{"id":"8db5ec34-5df4-41f8-9e67-9c41429d5d85","arxiv_id":"2506.13392","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For Z^d substitution subshifts, the paper gives conditions for radius-zero extended symmetries to shuffle supertiles, proves the height lattice is invariant, and describes irregular fibres, but the proofs have significant gaps.","lead":"The paper studies 'extended symmetries' of d-dimensional substitution subshifts, giving algorithms and structural constraints for radius-zero examples. It also claims a complete description of irregular fibres using new derived substitutions, but key proofs contain gaps.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 16's sufficiency direction uses an identity not implied by condition (2), and Example 18 contradicts the stated condition; the core algorithm is not established as written.","rationale":"The reader's weakest_assumption identified the Corollary 30 / Theorem 2 height-eigenfunction gap, which is also a real and significant problem. However, the most load-bearing concern for the central advertised contribution is Theorem 16, since it is the engine of the algorithm in Theorem 1. The algebraic error there is not a subtle missing justification but a direct contradiction with the paper's own Example 18: the displayed encodings make condition (2) impossible to satisfy, while the example claims it is satisfied with τ′=id. This can be verified by hand in a few lines, so it is a very concrete defect. I therefore agree with the reader's REJECT verdict, but for a concern that is even closer to the paper's main theorem than the one the reader singled out as weakest.","tokens_in":34016,"tokens_out":19783,"duration_ms":206510,"concrete_test":"Independently check Example 18 against the displayed definitions: compute ν and ν̄ from the idempotent columns θ_0 and θ_5 as in Section 3.2, then test whether any permutation τ′ of {0,1} satisfies τ′∘ν = ν̄∘τ for τ=(bc). The equations force τ′(0)=1 and τ′(1)=1, so no such τ′ exists, contradicting the example's claim that τ′=id works. This two-line check settles that condition (2), or the proof's use of it, is wrong; either way Theorem 16's sufficiency direction is currently unsupported.","verdict_should_be":"REJECT","load_bearing_attack":"Theorem 1 rests on Theorem 16, whose 'if' direction aims to prove θ_j(τ(a)) = τ(θ_{A^{-1}⊙j}(a)) from conditions (1)–(3). In the third equality of that proof, ν(τ(a)) is replaced by τ′(ν̄(a)) using 'Property (2)'. But condition (2) is τ′∘ν = ν̄∘τ, i.e. τ′(ν(x)) = ν̄(τ(x)). This gives ν̄(τ(a)) = τ′(ν(a)), hence ν(τ(a)) = (τ′)^{-1}ν̄(τ^2(a)) for involutive τ, not τ′ν̄(a). The displayed chain is therefore a non sequitur. The problem is not merely typographical: in Example 18, with ν = ν̄ given by a↦0, b,c↦1 and τ = (bc), condition (2) forces τ′(0)=1 and τ′(1)=1, so no permutation τ′ exists; yet the example asserts τ′=id satisfies it. Thus the statement of condition (2), or its use in the proof, is inconsistent with the paper's own worked example. Since Theorem 1 is the paper's main algorithmic claim, this gap is load-bearing: correctness of the algorithm depends on exactly this equivalence.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":34237,"tokens_out":21571,"duration_ms":210825,"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":[{"comment":"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.","section":"§3.3 (Theorem 16)"},{"comment":"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.","section":"§4.4 (Corollary 30)"}],"minor_comments":[{"comment":"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.","section":"§3.3 (Theorem 16, proof)"},{"comment":"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.","section":"§6.2 (Theorem 46)"},{"comment":"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.","section":"§3.1 (Proposition 13)"}],"recommendation":"major_revision","confidential_remarks":"The decisive issue is Theorem 16. If the authors can replace condition (2) with a correct compatibility relation and supply a valid proof of the converse, the paper's main claims are likely defensible; but if the characterization itself is wrong, the algorithmic claim collapses. I therefore recommend major revision rather than outright rejection, since no counterexample to the theorem is apparent and the required repair may be local to Section 3.3. The Corollary 30 point is, in my view, a repairable presentation gap rather than a fatal flaw."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The paper introduces a genuinely new tool, the J-derived substitution, and uses it to give the first higher-dimensional analogue of the CQY16 irregular-fibre description for non-bijective Z^d substitution shifts. That alone is a solid contribution. But the paper is not sound as written: the sufficiency direction of Theorem 16, which underpins Theorem 1, uses an identity that does not follow from the stated condition, and Theorem 3 is stronger than what Theorem 47 actually proves.\n\nWhat is new and good: the derived substitution construction (Defs 38, 41) is a real addition; Theorem 2's height-lattice invariance is a nice constraint and the proof strategy via eigenfunctions is sensible; Example 49 is honest about its limits; the paper cites prior work correctly and does not hide its dependencies.\n\nWhere it is soft:\n1. Theorem 16. In the 'if' direction, the third equality replaces nu(tau(a)) with tau' composed with bar-nu(a), citing Eq. (7). Eq. (7) is tau' composed with nu equals bar-nu composed with tau, which gives nu(tau(a)) = (tau')^{-1} composed with bar-nu(tau^2(a)) in general, not tau' composed with bar-nu(a). The rest of the chain depends on this move, so the algorithmic characterization is not established as written. The stress-test's secondary claim that Example 18 contradicts Eq. (7) is mistaken—tau'=id does satisfy the equation there—but the main gap in the proof is real. This looks correctable, but as it stands Theorem 1 is unsupported.\n2. Corollary 30. The proof applies Lemma 29 to a subgroup the lemma doesn't handle: Lemma 29 requires the first factor to be preserved, but the paper only establishes preservation of the torsion factor. Swapping factors gives the desired result, so this is a fix rather than a fatal flaw, but the current text is incomplete.\n3. Theorem 3 versus Theorem 47. Theorem 47 explicitly excludes integer-coordinate points z in Z^d, and the paper does not supply the missing argument that the sofic shift Z_theta also captures those fibres. The one-dimensional version in CQY16 needed extra work for fixed points, so this is not a negligible point.\n\nAll three are load-bearing because Theorems 1-3 are the paper's headline claims.\n\nFor whom: people working on automorphism and normaliser groups of substitution shifts, and on fibre structure of MEFs. They will find the constructions and examples useful even before the proofs are corrected.\n\nRecommendation: This deserves a serious referee, not a desk reject. I would send it back for major revision and re-review. If the authors fix the identity in Theorem 16 and close the integer-coordinate gap in Theorem 3, it is a worthwhile paper. I would not cite it for the theorems yet.","headline":"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.","tokens_in":34810,"tokens_out":6462,"would_cite":false,"duration_ms":54789,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37B10","52C23","37B05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["extended symmetries","Z^d substitution subshifts","radius-zero symmetries","height lattice","irregular fibres","derived substitutions","minimal sets","coincidence graph"],"falsifier":"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 $A\\Gamma=\\Gamma$.","tokens_in":33771,"feed_emoji":"🧩","tokens_out":5764,"duration_ms":51365,"temperature":0.7,"pith_summary":"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.","feed_headline":"Algorithm lists all shuffling symmetries of substitution shifts","feed_subtitle":"Height invariance and an explicit sofic shift pin down the normaliser and its irregular fibres.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the minimal-set encoding and the criterion that radius-zero automorphisms are supertile-shuffling, which the present algorithm generalises to extended symmetries.","marker":"[MY21]"},{"why":"Provides the one-dimensional irregular-fibre description via the coincidence graph that Theorem 3 extends to block substitutions.","marker":"[CQY16]"},{"why":"Gives the algorithm for admissible extended symmetries of bijective substitutions, which the non-bijective case here generalises.","marker":"[BLM23]"},{"why":"Defines extended symmetries, the normaliser cocycle, and the Curtis–Hedlund–Lyndon form that the paper uses throughout.","marker":"[BRY18]"},{"why":"Supplies the synchronising-substitution construction used to realise minimal sets as idempotent columns.","marker":"[LM20]"},{"why":"Establishes recognizability of aperiodic substitutions, which is used to desubstitute points and define tile coordinates.","marker":"[Sol98]"},{"why":"Provides the comparison example of large normalisers with all matrices in $\\mathrm{GL}(2,\\mathbb{Z})$, used to frame the height-lattice restrictions.","marker":"[CP24]"}],"fun_headline_variants":["Radius-zero symmetries of substitution shifts fully computed","Height invariance plus explicit sofic shift fix normaliser","Algorithm decides radius-zero extended symmetries via minimal sets","Irregular fibres described by derived substitutions and sofic shift","Normaliser structure: height lattice and sofic shift are key invariants"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Radius-zero symmetries of substitution shifts fully computed","Height invariance plus explicit sofic shift fix normaliser","Algorithm decides radius-zero extended symmetries via minimal sets","Irregular fibres described by derived substitutions and sofic shift","Normaliser structure: height lattice and sofic shift are key invariants"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000217,"raw_usage":{"total_tokens":1417,"prompt_tokens":910,"completion_tokens":507,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":427}},"tokens_in":526,"tokens_out":507,"duration_ms":5036,"temperature":1.0,"reasoning_tokens":427,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:04:04.482181+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $A\\Gamma=\\Gamma$.","supporting_citations":[],"review_version":1}