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REVIEW 3 major objections 5 minor 12 references

The Hat and Spectre aperiodic monotiles yield quantum error-correcting codes whose code spaces split into two sectors; under the physically natural symmetries, only the Hat keeps a superselected classical chirality bit.

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2026-08-02 00:28 UTC pith:6B3GNC25

load-bearing objection A genuine extension of the Li–Boyle construction to monotiles, with a nice superselection twist; the main gap is an unproved hull identification that the abstract's unconditional claims depend on. the 3 major comments →

arxiv 2607.15326 v1 pith:6B3GNC25 submitted 2026-07-16 quant-ph math.MG

Quantum error-correcting codes from aperiodic monotiles: the Hat and the Spectre

Josep Batle , Adam Bednorz This is my paper
classification quant-ph math.MG MSC 81P7052C2337B50 PACS 03.67.Pp
keywords quantum error correctionaperiodic monotilesHat tilingSpectre tilinglocal indistinguishabilitylocal recoverabilitycut-and-project schemesuperselection
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper extends to the aperiodic monotiles — the Hat and the Spectre — a construction that turned the Penrose tiling into a quantum error-correcting code: superpose a tiling over all its allowed positions and orientations, and no bounded erasure can read or damage the stored quantum information. For the Hat it establishes both required properties on the full family: every finite patch has the same frequency within each of the two chirality classes, and every nonsingular Hat tiling is uniquely determined by the complement of any bounded region. Because there are two local-indistinguishability classes rather than one, the code space splits into two sectors, and a local counter — the reflected-tile fraction — distinguishes them. Under the physically natural gauge group that hides position and orientation but not parity, the Hat code therefore also stores one classical bit, the global handedness of the tiling, alongside its protected quantum sectors; the chiral Spectre loses its analogous label because a 30° rotation swaps its classes. The paper supplies exact frequency data for both codes: reflected hats at densities (3∓√5)/6 and Spectre orientation classes at (5±√15)/10.

Core claim

Both aperiodic monotiles, the Hat and the Spectre, admit erasure-correcting quantum codes built from isometry-orbit superpositions. The Hat's tilings split into two chirality classes with uniform patch frequencies and local recoverability for nonsingular tilings, so erasure of any bounded region is correctable on each class. Under the physically natural gauge group SE(2), the two classes become superselection sectors — only reflections swap them — so the Hat code stores one classical handedness bit, readable with separation Δ = |K|√5/3. The Spectre's classes are swapped by a proper 30° rotation, so under SE(2) they merge; only under a smaller hexagonal subgroup does it form a two-sector code

What carries the argument

The argument rests on three interlocking pieces: the unique-hierarchy theorem, which gives every Hat tiling exactly one decomposition into ever-larger supertiles and turns patch frequencies into Perron–Frobenius limits, yielding local indistinguishability within each chirality class; the torus parametrization of the cut-and-project scheme, a continuous translation-equivariant map assigning each tiling a phase that is one-to-one on nonsingular tilings, so two tilings agreeing outside a bounded region share a phase and coincide, yielding local recoverability; and the census matrix [[1,1],[5,6]] for reflected versus unreflected hats, whose Perron–Frobenius data give the reflected-hat frequencie

Load-bearing premise

The load-bearing premise is that the cut-and-project description of the Hat hull gives a torus parametrization that is one-to-one on nonsingular phases — so two tilings agreeing outside a bounded region must share a phase and therefore be identical; if that fiber-to-singleton correspondence fails on a positive-measure set, or if the hull used is larger than the family of legal Hat tilings, local recoverability for all nonsingular Hat tilings is not established.

What would settle it

Seek a bounded singular pair: two distinct legal Hat tilings that agree everywhere outside some bounded region. Finding any tile-union region, inside any Hat tiling, that admits a second tiling by hats would produce such a pair and refute full local recoverability; extending the exhaustive retiling enumeration beyond the certified 2490-tile patch, or proving the border-forcing property of the inflation, would settle the open question.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Erasure of any bounded region is correctable on all nonsingular Hat tilings, with correctable radius growing as a φ^{2n} per deflation step; the same applies to nonsingular Spectre tilings with growth a(4+√15)^n.
  • The Hat code is a hybrid memory: two erasure-correcting quantum sectors plus one superselected classical bit; coherences between sectors are not protected, but the bit is readable in any bounded window via the reflected-tile fraction.
  • Under SE(2) the Spectre code has a single sector, whereas under the hexagonal subgroup G6 it splits into two sectors separated by orientation parity with Δ = |K|√15/5 tiles.
  • If the singular-pair question is settled in the affirmative — no bounded singular pairs — local recoverability extends from nonsingular to all Hat tilings; the certified 2490-tile patch already excludes every tile-union region occurring in that patch, including all first-corona and radius-2 neighbourhood classes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: The Hat–Spectre contrast suggests a general principle: in any tiling code, a sector label survives gauging by G exactly when the isometry exchanging the LI classes is outside G. Other reflexible aperiodic tilings with reflection-related classes might therefore carry similar classical bits, while chiral tilings under SE(2) generically will not.
  • Editorial inference: Because the chirality bit is invisible to entanglement entropy yet readable by a local counter, this code separates classical and quantum information in a way that could be exploited for combined classical-quantum storage; a lattice implementation on the kisrhombille substrate is a concrete testbed.
  • Editorial inference: The singular-pair question could be attacked by exact retiling enumeration on substitution-grown patches well beyond 2490 tiles; a counterexample would produce an uncorrectable erasure class, while a proof of unique retiling for all tile-unions would complete the code on the full hull including singular tilings.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper extends Li and Boyle's construction of quantum error-correcting codes from aperiodic tilings to the Hat and Spectre monotiles. For the Hat it claims strong local indistinguishability within each of two chirality classes, local recoverability for all nonsingular tilings via the torus parametrization of the underlying cut-and-project scheme, and a reduction of the remaining singular case to a sharply posed retiling question supported by computational evidence on a 2490-tile patch. It further claims that under the physically natural gauge group SE(2) the Hat code is a hybrid memory: two erasure-correcting quantum sectors plus a superselected classical handedness bit, while the Spectre's analogous two-class label is gauged away by a 30-degree rotation. Exact Perron--Frobenius frequencies are derived for both codes, and a comparison across the Li--Boyle family is given.

Significance. If the central recoverability claim can be made fully self-contained, this is a worthwhile contribution: it is the first extension of the Li--Boyle construction to aperiodic monotiles, it identifies a new structural feature (two local-indistinguishability classes leading to a hybrid quantum--classical memory), and it provides exact frequency data with no fitted parameters. The paper is careful to distinguish theorems from finite computational evidence, and the certified SAT enumeration with exact integer arithmetic is a concrete strength. The main obstacle is the unproved identification between the family of nonsingular Hat tilings and the hull on which the torus parametrization is defined; until that is resolved, the 'unconditional' recoverability statement is conditional on an external finite-check assertion. The paper's honest treatment of the open singular-pair question is also a positive feature.

major comments (3)
  1. [§3.2 Remark; §3.3.2; Eq. (8); Theorem 9(i)] Theorem 6 proves recoverability only for tilings in the hull Ω_Hat, where the torus parametrization β is defined (Theorem 6, Steps 1–2). The code, however, is defined on all nonsingular tilings of each class through T^{±,ns}_Hat in Eq. (8), and both the abstract and Theorem 9(i) state erasure correction for all nonsingular Hat tilings. The equality T^+_Hat = Ω_Hat is asserted in the Remark of §3.2 with a sketch ('unique-hierarchy theorem together with the standard finite check ... implicit in [4]') but no proof and no theorem number. Without this equality, a nonsingular tiling outside Ω_Hat has no β-parameter and Theorem 6 does not apply to it. This is the central scope gap: either prove the equality as a lemma (e.g. via the unique hierarchy and primitivity, showing every finite patch of a Hat tiling occurs in a substitution tiling) or state the code and Theorem 9 on Ω_Hat only.
  2. [§3.3.4, Proposition 8 and following paragraph] The computational certificate does not supply the missing ‘standard finite check’ for T^+_Hat = Ω_Hat. Proposition 8 verifies uniqueness of tiling for one 2490-hat patch and, as a consequence, for all tile-union subregions of that patch; it does not enumerate all legal local configurations of the Hat tiling space. The sentence ‘the same enumeration doubles as a finite adjacency check ...’ verifies only the 66 radius-2 and 30 flower classes occurring in the certified patch. A finite patch, however large, cannot by itself rule out a bounded singular pair living outside it, nor prove that every legal local configuration extends uniquely, unless an independent finiteness/completeness argument is supplied. The open status of Lemma 7 is stated correctly, but the use of Proposition 8 as evidence for the hull identification should be removed or supplemented by a completeness argument.
  3. [§4.3; Theorem 12] The Spectre recoverability claim inherits the same hull-identification problem. The text says ‘the proof of Theorem 6 applies verbatim within each LI class,’ but Theorem 12 defines sectors from nonsingular tilings of LI_k, and recoverability is again established only on the relevant hull. Unless LI_k is proved to equal the relevant hull component (or is defined that way), the same gap affects the Spectre sectors under G_6. This should be addressed explicitly in the revision, not merely inferred from the Hat discussion.
minor comments (5)
  1. [§2, Eq. (1)] The group-average state in Eq. (1) is an integral over the noncompact group SE(2) (or E(2)) and is not normalizable as written. This is inherited from [1], but a short sentence stating the formal convention used for the Knill–Laflamme conditions would help readers not already familiar with that construction.
  2. [§4.1, Eqs. (11)–(12)] The terminology around λ_+ = 4+√15 could confuse: it is called ‘area inflation per substitution step’ and later ‘linear inflation per σ² step’. Please state explicitly that Eq. (11) is the single-step substitution matrix, whose PF eigenvalue is the area inflation for one step, while the linear inflation for the chirality-preserving step σ² is 4+√15.
  3. [Table 2 and Table 3, E(2) row for the Spectre] The Spectre tiling family is not closed under reflections (indeed the physical tile is curved so reflected copies cannot fit). The E(2) row for the Spectre is therefore not a valid Li–Boyle code as defined in §2. Please mark that entry as N/A or explicitly define an enlarged family that is E(2)-closed.
  4. [§3.1, Eq. (4)] The two-type census matrix M is asserted without derivation. Since the reflected-hat frequency in Eq. (5) and the bit separation in Eq. (9) depend on it, include the aggregation of the four-metatile substitution of [2] that yields Eq. (4), or give a precise citation to the location in [2] where this census appears.
  5. [§3.4, Eq. (9)] The notation |K|_tiles is used for the expected number of tiles in the window K, but it is not defined. Also, ‘readable in any bounded window’ should be phrased as an expectation separation: a single-shot measurement with finite variance will not distinguish the two classes reliably unless the window is large enough.

Circularity Check

0 steps flagged

No significant circularity: the code properties are derived from external hierarchy and model-set results, not assumed into the target.

full rationale

Walking the derivation chain: Theorem 2 computes patch frequencies from the unique-hierarchy theorem [2, Thm. 5.1] plus the Perron–Frobenius theorem, so strong local indistinguishability is derived rather than assumed. Theorem 6 uses the torus parametrization of regular model sets from [7] (with the model-set description from [4]); its proof is a standard almost-automorphic rigidity argument and does not feed the target recoverability claim into its assumptions. The code spaces in Eq. (8) are defined only after these hypotheses, and Theorem 9 applies the Knill–Laflamme conditions directly. The superselected-bit separation in Eq. (9) is the difference of exact Perron–Frobenius frequencies (3∓√5)/6, not a fitted parameter renamed as a prediction. The Spectre results similarly transfer from [3] and [5] verbatim. No self-citation by the present authors is load-bearing anywhere. The one item that is not fully proved in-text is the identification T^+_Hat = Ω_Hat asserted in the Remark in §3.2; this is a gap affecting the abstract's phrasing over all nonsingular Hat tilings, but it is not a circular reduction—the paper explicitly describes it as a standard finite check implicit in [4] and additionally supports it by the certified two-corona-forcing check in Proposition 8. There is no step where an equation reduces to its own input by construction, and no fitted parameter is presented as a prediction.

Axiom & Free-Parameter Ledger

0 free parameters · 8 axioms · 0 invented entities

The code construction imports the entire tiling-space machinery (unique hierarchy, cut-and-project regularity, LI-class structure) from [2]-[7]; no new physical entities are introduced. The most exposed assumptions are the hull identification T^+_Hat = Omega_Hat and the un-audited computational certificate.

axioms (8)
  • domain assumption Unique-hierarchy theorem for the Hat: every Hat tiling has a unique infinite hierarchy of supertiles and all Hat tilings arise from the substitution [2, Thm 5.1].
    Basis of Theorem 2 frequency computation and Proposition 5 reduction; not re-derived.
  • domain assumption Unique-hierarchy theorem for the Spectre [3, Thm 2.2].
    Plays the same role for Section 4 as the Hat hierarchy plays in Section 3.
  • domain assumption The Hat hull is topologically conjugate to the CAP tiling hull, a regular model set with pure-point spectrum; the system is almost automorphic with torus parametrization [4],[7].
    Underwrites Theorem 6 and the claim of singleton fibers over nonsingular parameters.
  • domain assumption The Spectre hull arises from a 4:2 cut-and-project scheme with regular windows and carries a torus parametrization [5],[7].
    Transfers the nonsingular recoverability argument to the Spectre.
  • domain assumption The Spectre has two LI classes exchanged by a 30-degree rotation, with orientation-parity frequencies (5 ± sqrt(15))/10 [5].
    Used in Theorem 11 and Theorem 12; the class structure is imported from [5].
  • domain assumption T^+_Hat = Omega_Hat: the full right-handed tiling family equals the translation hull.
    Asserted in the Remark in §3.2 as following from the unique-hierarchy theorem plus a finite check; not proved in this text, yet needed to move Theorem 6 from the hull to 'all nonsingular Hat tilings'.
  • domain assumption The Li-Boyle formal code framework: group-averaged states over the noncompact isometry group and the Hilbert space of infinitely many tilings are well defined and satisfy Knill-Laflamme formally.
    Inherited from [1]; no regularization or finite-size limiting procedure is supplied.
  • domain assumption The computational verification pipeline (exact cover/SAT, exact integer lattice arithmetic, Che?ritat datasets) is correct and exhaustive as implemented.
    The Proposition 8 certificate cannot be audited from the submitted text; its soundness is assumed from the described methods and ancillary files.

pith-pipeline@v1.3.0-alltime-deepseek · 17294 in / 23194 out tokens · 239680 ms · 2026-08-02T00:28:54.511363+00:00 · methodology

0 comments
read the original abstract

Li and Boyle showed that the Penrose tiling defines a quantum error-correcting code: superpositions of tilings over isometry orbits protect quantum information against erasure of any bounded region. We extend the construction to the aperiodic monotiles discovered by Smith, Myers, Kaplan and Goodman-Strauss. For the Hat, we prove strong local indistinguishability for all Hat tilings, and we prove local recoverability unconditionally for all nonsingular Hat tilings via the torus parametrization of the underlying cut-and-project scheme. The remaining singular case reduces to one sharply posed geometric question -- can a region that is a union of hats be retiled a second way? -- which we verify computationally has no counterexample up to a substantial scale: a certified $2490$-tile patch admits precisely one tiling by hats, so all $2^{2490}$ of its tile-subregions retile uniquely. Unlike the Penrose, Ammann-Beenker and Fibonacci tilings, both monotiles form two local-indistinguishability classes, so their code spaces split into two erasure-correcting sectors carrying a superselected classical label. Whether the label survives depends on which isometries are gauged: the Spectre's classes are exchanged by a $30^{\circ}$ rotation and merge once all proper isometries are gauged, whereas the Hat's are exchanged only by reflections. Under the physically natural gauge group $SE(2)$, the Hat code therefore stores one robust classical bit -- the handedness of its long-range order, readable in any bounded window with separation $\Delta = |K|\sqrt{5}/3$ -- alongside its protected quantum sectors. It is the reflexible monotile, not the chiral one, that carries the chirality bit. We give the exact Perron-Frobenius data of both codes, including the per-class reflected-Hat frequencies $(3\mp\sqrt{5})/6$ and the Spectre orientation-class frequencies $(5\pm\sqrt{15})/10$.

Figures

Figures reproduced from arXiv: 2607.15326 by Adam Bednorz, Josep Batle.

Figure 1
Figure 1. Figure 1: The prototiles. The hat (blue) is a union of eight [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Convergence of the reflected-hat fraction [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Why recoverability can fail: the Fibonacci mecha [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The reduction of Proposition 5: composing n times, the region where the two tilings’ supertile structures may disagree shrinks relative to the supertile size and con￾verges to the universal fixed point R ∗ = R0/φ, independently of the size ρK of the erased region. Deflation compresses ev￾ery bounded erasure to this scale — and no further. proof. The constant is explicit: the classification rules of [2, §4]… view at source ↗
Figure 5
Figure 5. Figure 5: One of the region-retiling tests, shown on the [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Handedness census of the certified 2490-hat patch of §3.3: 2177 hats (blue), 313 anti-hats (orange); reflected fraction 0.1257 against the asymptotic (3− √ 5)/6 = 0.1273. This tiling belongs to T + Hat; in its mirror image the colours — and the value of the superselected bit — are exchanged. Any bounded window estimates the reflected fraction and thereby reads the bit, with the macroscopic separation of Eq… view at source ↗
Figure 7
Figure 7. Figure 7: A Spectre tiling generated by the Γ–Ψ cluster sub￾stitution of [3] (four inflation steps, 4401 tiles; central disk shown), coloured by orientation parity as in [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The twelve Spectre orientations, coloured by ori [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗

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Reference graph

Works this paper leans on

12 extracted references · 4 linked inside Pith

  1. [1]

    Li and L

    Z. Li and L. Boyle,The Penrose tiling is a quantum error-correcting code, preprint (2023); arXiv:2311.13040

    Pith/arXiv arXiv 2023

  2. [2]

    Smith, J

    D. Smith, J. S. Myers, C. S. Kaplan, and C. Goodman-Strauss,An aperiodic monotile, Combinatorial Theory4(1) (2024), #6, 91 pp.; doi:10.5070/C64163843; arXiv:2303.10798

    Pith/arXiv arXiv 2024

  3. [3]

    Smith, J

    D. Smith, J. S. Myers, C. S. Kaplan, and C. Goodman-Strauss,A chiral aperiodic monotile, Combinatorial Theory4(2) (2024), #13, 25 pp.; doi:10.5070/C64264241; arXiv:2305.17743

    Pith/arXiv arXiv 2024

  4. [4]

    270(1) (2025), 449–485; doi:10.1007/s11856-025- 2780-8; arXiv:2305.05639

    M.Baake, F.Gähler, andL.Sadun,Dynamics and topology of the Hat family of tilings, IsraelJ.Math. 270(1) (2025), 449–485; doi:10.1007/s11856-025- 2780-8; arXiv:2305.05639

    arXiv 2025

  5. [5]

    Baake, F

    M. Baake, F. Gähler, J. Mazáč, and L. Sadun,On the long-range order of the Spectre tilings, Discrete Comput. Geom. (2025), doi:10.1007/s00454-025- 00756-z; arXiv:2411.15503

    Pith/arXiv arXiv 2025

  6. [6]

    Baake, F

    M. Baake, F. Gähler, J. Mazáč, and A. Mitchell, Diffraction of the Hat and Spectre tilings and some of their relatives, J. Math. Phys.66(9) (2025), 092707, 26 pp.; doi:10.1063/5.0264955; arXiv:2502.03268

    arXiv 2025

  7. [7]

    Schlottmann,Generalized model sets and dy- namical systems, inDirections in Mathematical Quasicrystals, eds

    M. Schlottmann,Generalized model sets and dy- namical systems, inDirections in Mathematical Quasicrystals, eds. M. Baake and R. V. Moody, CRM Monogr. Ser. 13, Amer. Math. Soc., Provi- dence, RI (2000), 143–159

    2000

  8. [8]

    C. S. Kaplan,hatviz: visualization tools for hat tilings, software repository, github.com/isohedral/hatviz (2023); the canon- ical hat outline used in §3.3 ishat_outlinein geometry.js

    2023

  9. [9]

    Solomyak,Nonperiodicity implies unique com- position for self-similar translationally finite tilings, Discrete Comput

    B. Solomyak,Nonperiodicity implies unique com- position for self-similar translationally finite tilings, Discrete Comput. Geom.20(2) (1998), 265–279; doi:10.1007/PL00009386

    doi:10.1007/pl00009386 1998

  10. [10]

    Solomyak,Dynamics of self-similar tilings, Ergodic Theory Dynam

    B. Solomyak,Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems17(3) (1997), 695–738; doi:10.1017/S0143385797084988

    doi:10.1017/s0143385797084988 1997

  11. [11]

    Baake and U

    M. Baake and U. Grimm,Aperiodic Order. Vol. 1: A Mathematical Invitation, Encyclopedia of Mathematics and its Applications, Vol. 149, Cambridge University Press, Cambridge (2013); doi:10.1017/CBO9781139025256

    doi:10.1017/cbo9781139025256 2013

  12. [12]

    Queffélec,Substitution Dynamical Systems — Spectral Analysis, 2nd ed., Lecture Notes in Math- ematics 1294, Springer (2010)

    M. Queffélec,Substitution Dynamical Systems — Spectral Analysis, 2nd ed., Lecture Notes in Math- ematics 1294, Springer (2010). 13

    2010