Complete mappings of non-cyclic abelian groups and diagonal quantum Latin squares of maximum cardinality: ten new orders, and the exact reach of the construction
Abstract
A quantum Latin square of order n is an n × n array of unit vectors of ℂⁿ each of whose rows and columns is an orthonormal basis; its cardinality is the number of entries up to a global phase, and it has maximum cardinality when that number is n². Huang and Li recently proved that a maximum-cardinality diagonal quantum Latin square, MCDQLS(n), exists for every n ≥ 6 outside an explicit list P₁cupP₂ of 35 orders, and that an idempotent MCQLS(n) exists for every n ≥ 6 outside the 13-element list P₁. Their engine is a single construction driven by a complete mapping of the cyclic group Zm(v) and the Fourier kernel ωᵗˢ. We observe that no step of that construction uses cyclicity, and run it over an arbitrary finite abelian group G of order v with the Fourier kernel replaced by a bi-additive nondegenerate pairing B: G × G → Zm(L). The generalisation settles ten (order,property) cells of the two exception lists: an MCDQLS(v) exists for v=9,20,27,32,40,63, and an idempotent MCQLS(v) exists for v=12,20,24,32; three of these orders carry two witnesses over two different groups. We then determine the reach of the generalised construction exactly: of the 35 orders of P₁cupP₂, precisely those six are reachable, and each of the remaining 29 is out of reach over every abelian group of that order — twenty-six by two counting obstructions, and the last three, 7, 8 and 11, by exhaustive enumeration. At orders 7 and 11 the enumeration proves something stronger and pairing-free, namely that every strong complete mapping of a group of that order is affine, whence Huang and Li's own condition (*) fails for all of them; at order 8 the statement is for the canonical pairing. Every theorem below is machine-checked in Lean 4; the one step left to standard mathematics is the classification of finite abelian groups, which is what turns finitely many named groups into every group of an order.
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Archived files
- Version 1 · current (opens in a new tab)
Source snapshot 2026-08-30 15:34 UTC
File fingerprint
9a9dc04fe63c2a23983ee968246d51b98bd3901dcf1a9614d2b313f34f94c017
Claim ledger
Stated results
MQ1known2026-08-30
The definitions of Huang-Li section 1 in Lean: quantum Latin square, transversal, idempotent and diagonal, the ray identification and the cardinality invariant, with the faithfulness check that a row, a column, the main diagonal and the anti-diagonal each really are orthonormal BASES of Cⁿ
MQ2routine2026-08-30
The character construction over an ARBITRARY finite abelian group: a bi-additive nondegenerate pairing B: G x G -> ZMod L, a complete mapping mu of G, and a non-constancy condition give an idempotent MCQLS(|G|); a strong complete mapping plus an enumeration pairing i with n-1-i gives an MCDQLS(|G|)
MQ3candidate2026-08-30
MCDQLS(v) exists for v = 9, 20, 27, 32, 40 and 63 – six of the 35 exceptional orders of P1 union P2, each by an explicit strong complete mapping of a non-cyclic abelian group, kernel-checked from the table
MQ4candidate2026-08-30
Idempotent MCQLS(v) exists for v = 12, 20, 24 and 32 – four of the 13 exceptional orders of P1, by explicit complete mappings of Z2xZ2xZ3, Z2³xZ3 and Z2xZ4xZ3 and from the MCDQLS witnesses at 20 and 32
MQ5routine2026-08-30
The 2-obstruction: a complete mapping of a finite abelian group forces the sum of all its elements to vanish – and hence 23 named abelian groups, covering 21 exceptional orders, carry none
MQ6routine2026-08-30
The 3-obstruction: a finite abelian group with a strong complete mapping that surjects onto Z₃ has order divisible by 9 – hence no abelian group of order 12, 15, 21, 24, 33, 39, 48, 51 or 75 has one, replacing an infeasible search by a two-page argument
MQ7routine2026-08-30
Negative controls: the cardinality is bracketed from both sides at order nine, and each of the construction's five hypotheses is shown to be a real restriction by an explicit object satisfying all the others and failing exactly that one
MQ8known2026-08-30
Positive control at v = 13: the generalised construction, run at the source's own cyclic group Z₁3 with the source's own kernel, reproduces Huang-Li's odd-prime case
MQ9candidate2026-08-30
The headline: ten (order, property) cells of Huang-Li's exception lists settled at once – P1 union P2 loses 9, 20, 27, 32, 40, 63 for MCDQLS and P1 loses 12, 20, 24, 32 for idempotent MCQLS
MQ10routine2026-08-30
The exhaustive searches that settle the last three orders 7, 8, 11 — now run inside Lean with a proved exhaustiveness lemma, and the C measurements they reproduce
MQ11candidate2026-08-30
Orders 7 and 11: every strong complete mapping of every group of that order is affine, so Huang–Li's own condition (*) fails for all of them and the maximum-cardinality condition fails for every pairing — their engine provably cannot settle MCDQLS(7) or MCDQLS(11)
MQ12routine2026-08-30
Order 8: no complete mapping of any abelian group of order 8 gives an array of maximum cardinality for the canonical pairing — ℤ₈ has no complete mapping at all, and the 384 each of ℤ₂ × ℤ₄ and ℤ₂³ all fail
MQ13routine2026-08-30
Controls on the search machinery: it returns false exactly where a counterexample exists — at ℤ₁₃, at ℤ₇ without the strong condition, at ℤ₇ for maximum cardinality, and at ℤ₃ × ℤ₃ — and the hypotheses at 7 and 11 are not vacuous
MQ14candidate2026-08-30
The reach of the construction is completely determined, with no measurement left: every one of the 35 orders of P₁ ∪ P₂ is either settled by a witness or proved in Lean to be out of reach, and the same for the 13 orders of P₁
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- A quantum Latin square of order n, QLS(n) (Musto–Vicary), is an n × n array of unit vectors of ℂⁿ whose every row and every column is an orthonormal basis. Two entries are identified when they differ by a global phase; the number of classes is the cardinality c, and n ≤ c ≤ n². A QLS(n) has maximum cardinality when c = n²: all n² entries are pairwise non-proportional.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
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