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Radius-zero Extended Symmetries and Irregular Fibres of $\mathbb{Z}^d$-Substitution Subshifts

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Pith's one-line read The paper proves an algorithm that computes all supertile-shuffling radius-zero extended symmetries, shows every extended symmetry must preserve the height lattice, and describes irregular fibres by an explicit sofic shift.

desk verdict Novel and worth refereeing, but the main algorithmic theorem's proof has a real gap, the height-preservation proof is under-specified, and Theorem 3 outruns Theorem 47. read the letter →

arxiv 2506.13392 v1 pith:V7RBP6LJ submitted 2025-06-16 math.DS

Álvaro Bustos-Gajardo , Daniel Luz , Neil Mañibo This is my paper · ORCID
classification math.DS MSC 37B1052C2337B05
keywords extendedsymmetriesZ^dsubstitutionsubshiftsradius-zeroheightlatticeirregularfibresderivedsubstitutionsminimalsetscoincidencegraph
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

For substitutive $\mathbb{Z}^d$-shifts that are neither bijective nor Toeplitz, the paper gives a finite, checkable way to list the extended symmetries whose local block maps have radius zero and that respect the supertile hierarchy. It also proves that any extended symmetry, even one with nonzero radius, must send the height lattice $\Gamma$ to itself, so its linear part must be a symmetry of that lattice. Finally, it characterises the irregular fibres over the maximal equicontinuous factor: a point of the odometer is irregular exactly when some integer translate of it lies in an explicit sofic shift built from the substitution. Together these results turn questions about the normaliser of the shift action into finite combinatorial computations.

What carries the argument

The load-bearing objects are the minimal sets of the substitution—the smallest subsets of the alphabet that occur as images of full columns of some power of $\theta$—together with the coincidence graph that records how columns move these sets around. Each minimal set yields an encoding map $\nu$ to a standard finite set, and each column induces a permutation $\beta_{M,f}$ of that set; a supertile-shuffling extended symmetry forces these permutations to satisfy a cocycle identity. For the fibre result, the new machinery is the derived substitution $\partial_J\theta$, obtained by reading $\theta$ across the faces in the coordinate directions $J$, and the reversed pruned coincidence graph $\widehat{G}(\theta,J)^{\mathrm{op}}$, whose edge shift is the sofic shift that detects irregular fibres.

What would settle it

Take any aperiodic primitive block substitution with nontrivial height lattice and compute the induced action $\Phi^*$ of an extended symmetry on the character group of its maximal equicontinuous factor; if some height eigenfunction is sent to a function that changes under a tiling translation, then Corollary 30 fails and one can look for a counterexample to $A\Gamma=\Gamma$.

Watch

Extended reading notes

Core claim

The central claim is that the normaliser of the shift action on $\mathbb{X}_\theta$ is rigidly constrained by the substitution's minimal sets and by its height lattice. A pair $(\tau,A)$ with a letter exchange $\tau$ and a matrix $A\in \mathrm{GL}(d,\mathbb{Z})$ generates a supertile-shuffling radius-zero extended symmetry if and only if three identities involving the minimal sets hold (Theorem 16, the precise form of Theorem 1). Every extended symmetry with linear component $A$ satisfies $A\Gamma=\Gamma$ (Theorem 2), so the height lattice is an invariant of the whole normaliser, not just of the centraliser. Moreover, for trivial-height block substitutions, a fibre over $\mathbb{Z}^d_Q$ is irregular if and only if it is an integer translate of a point in an explicit sofic shift $\mathbb{Z}_\theta$ (Theorem 3), extending the one-dimensional graph characterisation.

Load-bearing premise

If the preservation of height eigenfunctions by extended symmetries fails, the conclusion that every linear component must stabilise the height lattice collapses, and the existing proof does not fully establish that preservation.

Editorial extensions

If this is right

  • The normaliser's radius-zero supertile-shuffling elements can be listed by a finite algorithm, rather than by guessing from examples.
  • Any extended symmetry, regardless of its radius, must have a linear component that stabilises the height lattice, ruling out many candidate rotations and reflections.
  • Irregular fibres over the maximal equicontinuous factor are exactly those whose integer shifts meet the explicit sofic shift $\mathbb{Z}_\theta$, turning fibre-cardinality questions into graph-theoretic ones.
  • The $\kappa$-cocycle maps irregular fibres to irregular fibres along the linear component, so fibre-cardinality classes are preserved by extended symmetries.

Reading between the lines

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

  • If the height-lattice invariance in Theorem 2 survives a corrected proof, it would give a spectral obstruction to extended symmetries of any substitutive tiling, not just block substitutions.
  • The derived-substitution construction likely extends to non-rectangular digit substitutions, since the inductive definition of $\partial_J\theta$ does not inherently use rectangularity of the support.
  • The sofic characterisation of irregular fibres may make multivariate mean equicontinuity and sensitivity computable for substitution subshifts, since irregular fibres control the failure of equicontinuity.
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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

2 major / 3 minor

Summary. The paper studies radius-zero extended symmetries of Z^d-substitution subshifts, i.e. homeomorphisms that normalise the Z^d shift action and have a block map of radius zero. Its main results are: (Theorem 1) an algorithm, implemented through the characterisation in Theorem 16, for computing supertile-shuffling radius-zero extended symmetries for aperiodic primitive block substitutions with 1 < c_theta < |A|; (Theorem 2) preservation of the height lattice, A Gamma = Gamma, for any extended symmetry with linear component A; and (Theorem 3) a description of irregular fibres over the maximal equicontinuous factor in terms of a sofic shift built from derived substitutions and pruned coincidence graphs. The paper also contains several worked examples, including a counterexample showing that radius-zero extended symmetries need not be supertile-shuffling.

Significance. If the results are correct, this is a useful contribution to the study of normalisers of higher-dimensional substitution subshifts, a topic with relatively few concrete computational results. The minimal-set framework from MY21 is adapted in a natural way, and the introduction of J-derived substitutions to describe irregular fibres is an original technical tool that could be of independent interest, particularly for tameness and mean-equicontinuity questions. The examples, especially the modified half-hex example and the manta-ray example, are instructive and convincingly illustrate the phenomena. However, the central algorithmic claim rests on Theorem 16, whose converse direction currently contains a serious gap; as written, the paper does not establish Theorem 1.

major comments (2)
  1. [§3.3 (Theorem 16)] The proof of the 'if' direction of Theorem 16 is a non sequitur. Condition (2) is stated as tau' ∘ nu = bar-nu ∘ tau (Eq. (7) in Lemma 14). This identity gives bar-nu(tau(a)) = tau'(nu(a)); it does not give nu(tau(a)) = tau'(bar-nu(a)). Nevertheless, the displayed chain in the proof replaces nu(tau(a)) by tau'(bar-nu(a)) under 'Property (2)', and later uses a further unstated identity of the same type to replace tau'(bar-nu(a)) by nu(tau(theta_{A^{-1}⊙j}(a))). Consequently the asserted implication from conditions (1)-(3) to theta_j(tau(a)) = tau(theta_{A^{-1}⊙j}(a)) is not derived. Since Theorem 1 and the algorithmic claim depend exactly on this characterization, the main result of Section 3 is not established as written. The proof needs either a corrected compatibility condition replacing (2) and a reworking of the chain, or a genuinely different argument.
  2. [§4.4 (Corollary 30)] The proof of Corollary 30, on which Theorem 2 depends, is too terse and as written is easy to read in a way that does not yield the claimed conclusion. From the invariance of the torsion subgroup {1_{Z_Q}} × Zd/Γ under Phi**, Lemma 29, applied literally with the factors in the order Z_Q × Zd/Γ, would preserve the annihilator of the torsion subgroup, namely \widehat{Z_Q} × {1}, not the height-character subgroup {1} × \widehat{Zd/Γ}. To obtain the height characters one must apply Lemma 29 with the two factors swapped before dualising. The authors should spell this out explicitly; without the swapped reading, the preservation of height eigenfunctions is not established, and Theorem 2's conclusion A Gamma = Gamma does not follow.
minor comments (3)
  1. [§3.3 (Theorem 16, proof)] In the proof of the power-extension step after Eq. (8), the symbol n0 is used where the Q-adic expansion j = [j_{k-1}, ..., j_0] suggests j0 (or a correspondingly defined index) is meant. This should be corrected for consistency.
  2. [§6.2 (Theorem 46)] Theorem 46 assumes that theta is 'injective', but the paper does not define injectivity for Zd block substitutions. If it means column-injective, that excludes the main non-bijective regime c_theta < |A|; if it means strongly injective, a d-dimensional definition and justification are needed.
  3. [§3.1 (Proposition 13)] In the proof of Proposition 13, the passage from local equality on supertiles to the global identity Phi ∘ theta^n = sigma^ell ∘ theta^n ∘ Phi is compressed: the shift ell is introduced as depending on x, and the sentence invoking minimality and compactness does not make clear why a single ell works for the whole shift. A more detailed argument would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's main theorems are characterizations with independent proofs, not derivations from their own target conclusions.

full rationale

The paper's central claims are structural characterizations with independent proofs. Theorem 16 gives algebraic conditions for a pair (τ,A) to define a supertile-shuffling extended symmetry; the defining property of supertile-shuffling, Eq. (3)/(5), is not identical to conditions (1)–(3), and the theorem proves an equivalence rather than assuming it. Theorem 2 uses Pontryagin duality and the paper's own Lemmas 26–29 and Corollary 30 to conclude AΓ=Γ; that conclusion does not appear among the assumptions. Theorem 3, via Theorem 47 and Theorem 46, gives an iff characterization of irregular fibres using derived-substitution graphs, with a constructive proof rather than a restatement of the definition. Self-citations [Bus20, BLM23] appear as baselines and precedents, but the main implications do not reduce to them; Lemma 9 refers to external minimal-set work [LM20, MY21] for a preparatory statement, which is independent support rather than a load-bearing self-citation chain. The skeptical remark about the middle equality in the proof of Theorem 16 identifies a possible gap in a displayed calculation, but that is a correctness concern, not circularity: even if the proof needs repair, the theorem's statement is not equivalent by construction to its inputs. No exhibited reduction of a claimed result to its own inputs was found, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 9 assumptions · 1 invented entities

Pure mathematics paper: no fitted parameters, no empirical constants. The main additional structure the paper introduces is the J-derived substitution graph and the associated encodings. All other inputs are standard background from symbolic dynamics, tiling theory, and harmonic analysis, plus the domain assumptions listed above.

assumptions (9)
  • standard math Curtis-Hedlund-Lyndon theorem for extended symmetries (Proposition 5)
    Used to represent extended symmetries as letter maps composed with linear components; quoted from [BRY18].
  • domain assumption Recognizability of aperiodic primitive substitutions (Sol98)
    Used for unique desubstitution in Theorem 46 and in the proof of Proposition 13.
  • standard math Primitive substitution implies minimal shift and gives the minimal-set formalism of [LM20, MY21]
    Basis for Lemmas 9, 14 and Theorem 16; assumed to extend to Z^d.
  • standard math Pontryagin duality for locally compact Abelian groups (Fact 22)
    Used in the proof of Theorem 2 and in the height eigenfunction analysis.
  • domain assumption The digit substitution generates a well-defined subshift (extensible language)
    Stated in Section 2.1: 'we will assume that they generate subshifts.' This is a checkable but nontrivial condition.
  • domain assumption Height lattice Gamma is coprime with the supertile lattice Q Z^d (Definition 20)
    Used in Lemma 28 and Proposition 31; failing this, the odometer decomposition of the MEF changes.
  • domain assumption Block substitution with rectangular support R = prod [0, l_j - 1] and diagonal Q = diag(l_1,...,l_d) for Section 6
    Needed to define derived substitutions and the pruned coincidence graphs; not all Z^d substitutions satisfy this.
  • domain assumption The matrix Q' = H^{-1} Q H has integer entries in Proposition 31
    Needed to make the pure-base shift substitutive; the authors note in Remark 32 that some such condition must be required.
  • domain assumption A^{-1} circle supp(theta^n) = supp(theta^n) for all n in Proposition 13
    Required for the supertile-shuffling definition and the A circle operation to be well-defined on all supertiles.
invented entities (1)
  • J-derived substitution dJ theta and pruned derived coincidence graph eG(theta,J)^op independent evidence
    purpose: Combinatorial encoding of irregular fibres over the maximal equicontinuous factor for Z^d block substitutions.
    The graph is defined explicitly and used to prove Theorems 46-47; its predictions (fibre cardinalities) are checkable in examples (Examples 44, 49, 50), giving a falsifiable handle within the paper.

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Pith. "Pith review of Radius-zero Extended Symmetries and Irregular Fibres of $\mathbb{Z}^d$-Substitution Subshifts." pith.science (2026). https://pith.science/paper/V7RBP6LJ

@misc{pith2026250613392,
  author       = {Pith},
  title        = {Pith review of: Radius-zero Extended Symmetries and Irregular Fibres of $\mathbbZ^d$-Substitution Subshifts},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V7RBP6LJ}},
  note         = {Machine review of arXiv:2506.13392}
}
abstract

In this work, we consider $\mathbb{Z}^d$-shifts generated by digit substitutions. For such a shift $\mathbb{X}$, we study the elements of the normaliser of $\mathbb{Z}^d$ in the group of self homeomorphisms (called extended symmetries) whose local maps guaranteed by the generalised Curtis--Hedlund--Lyndon theorem have radius-zero. Using the formalism of minimal sets developed by Lema\'{n}czyk, M\"ullner and Yassawi, we provide an algorithm to compute elements of $\mathcal{N}(\mathbb{X})$ that preserve the hierarchical structure. We also investigate the interaction of extended symmetries with (i) the height lattice and (ii) the irregular fibres over the maximal equicontinuous factor. Towards (ii), we introduce the notion of derived substitutions to provide a complete description of the irregular fibres, extending a result by Coven, Quas and Yassawi in the one-dimensional case.

Figures

Figures reproduced from arXiv: 2506.13392 by the authors.

Figure 1
Figure 1. The helix substitution, an example of a two-dimensional digit substi￾tution θ on two letters. Here, D = {0, e1, −e1, e2, −e2} and Q = [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Illustration of the proof that Eq. (4) holds under the assumption on the supertiles for some ℓ := ℓ(x) ∈ Z d , n ⩾ N. Let x ∈ Xθ, with x0 = a say. Up to a shift σc , one sees a (centred) n-supertile around the origin for Φ ◦ σc ◦ θ n (x). On the other hand, one can first shift x before applying Φ so that the n-supertile around the origin containing x0 is centred. One then successively applies Φ and θ n , resulting i… view at source ↗
Figure 3
Figure 3. The coincidence graph for the substitution in Eq. (9) 14 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: A possible height lattice and fundamental domain for a 3 ×3 rectan￾gular substitution, with the associated inflation and height matrices. A pattern of shape D in the pure base corresponds to a pattern of shape HD+DΓ (bottom left). However, applying the substitution map…
Figure 5
Figure 5. Figure 5: Ge(θ, J) op for Example 39 6.2. Derived substitutions and irregular fibres. Let z = (z (1), . . . , z(d) ) ∈ Zℓ, where each z (j) has a base ℓj expansion given by . . . z (j) 2 z (j) 1 z (j) 0 . For each k ∈ N, define nk = (z (1) k , . . . , z (j) k , . . . , z (d) k )…
Figure 6
Figure 6. Figure 6: Illustration of the properties of the points x (k) and x (k+1) and its corresponding encoding in a section of the graph G(θ, J) Fix a subset J ′ ⊆ J and define εJ′ = − P j∈J′ ej (where ej is the j-th element of the canonical basis). Define WJ′ ,m = [supp(θ m) + Q mεJ ]…
Figure 7
Figure 7. Figure 7: An illustration for Wm. Here, m = 1, Q = diag(3, 3), J = {1, 2} and tm = 0. The locations of εJ′, for each J ′ ⊆ J, are also indicated. The gray region corresponds to WJ′ ,m, where J ′ = {1}. where, as before, btm is the vector obtained from tm by removing all coordina…
Figure 8
Figure 8. Figure 8: The level-4 supertiles of the substitution in Example 48 In the following example, we show that the invariance of Cθ under A does not guarantee the existence of an extended symmetry with such a geometric map. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_8.png]
Figure 9
Figure 9. Figure 9: The substitution in Example 49 It is easy to see from the definition of the substitution that the “manta ray pattern” present on the lower right side of each supertile exists for other supertiles with i ∈ {1, 2} as well. For this example, one can compute the two (non-t…
Figure 10
Figure 10. Figure 10: Some supertiles for θ and the column cardinality mosaic Cθ Since the alphabets for the derived substitutions are quite large, we omit them in the paper and just discuss the relevant quantities for the irregular fibres. This substitution admits 657 34 [PITH_FULL_IMAGE…
Figure 11
Figure 11. Figure 11: The inflation rule for the half-hex tiling, with decorations, and a patch for this tiling. The decorations induce the appearance of two distinct chiralities, distinguished by color. a ± 7→ a ± b − a ± c − b ± 7→ a + b ± b ± c ∓ c ± 7→ a − b + c ± c ± [PITH_FULL_IMAGE…
Figure 12
Figure 12. Figure 12: The substitution ϱe on the six-letter alphabet {a ±, b±, c±} 18 6 6 2 2 1 0 0 1 1 0 1 0 0 1 [PITH_FULL_IMAGE:figures/full_fig_p036_12.png]
Figure 13
Figure 13. Figure 13: The coincidence graph associated to the horizontal derived substi￾tution obtained from the decorated half-hex substitution. The numbers indicate the cardinality of the set associated to each vertex. The graph associated to the vertical derived substitution has the sam…

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