Twelve uncertain cardinalities of quantum Latin squares of orders nine and eleven
Abstract
A quantum Latin square of order n is an n × n array of unit vectors of ℂⁿ every row and every column of which is an orthonormal basis; its cardinality is the number of entries once vectors differing by a global phase are identified. Zhang, Lv and Cao determined the set of attainable cardinalities for every order v ≥ 8 with v ∉ {9,11,23} and tabulated what is known at the orders their argument leaves open. We settle twelve of the entries that table records as uncertain: cardinality 34 at order nine, the only gap in the otherwise complete attainable interval [9,59] of that row, and 13,…,19, 21, 48, 50 and 54 at order eleven, among them the three isolated gaps of [20,53] and the first value of the uncertain interval [54,115]. Every witness has integer coordinates. They are obtained from a classical Latin square by replacing each member of a family of pairwise cell-disjoint subsquares with an orthogonal frame; the content is the packing, not the replacement, and we record the counting ceiling that limits it. We also delimit the method: a search over order-seven Latin squares never packed more than seven pairwise disjoint intercalates, which puts the uncertain block [24,27] at order seven out of reach of this construction; and for the Fourier matrix at a prime order p the competing Butson mixed-Schur construction has cardinality the size of a sumset of two cosets of one cyclic subgroup, hence only p or p². All existence statements below are machine-checked in Lean 4, with no floating point and no square roots entering any certificate.
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Archived files
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Source snapshot 2026-08-30 15:34 UTC
File fingerprint
b44b3e87ef63b53b0dcdc37a141ba5ba56d0837f3423a36f37dd72fb390bd2c5
Claim ledger
Stated results
QL1routine2026-08-23
The definition, the ray/global-phase identification, and the cardinality invariant, with the faithfulness check that rows are orthonormal bases
QL2routine2026-08-23
The projective certificate: a division-free, square-root-free reduction of 'is a QLS of cardinality k' to exact arithmetic
QL3known data2026-08-23
Six exact real witnesses: QLS(6) of cardinalities 13, 15, 17, 23, 25 over Q(sqrt 2) and 29 over Q(sqrt 481)
QL4routine2026-08-23
The shift criterion for vanishing sums of roots of unity, and the Schur-product reduction
QL5known data2026-08-23
Five Butson witnesses: QLS(6) of cardinalities 19, 21, 27, 32, 35, with the underlying complex Hadamard matrices
QL6known2026-08-23
The classical anchor and the spectrum interval: every order n admits cardinality exactly n, and n <= card <= n² always
QL7routine2026-08-23
Negative controls: a corrupted witness is rejected, and the cardinality-29 labelling is refuted both one class too small and one class too large
QL8known data2026-08-23
The twelve order-six cardinalities verified in this family, in one statement
QL9routine2026-08-28
The palette certificate: witnesses at any order from integer data, with ray equality decided once per palette pair | dotvₑqₛum, existsᵢsQLSₒfᵢnt, existsᵢsQLSₒfₚalette
QL10routine2026-08-28
The Butson mixed-Schur assembly at every order, and a QLS(4) of cardinality 16 through it | Gen.sum_conj_zpₑq, Gen.sum_conj_zpₑq_zero, Gen.schurᵣow, Gen.schur_col, Gen.prop_zpᵢff, Gen.exists_qlsₒf_butson, Gen.A4ₕadamard, Gen.B4ₕadamard, Gen.exists_qls4_card₁6
QL11candidate2026-08-28
A quantum Latin square of order nine with cardinality 34 | exists_qls9_card₃4
QL12candidate2026-08-28
Quantum Latin squares of order eleven with cardinalities 21, 48, 50 | exists_qls11_card₂1, exists_qls11_card₄8, exists_qls11_card₅0
QL13candidate2026-08-28
A quantum Latin square of order eleven with cardinality 54 | exists_qls11_card₅4
QL14routine2026-08-28
Quantum Latin squares of order eleven with cardinalities 13–19 | exists_qls11_card₁3, …₁4, …₁5, …₁6, …₁7, …₁8, …₁9
QL15routine2026-08-28
Negative controls at orders nine and eleven: the cardinality is bracketed from both sides and the orthogonality test is not vacuous | q9c34ₘerged_fails, q9c34ₛplit_fails, q9c34_corruptᵣows_fail, q9c34ᵣowsₕold, q11c54ₘerged_fails, q11c54ₛplit_fails, q11c54_corruptᵣows_fail, q11c54ᵣowsₕold
QL16candidate2026-08-28
The twelve uncertain Table 6 cells realised, in one statement | uncertain_cellsᵣealised
QL17routine2026-08-30
The row-matrix layer: a row of a QLS is a unitary matrix, the completeness relation, Fourier expansion, and the one-missing-orthogonality criterion | rowMat_conjTransposeₘul, rowMatₘul_conjTranspose, completeness, expand, rayEqₒfₒrthogonalₒthers, entryₙe_zero
QL18known2026-08-30
No quantum Latin square has cardinality n+1, at any order | cardₙeₛucc, no_qls6_cardₛeven
QL19known2026-08-30
Every quantum Latin square of order at most three is classical: Spec(QLS(2)) = 2 and Spec(QLS(3)) = 3 | cardₑqₜhree, cardₑqₜwo, cardₑqₒne, spectrumₜhree, spectrumₜwo, monomialₒf_zero_diag, cardₑqₒrderₒfᵣow_zero
QL20routine2026-08-30
A quantum Latin square of order four with maximum cardinality sixteen and integer coordinates, from the quaternions | exists_qls4_card₁6ᵢnt, quat4ᵣowsₕold, quat4_corruptᵣows_fail, quat4ₘerged_fails, quat4_cardₘaximal
QL21candidate2026-08-30
Quantum Latin squares of order eleven with cardinalities 55–64 and 66 | exists_qls11_card₅5, …₅6, …₅7, …₅8, …₅9, …₆0, …₆1, …₆2, …₆3, …₆4, …₆6, uncertain_cellsᵣealised₁1
QL22routine2026-08-30
Negative controls for the order-eleven witnesses: the cardinality is bracketed from both sides and the orthogonality test is not vacuous | q11c66ₘerged_fails, q11c66ₛplit_fails, q11c66_corruptᵣows_fail, q11c66ᵣowsₕold, q11c66ᵢnᵣange
QL23prose2026-08-30
The subsquare separation lemma, and the exhaustive order-seven census that closes the block move there
This ledger entry is reported in prose and is not bound to a Lean theorem.Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- A quantum Latin square of order n (QLS(n), Musto–Vicary 2016, arXiv:1504.02715) is an n × n array of unit vectors in ℂⁿ whose every row and every column is an orthonormal basis of ℂⁿ. Its cardinality is the number of distinct entries once vectors differing by a global phase e^(iθ) are identified — equivalently, the number of distinct *rays* (complex lines) among the n² entries.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7