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Exact renormalisation for patch frequencies in inflation systems

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2507.07753 v1 pith:HDXLTDWL submitted 2025-07-10 math.DS

Jan Mazáč This is my paper · ORCID
classification math.DS MSC 37B1037A3052C23
keywords substitutiondynamicalsystemsinflationtilingspatchfrequencyrenormalisationequationssymbolicdynamicssuspensionflowsFibonacciPerron-Frobeniuseigenvector
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

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.

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 (4)
  1. [Section 3, Eq. (2)] 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.
  2. [Section 3, proof of Theorem 4] 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.
  3. [Section 3, proof of Theorem 4] 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.
  4. [Section 4.2, Theorem 12] 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.
minor comments (4)
  1. [Section 4.1, Proposition 7] 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.
  2. [Section 4.1, Lemma 8] 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.
  3. [Section 2 and Section 3] 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).
  4. [Section 3, after Proposition 1] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: renormalisation relations are derived from self-similarity and local recognisability, not assumed; the main uniqueness argument relies on an external result [2] and standard Perron-Frobenius theory.

full rationale

The derivation of the exact renormalisation relations (Theorem 2, Eq. (2)) starts from the definition of patch frequencies (Eq. (1)) and uses local recognisability and the inflation symmetry of the control-point set; the proof explicitly derives the relation by summing over supertiles and displacements rather than postulating the target frequencies. The uniqueness proof (Theorem 4) invokes Lemma 3 from Baake-Gähler-Mañibo [2] as an external result for m=2, then extends it by induction; it does not assume the conclusion. The author's own PhD thesis [23] is cited only for background on the measure-extension / Eberlein-convolution analogy and is not used as a load-bearing premise. Theorems 10 and 12 are transfer statements: they express symbolic/suspension patch frequencies as finite sums of the already-computed geometric pair correlations, using rational independence (Lemma 9, proved in the text), so no input quantity is renamed as a prediction. The skeptic's concern about Theorem 4 -- that the asserted normal form (5)-(6) and Lemma 3 would force strict positivity, while actual frequency vectors can contain zero entries for illegal patches -- is a genuine internal correctness gap (the proof says 'it is clear' without proving the normal form or specifying the support of legal patches), but it is not a circularity: it is an omitted/incorrect proof step, not an equivalence of the conclusion with an input. For this reason no circular step is recorded and the circularity score is minimal (1, reflecting only the peripheral self-citation [23]).

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted in this note; all quantities come from the substitution rule and the dynamics. No new entities are introduced. The main axioms are standard substitution-tiling facts plus an irreducibility condition used only for the symbolic transfer.

assumptions (4)
  • domain assumption The substitution ϱ is primitive and aperiodic; the geometric hull is strictly ergodic.
    Stated in Section 2 and used throughout; ensures patch frequencies exist and are independent of the hull element.
  • domain assumption Local recognisability: every tile in an aperiodic primitive inflation tiling belongs to a unique level-1 supertile.
    Used in the proof of Theorem 2 to map a patch in Λ to a patch in the inflated hull.
  • domain assumption The characteristic polynomial of the substitution matrix is irreducible, so natural tile lengths are rationally independent (Lemma 9).
    Assumed before Theorem 10; needed to identify symbolic words with unique geometric distances. Not needed for the geometric renormalisation Theorem 2.
  • standard math Perron-Frobenius theory and the Gantmacher normal form of non-negative matrices (Lemma 3 from [2]).
    Used in Theorem 4 to prove uniqueness of the positive solution of the self-consistent equations.

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Cite this review

Pith. "Pith review of Exact renormalisation for patch frequencies in inflation systems." pith.science (2026). https://pith.science/paper/HDXLTDWL

@misc{pith2026250707753,
  author       = {Pith},
  title        = {Pith review of: Exact renormalisation for patch frequencies in inflation systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HDXLTDWL}},
  note         = {Machine review of arXiv:2507.07753}
}
read the original abstract

This note provides an explicit way of calculating the patch frequencies in geometric realisations of primitive substitutions using exact renormalisation relations. Further, we profit from these results to obtain the patch frequencies in the symbolic case as well as in other suspensions (under mild assumptions on the substitution). We illustrate this procedure on the Fibonacci example.

Figures

Figures reproduced from arXiv: 2507.07753 by the authors.

Figure 1
Figure 1. Illustration of the renormalisation scheme. Every tile ai in a patch in Λ belongs to a unique level-1 supertile αi , and within this supertile, it has a relative distance r (i) . This allows one to relate the patch frequencies at the level of supertiles, i.e., in ϱ(Λ), with the patch frequencies in the original point set Λ. a1 , . . . , am and all possible displacements within each supertile gives the claim together… view at source ↗

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