for cylinder sets cannot do for arbitrary patches.","feed_headline":"Renormalisation yields exact patch frequencies for inflation tilings","feed_subtitle":"New exact renormalisation relations compute patch frequencies for geometric, symbolic, and suspended substitution systems.","key_machinery":"The central object is the patch frequency function $\\nu_{a_1\\cdots a_m}(x_1,\\ldots,x_{m-1})$, which measures how often $m$ control points of specified tile types are found with separations $x_i$. The argument runs on the exact renormalisation relation (Eq. (2)), which expresses a large-scale patch frequency in terms of the smaller rescaled patches inside level-one supertiles; the displacement matrix $(t_{ij})$ records where each tile sits in a supertile and supplies the shifts in the relation. Because the rescaled separations are generally smaller than the originals, the equations split into a finite self-consistent subsystem and a recursive part, and the self-consistent subsystem is an eigenvector equation $\\nu^{(m)} = \\frac{1}{\\lambda} M^{(m)} \\nu^{(m)}$ for a non-negative matrix. The uniqueness argument uses the block normal form of that matrix together with a Perron–Frobenius eigenvector lemma.","core_discovery":"The central claim is that patch frequencies in a primitive inflation system obey exact self-similarity: the frequency of an m-tile patch with specified separations equals a $\\lambda^{-1}$ sum of frequencies of the same patch rescaled into level-one supertiles, with the separations shifted by the displacement data coming from the inflation rule (Eq. (2)). From these relations the paper proves that the finite self-consistent subsystem has a unique positive solution, so the recursion determines every patch frequency, not just pair correlations. For substitutions whose substitution matrix has irreducible characteristic polynomial, it then shows that symbolic patch frequencies are obtained by summing geometric patch frequencies over all tile-length decompositions of the symbolic distance, and that the same law governs any suspension flow built from other tile lengths. The Fibonacci substitution is worked out explicitly, yielding closed formulas for pair correlations.","pith_inferences":["A direct algorithmic reading of the self-consistent subsystem would compute $M^{(m)}$ for any given substitution and solve the eigenvalue problem exactly; the only practical barrier is enumerating the finitely many patch positions below the bound (3), so a complexity estimate in terms of $m$ and the tile lengths is a natural next step.","The same renormalisation scheme should work in higher-dimensional inflation tilings, where finite local complexity still makes the self-consistent part finite, but the normal-form argument would need to handle vector-valued displacements.","The paper leaves implicit that the exact equations convert the legality question into a decision procedure; checking the asserted block triangular form of $M^{(3)}$ and $M^{(4)}$ on concrete substitutions such as Fibonacci would test whether the uniqueness proof needs strengthening."],"forward_implications":["For any primitive aperiodic substitution with irreducible characteristic polynomial, every finite word in the symbolic hull gets an exact frequency computed by a finite sum of geometric patch frequencies, and a zero outcome tells the reader the word is illegal.","The transfer theorem covers all suspension flows: rationally independent tile lengths preserve geometric pair-correlation values directly, while rationally dependent lengths sum the geometric values over every decomposition of the distance.","In the Fibonacci case the symbolic pair correlations reduce to at most two geometric terms via the model-set window, giving formulas such as $\\nu^s_{aa}(4) = \\tau^{-3}$.","The same summation principle extends from pair correlations to arbitrary m-tile patches, so the method is not limited to two-tile statistics."],"supporting_citations":[{"why":"Establishes the pair-correlation case whose renormalisation scheme Theorem 2 extends to all m-tile patches.","marker":"[2, Thm. 3.19]"},{"why":"Provides the Perron–Frobenius eigenvector lemma used in Theorem 4 to get uniqueness of the positive solution.","marker":"[2, Lemma 3.18]"},{"why":"Supplies the normal-form theory for non-negative matrices underlying the block decomposition of $M^{(m)}$.","marker":"[17, Sec. 13.4]"},{"why":"Gives minimality and strict ergodicity of substitution hulls, guaranteeing patch frequencies exist uniformly.","marker":"[3]"},{"why":"Establishes strict ergodicity for self-similar tiling hulls, justifying the geometric frequency formula (1).","marker":"[28]"},{"why":"Supplies the irreducible-characteristic-polynomial condition used for rational independence of natural tile lengths in Theorems 10 and 12.","marker":"[30, Prop. 2.17]"},{"why":"Gives the Fibonacci pair-correlation renormalisation equations that the worked example starts from.","marker":"[1]"},{"why":"Backs the model-set representation of patch frequencies as integrals over windows, used in Proposition 5 and Lemma 8.","marker":"[24]"}],"fun_headline_variants":["Exact renormalisation yields patch frequencies for all inflation systems","Renormalisation solves patch frequencies exactly for inflation tilings","Self-similar renormalisation pins down every patch frequency","Patch frequencies from exact renormalisation: Fibonacci and beyond"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The uniqueness proof relies on an unproved assertion about the renormalisation matrix -- that, after a suitable rearrangement of its rows and columns, the diagonal blocks start with the substitution matrix and every lower block misses the largest eigenvalue; the text says this is clear rather than proving it.","fun_headline_variants_meta":{"raw":{"variants":["Exact renormalisation yields patch frequencies for all inflation systems","Renormalisation solves patch frequencies exactly for inflation tilings","Self-similar renormalisation pins down every patch frequency","Patch frequencies from exact renormalisation: Fibonacci and beyond"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001465,"raw_usage":{"total_tokens":5792,"prompt_tokens":741,"completion_tokens":5051,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":357,"completion_tokens_details":{"reasoning_tokens":4983}},"tokens_in":357,"tokens_out":5051,"duration_ms":41164,"temperature":1.0,"reasoning_tokens":4983,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:33:29.396625+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a concrete substitution such as Fibonacci, write out the matrix $M^{(3)}$ for the three-tile self-consistent subsystem and test whether it can be permuted into the asserted block triangular normal form with no lower block carrying the largest eigenvalue; a lower block that carries that eigenvalue, or a second positive eigenvector, would refute the uniqueness claim.","supporting_citations":[{"cited_title":"Baake and U","cited_arxiv_id":null,"evidence_quote":"Gives minimality and strict ergodicity of substitution hulls, guaranteeing patch frequencies exist uniformly."},{"cited_title":"Solomyak, Dynamics of self-similar tilings, Ergod","cited_arxiv_id":null,"evidence_quote":"Establishes strict ergodicity for self-similar tiling hulls, justifying the geometric frequency formula (1)."},{"cited_title":"Moody, Uniform distribution in model sets, Can","cited_arxiv_id":null,"evidence_quote":"Backs the model-set representation of patch frequencies as integrals over windows, used in Proposition 5 and Lemma 8."}],"review_version":1}