{"id":"a9ff6f98-9b18-43e3-8331-ebcc8985f873","arxiv_id":"2607.11800","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"Explicit six-by-six quantum Latin squares with 19, 21, 23, 25, and 27 distinct states are constructed, completing the order-six cardinality list through 28.","lead":"Quantum Latin squares are tables of quantum states in which each row and each column forms a complete set of mutually perpendicular states. This paper constructs five new six-by-six examples with 19, 21, 23, 25, and 27 distinct states, closing previously open entries in the order-six spectrum.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Cardinality 25 depends on Theorem 4's unshown inner-product audit; no error found in spot checks, but exact verification would settle it.","rationale":"I read the paper in good faith and audited the main finite certificates. The H19 exponent matrix and Proposition 3 residue patterns are consistent; the only intended repetition among the 21 Schur signatures is (0,1),(2,5),(3,4), giving cardinality 19. The Tao matrix signatures in Appendix B are pairwise distinct, giving cardinality 21. For the direct-sum family, I verified all twelve orthonormal bases in Proposition 5 by direct dot products, and the phase-class certificates C23 and C25 are label-consistent with the array. The H27 signature table gives exactly 27 unique normalized signatures, with only the nine predicted 3x3-block repetitions. No internal inconsistency emerged. The single most load-bearing step is Theorem 4's finite inner-product audit: the paper asserts the exact set of 210 inner products but does not show which pairs realize each value. If that audit contained an unnoticed pair of parallel vectors, cardinality 25 would fail. However, spot checks and the printed C25 certificate support the stated result, so I do not regard this as a demonstrated error. The reader's weakest assumption identified the same step, and my read does not change the ACCEPT verdict; the recommended action is to run an exact-arithmetic verification of that audit, and optionally to confirm [18, Table 6] covers the remaining attainable values in Corollary 1.","tokens_in":9679,"tokens_out":37551,"duration_ms":288750,"concrete_test":"Recompute with exact arithmetic all 210 dot products among A,...,U at (4/5,3/5); if the maximum off-diagonal modulus is exactly 24/25 (<1) and the resulting phase-class labeling matches C25, the cardinality-25 claim holds. Any modulus-1 value would invalidate Theorem 4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The new value 25 rests on Theorem 4's one-line 'direct calculation' that the off-diagonal inner products among the 21 vectors A,...,U at (a,b)=(4/5,3/5) have maximum modulus 24/25 < 1. If an omitted pair also satisfied |<X,Y>| = 1, two U-rays would coincide and card(Φ25) would fall below 25, contradicting the printed certificate C25. This is not an identified error: spot checks confirm F·T = J·U = 24/25, the listed basis sets in Proposition 5 are orthonormal, and C25's 25 labels are consistent with the array. However, the proof as written does not display the full 210-pair audit, so it is the least machine-checked step. The other constructions (H19 residue patterns, H21 signatures, H27 signature table, Proposition 5 Gram matrices) are fully listed or easily verified and showed no issues. The completeness half of Corollary 1 additionally depends on [18, Table 6] for the previously known values, which is external to this manuscript.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":9910,"tokens_out":16170,"duration_ms":136324,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"§6.2, Theorem 4"},{"comment":"The sentence 'pairwise distinct in Z6_3' should read '(Z_3)^6' or 'Z_3^6'; the current notation is ambiguous.","section":"§5"},{"comment":"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.","section":"Remark 2"},{"comment":"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.","section":"Corollary 1"},{"comment":"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.","section":"Eq. (11)"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a careful computational construction with all main claims explicitly certified or easily verifiable. My only substantive request is for a fuller audit of the Theorem 4 inner-product calculation; this is a local, fixable issue rather than a correctness problem. The referee's spot checks found no errors. The editor may wish to confirm that the cited prior spectrum [18, Table 6] indeed contains all values asserted as previously known, since the completeness claim in Corollary 1 depends on that external summary."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper does what it says. It gives explicit QLS(6) constructions for cardinalities 19, 21, 23, 25, and 27, and 23/25/27 were open in [18, Table 6]. That closes the spectrum through 28 (with 7 impossible). I checked the finite certificates — H19 residue patterns, H21 signatures, the signature table for 27, and the Gram matrices for the direct-sum bases — and found nothing wrong. This is a real advance, not a restatement.\n\nThe nicest piece is the direct-sum family: one array Φ(a,b) in C4⊕C2 that gives 23 at the symmetric point and 25 at (4/5,3/5) by splitting two ray coincidences. The row-permutation trick for 27 is also clever. The paper is honest about its certificates and the mechanisms are reusable.\n\nSoft spots, in proportion. Theorems 3 and 4 assert a direct calculation of all off-diagonal inner products among 21 vectors. Theorem 4's conclusion max = 24/25 < 1 is load-bearing for cardinality 25, and the full 210-pair audit isn't printed. I spot-checked several pairs and the C25 certificate is consistent, so this is a verifiability gap, not an identified error. A referee should ask for the full table or a short verification script. Minor: Corollary 1 leans on the spectrum summary in [18, Table 6]; if that summary has a gap, the completion claim weakens, though the three new values stand on their own. Also no code or data files are shipped, but the explicit matrices are enough for exact verification.\n\nThis is a niche paper for people tracking QLS cardinalities or complex Hadamard matrices. It deserves a serious referee; the main request is to make the inner-product audit in §6.2 fully explicit. I'd accept with light-to-moderate revision.","headline":"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.","tokens_in":10430,"tokens_out":2219,"would_cite":true,"duration_ms":20614,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05B15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs explicit quantum Latin squares of order 6 with five new cardinalities, completing the attainable spectrum through 28.","keywords":["quantum Latin square","cardinality","complex Hadamard matrix","Butson matrix","Schur product","direct-sum construction","order six"],"falsifier":"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.","tokens_in":9519,"feed_emoji":"","tokens_out":4447,"duration_ms":35727,"temperature":0.7,"pith_summary":"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.","feed_headline":"Quantum Latin squares of order six hit every size 6 to 28 except 7","feed_subtitle":"Explicit arrays give cardinalities 19, 21, 23, 25, 27, leaving only 29, 32, 35 open.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Quantum Latin squares of order six now cover all sizes 6-28 except 7","Explicit quantum Latin squares fill gaps 19,21,23,25,27 for order 6","Order-six quantum Latin squares: five new cardinalities complete spectrum to 28 except 7","Quantum Latin squares hit 19,21,23,25,27, leaving only 7,29,32,35 unresolved","New explicit arrays give quantum Latin squares of sizes 19,21,23,25,27"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Quantum Latin squares of order six now cover all sizes 6-28 except 7","Explicit quantum Latin squares fill gaps 19,21,23,25,27 for order 6","Order-six quantum Latin squares: five new cardinalities complete spectrum to 28 except 7","Quantum Latin squares hit 19,21,23,25,27, leaving only 7,29,32,35 unresolved","New explicit arrays give quantum Latin squares of sizes 19,21,23,25,27"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00227,"raw_usage":{"total_tokens":8639,"prompt_tokens":814,"completion_tokens":7825,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":558,"completion_tokens_details":{"reasoning_tokens":7698}},"tokens_in":558,"tokens_out":7825,"duration_ms":50702,"temperature":1.0,"reasoning_tokens":7698,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T06:47:37.940612+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":2}