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REVIEW 5 minor 18 references

This paper constructs explicit quantum Latin squares of order 6 with five new cardinalities, completing the attainable spectrum through 28.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-02 06:47 UTC pith:RX62KO62

load-bearing objection The three new order-six cardinalities (23, 25, 27) are genuine and the paper completes the spectrum through 28; the only real soft spot is the one-line inner-product audit behind cardinality 25, which is checkable but not shown.

arxiv 2607.11800 v2 pith:RX62KO62 submitted 2026-07-13 math.CO

Quantum Latin Squares of Order Six with Cardinalities Nineteen, Twenty-One, Twenty-Three, Twenty-Five, and Twenty-Seven

Zhipeng Xu This is my paper
classification math.CO MSC 05B15
keywords quantum Latin squarecardinalitycomplex Hadamard matrixButson matrixSchur productdirect-sum constructionorder six
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 establishes that every cardinality from 6 through 28, with the single exception of 7, is attained by some quantum Latin square of order 6. The new content is a set of five explicit arrays realizing cardinalities 19, 21, 23, 25, and 27, built from three mechanisms: pairwise Schur products of columns of complex Hadamard matrices, a one-parameter direct-sum family in C^4 ⊕ C^2, and mixed Schur products of a Hadamard matrix with a row-permuted copy. Each construction comes with a finite certificate—modular exponent signatures for the Butson cases and explicit inner-product lists for the direct-sum cases—so the phase-class counts are checkable by hand. If the calculations hold, the only remaining order-six cardinalities are 29, 32, and 35.

Core claim

Each of the five arrays is a 6×6 grid of unit vectors in C^6 whose rows and columns form orthonormal bases, with the number of distinct vectors after identifying global phases equal to the claimed cardinality. Cardinality 19 arises from an eighth-root-of-unity Butson matrix whose twenty-one unordered column-pair Schur products have exactly one triple coincidence; cardinality 21 comes from a third-root-of-unity Butson matrix whose twenty-one unordered products are all distinct, attaining the symmetric upper bound. Cardinalities 23 and 25 come from the same direct-sum design: at the symmetric parameter value a=b=1/√2 two pairs of vectors in the four-dimensional subspace coincide, giving 19 ray

What carries the argument

The phase-class counting is done with exponent signatures: for entries in Z_q, the Schur product of two columns is encoded as an exponent vector, and after dephasing two products are equal exactly when their signature vectors coincide. For the direct-sum family, the machinery is a fixed set of twelve orthonormal bases of R^4 that are continuous in a parameter (a,b) on the unit circle; changing the parameter splits coincidences among the labeled vectors without altering the bases. The mixed construction exploits a row permutation that preserves a column multiplier d, forcing one 3×3 block to repeat while all other signatures stay distinct.

Load-bearing premise

The cardinality-25 claim rests on a single 'direct calculation' that no two of the 21 labeled four-dimensional rays are phase-equivalent; if that finite list of inner products contains a missed pair with modulus exactly 1, the cardinality would drop below 25.

What would settle it

Check the 21 vectors A,…,U in the (a,b)=(4/5,3/5) direct-sum construction: compute all 210 off-diagonal inner products and verify that the maximum modulus is 24/25 < 1. Alternatively, verify that the signature table for the cardinality-27 construction has exactly nine repeated five-tuples and eighteen singletons.

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

If this is right

  • If correct, Spec(QLS(6)) ∩ [6,28] is exactly {6} ∪ {8,9,…,28}, with 7 proven impossible.
  • The cardinality-21 construction attains the upper bound of 21 for symmetric Schur products, showing the bound is sharp.
  • The direct-sum family demonstrates that cardinality can be tuned by splitting ray coincidences while keeping the row and column bases unchanged.
  • The mixed Hadamard construction removes the symmetry constraint v_ij = v_ji, opening a concrete search path for the unresolved values 29, 32, and 35.
  • Every construction is accompanied by a finite certificate—signature tables or inner-product lists—making the claims independently verifiable.

Where Pith is reading between the lines

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

  • The rational parameter choice (4/5,3/5) suggests that other Pythagorean triples in the same direct-sum family might yield additional distinct cardinalities, possibly approaching 29 without changing the base structure.
  • The row-permutation enumeration for the cardinality-27 matrix shows that most permutations give cardinality 36; a systematic search over row permutations of other Butson matrices may resolve 29 and 32.
  • The exponent-signature method over Z_q is quite general and could be applied to other orders where Butson matrices exist, potentially yielding new spectrum results beyond order six.
  • The completeness of the spectrum through 28 depends on the accuracy of a prior classification table; an independent audit of that table would strengthen the corollary.

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

0 major / 5 minor

Summary. The paper gives explicit quantum Latin squares of order 6 with cardinalities 19, 21, 23, 25, and 27, where global phase is modded out. The constructions are: (1) a symmetric Schur-product square built from a BH(6,8) matrix, with one deliberate triple coincidence among the 21 unordered products; (2) the same construction using Tao's isolated BH(6,3), whose 21 unordered Schur products are shown to be pairwise phase-inequivalent via an exponent-signature table; (3) a one-parameter direct-sum design in C^4⊕C^2 that gives cardinality 23 at a=b=1/√2 and cardinality 25 at (a,b)=(4/5,3/5); and (4) a mixed Schur-product square from a BH(6,6) and a row-permuted copy, with an explicit Z_6 signature table giving 27 phase classes. The paper then combines these values with the prior spectrum summary in [18, Table 6] to conclude that every c in {6}∪{8,...,28} occurs and c=7 does not, leaving only 29, 32, and 35 unresolved.

Significance. If the computations are correct, the paper closes three previously open order-six cardinalities (23, 25, and 27) and gives explicit certificates for all its claims. Its main strengths are concreteness and verifiability: the Hadamard matrices, signature tables, direct-sum bases, and phase-class certificates C23 and C25 are printed, and the arithmetic is exact (roots of unity and rational/quadratic-surd inner products). The direct-sum construction with fixed incidence pattern and two parameter values is a clean mechanism for changing cardinality by ray splitting, and the row-permutation census in Remark 2 is a useful addition. The result is a solid computational contribution to the classification of QLS(6) cardinalities, though it does not change the broader conceptual landscape.

minor comments (5)
  1. [§6.2, Theorem 4] The cardinality-25 claim rests on the assertion that the off-diagonal inner products among A,...,U have the displayed 13-value set and, in particular, maximum 24/25<1. The proof only says 'direct calculation'. Since this is the least visibly documented step, please provide a reproducible audit: a table of the 210 inner-product values, a short code snippet, or an appendix listing the pairs attaining each value. I spot-checked F·T and J·U (both 24/25) and the orthonormal sets in Proposition 5, and found no error; the request is for completeness and reader confidence rather than a correction.
  2. [§5] The sentence 'pairwise distinct in Z6_3' should read '(Z_3)^6' or 'Z_3^6'; the current notation is ambiguous.
  3. [Remark 2] The census of all 720 row permutations is stated without detail. If this enumeration is to be part of the record, please include the counting method or a script; otherwise it can be labeled as a computational observation.
  4. [Corollary 1] The completeness half of the corollary depends on the accuracy of [18, Table 6] for previously known attainable values. The dependence is stated, but it would be helpful to explicitly separate 'new values proved here' from 'values taken from the literature' in the proof of Corollary 1.
  5. [Eq. (11)] The typesetting of H27 makes the block structure hard to read; a display closer to the block form [[F_3, D F_3],[F_3, -D F_3]] would improve clarity.

Circularity Check

0 steps flagged

No circularity: the five cardinalities are derived from printed finite certificates (signature tables, basis lists, label arrays), not from the target values; prior spectrum results are external citations, not self-imported premises.

full rationale

Each construction supplies explicit data and the cardinality is counted from that data: the order-19 and order-21 signatures are fully listed in Appendices A and B; the order-27 signature table (15) is displayed in full; the order-23 and order-25 label arrays C23 and C25 are given; and Proposition 5 lists the twelve orthonormal bases. The parameter (a,b) is chosen at a rational point, and the claimed inner-product maximum 24/25 < 1 is a finite arithmetic assertion independent of the conclusion. This is the least audited step (Theorem 4's 'direct calculation' is not printed pair-by-pair), but a missing or imperfect computational audit is a correctness risk, not circularity: the assertion is not defined in terms of the cardinality target, and the certificate is independently checkable. The dependence on [18, Table 6] and on [14,16] for the impossibility of cardinality 7 is external to this manuscript and does not involve self-citation. No step reduces by construction to its own inputs, so the circularity score is 0.

Axiom & Free-Parameter Ledger

1 free parameters · 3 axioms · 0 invented entities

The only genuinely free design parameter is (a,b) in the direct-sum family; its two chosen values are hand-picked to yield the desired ray coincidences and splittings, and the proofs independently verify the consequences. No physical constants, new particles, or invented entities are introduced. The main external input is the prior spectrum summary [18], which is a cited domain assumption rather than a free parameter.

free parameters (1)
  • a,b in the direct-sum family (a²+b²=1, a,b>0) = a = b = 1/√2 for cardinality 23; a = 4/5, b = 3/5 for cardinality 25
    One-parameter family of linked orthonormal bases in C^4 ⊕ C^2. The symmetric point produces the pair coincidences F=T and J=U; the rational 3-4-5 point splits both pairs. These values are chosen by hand to realize the target cardinalities, and the proofs verify the resulting distinctness rather than fitting a model to data.
axioms (3)
  • domain assumption The previously attainable order-six cardinalities are exactly as summarized in [18, Table 6].
    Corollary 1 combines this external summary with the new values 23, 25, 27. If that table is incomplete or incorrect, the claimed completeness through 28 fails.
  • domain assumption A QLS(n) cannot have cardinality n+1, as cited to [14,16].
    Used in Corollary 1 to exclude cardinality 7 for n=6; the theorem is not proved in this paper.
  • standard math Standard properties of complex Hadamard and Butson matrices: row/column permutations preserve the Hadamard property, and Schur products of columns of dephased Hadamard matrices form orthonormal bases.
    Used throughout Propositions 1, 2, and the constructions in Sections 3 and 7.

pith-pipeline@v1.3.0-alltime-deepseek · 9314 in / 31758 out tokens · 277519 ms · 2026-08-02T06:47:37.940612+00:00 · methodology

0 comments
read the original abstract

We give explicit quantum Latin squares of order $6$ with cardinalities $19$, $21$, $23$, $25$, and $27$, where vectors differing only by a global phase are identified. Cardinality $19$ is obtained from a $BH(6,8)$ matrix whose unordered Schur products have one controlled triple coincidence. Cardinality $21$ is obtained from Tao's $BH(6,3)$ matrix; the twenty-one unordered Schur products have pairwise distinct exponent signatures over $\mathbb{Z}_3$. Cardinalities $23$ and $25$ arise from one parameterized direct-sum design in $\C^6=\C^4\oplus\C^2$: at the symmetric parameter value two pairs of four-dimensional rays coincide, while the rational point $(a,b)=(4/5,3/5)$ splits both pairs without changing any row or column basis. Finally, cardinality $27$ is obtained by mixed Schur products of a $BH(6,6)$ matrix and a copy with two rows exchanged; the corresponding signatures over $\mathbb{Z}_6$ form exactly nine two-element phase classes and eighteen singleton classes. Together with previously known attainable values and the general impossibility of cardinality $7$, these constructions complete the order-six spectrum through $28$ and leave $29$, $32$, and $35$ as the remaining unresolved values.

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