Connected noninvolutory symmetric quandles of order less than 48
Abstract
A CNS-quandle is a connected, noninvolutory quandle equipped with a good involution. Ta asked for which integers k a CNS-quandle of order k exists, and answered the question for k ≤ 8, where a single one occurs, the symmetric Galkin quandle of order 6. We settle the question for every k ≤ 47: a CNS-quandle of order k exists exactly when k ∈ {6,12,18,20,24,30,36,40,42}, and we give the number of isomorphism classes at each such order, namely 1,3,2,4,8,4,10,5,7. Two ingredients make this possible. First, an elementary argument shows that a good involution of a connected noninvolutory quandle is fixed-point-free, so that no CNS-quandle has odd order; the question is therefore only about even k. Second, a CNS-quandle is connected by definition, and connected quandles are classified up to order 47, so it suffices to decide, for each classified quandle, whether it carries a good involution. We show that this last test costs n trials rather than nⁿ, because a good involution of a connected quandle is determined by its value at a single point. The new negative orders are 16, 28, 32 and 44; all nine positive orders were already supplied by known constructions. The same computation refutes Ta's closing conjecture, that a connected noninvolutory quandle has at most one good involution, at order 24 — one tenth of the order of the counterexample of Nakajima and Yamaguchi — and shows that 24 is the smallest order at which the conjecture can fail. All statements below have been verified with a proof assistant.
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Claim ledger
Stated results
Q1routine2026-08-22
No CNS-quandle has odd order: the good involution of a connected noninvolutory symmetric quandle is fixed-point-free
Q2known data2026-08-22
Validation of Proposition 1.5 and of the source's SL(2,5) example: six CNS-quandles, at orders 6, 12 (four of them) and 20
Q3routine2026-08-22
No CNS-quandle of order 2 or 4, by an in-Lean exhaustion whose completeness is proved
Q4routine2026-08-22
Negative controls: a connected symmetric quandle that is a kei, a connected noninvolutory quandle with no good involution at all, and uniqueness of the order-6 good involution
Q5routine2026-08-22
External: the CNS order spectrum extended past the source's frontier – order 10 admits none, order 12 admits exactly three
This ledger entry is reported in prose and is not bound to a Lean theorem.Q6routine2026-08-22
Gate outcomes for scout-algebra rows 7 and 8: numbers reproduce, framings do not
This ledger entry is reported in prose and is not bound to a Lean theorem.R1routine2026-08-22
The per-quandle CNS test: on a connected quandle a good involution is determined by ρ(0), so the search space is n and not nⁿ — correctness proved in both directions, kernel-clean
R2known data2026-08-22
Gate: the classification route reproduces Proposition 1.5 and the family's own order-10 and order-12 values, and the family's group-built witnesses land on the RIG entries the test flags
R3candidate2026-08-23
No CNS-quandle of order 16, 28, 32 or 44 — four new negative answers to Problem 1.7, each a per-quandle exhaustion over the classification's list; and orders 14, 22, 26, 34, 38, 46 recorded as closed by McCarron 2012 rather than by this route
R4candidate2026-08-23
Exact CNS counts past the family's frontier: orders 18, 20, 24, 30, 36, 40, 42 admit 2, 4, 8, 4, 10, 5, 7 CNS-quandles among the classification's 12, 10, 42, 24, 73, 33, 26 connected quandles
R5candidate2026-08-23
Problem 1.7 is settled for every k ≤ 47, and the connectivity clause is exactly what makes that possible — with a noninvolutory symmetric quandle of order 12 that is *not* connected, showing what the route does not cover
R6candidate2026-08-23
The source's closing conjecture — a connected noninvolutory quandle has at most one good involution — is false at order 24, one tenth of the published counterexample's order, and 24 is the smallest order where it can fail
S1candidate2026-08-30
The inverse-translation obstruction, made decidable, accounts for every negative order below 48: no connected noninvolutory quandle of order 4, 8, 10, 16, 28, 32 or 44 has s₀ conjugate to s₀⁻¹ in Inn(Q), and across all 673 noninvolutory connected quandles of order < 48 exactly 46 pass the test of which 44 are CNS-quandles
S2candidate2026-08-30
A quandle whose translations each have a unique fixed point has no good involution unless it is a kei; in particular no latin quandle — hence no finite connected Alexander quandle — is a CNS-quandle, and the family's 4⁴ search for the tetrahedral quandle becomes an argument over all maps
S3known2026-08-30
Formalization of the source's Proposition 5.1 (a faithful quandle has at most one good involution), and the family's data on how far non-faithfulness is from sufficient
S4candidate2026-08-30
In a CNS-quandle every translation has an even number of fixed points — the local counterpart of "no CNS-quandle has odd order"
S5prose2026-08-30
For a finite connected quandle, #Good(Q) ≤ #Q / #Ret(Q) — the quantitative form of the source's Proposition 5.1
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
- Problem 1.7 of *Good involutions of conjugation subquandles*, arXiv:2505.08090 (May 2025):
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
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