Ta Problem 11.6: three invariants do not determine the count

Lực Ta, Good involutions of conjugation subquandles, arXiv:2505.08090v5, Section 11, Problem 11.6, asks:

Let G be a finite nonabelian group of order n. Is the number of good involutions of Conj G completely determined by n, the order of the center Z(G), and the class number k(G) of G? If so, is there an exact formula to compute this number?

Here s_x(y)=xyx^-1. A good involution is a function rho with rho squared equal to the identity, commuting with every s_x, and satisfying s_(rho(x))=s_x^-1. The count is of all such functions on the whole group, without quotienting by quandle isomorphism. This is separate from 11.7's question about a bound for conjugation subquandles.

The requested determination would force any two finite nonabelian groups with identical (order, center order, class number) to have identical good-map counts. The formal claim : Prop states exactly that pairwise assertion. Its negation also excludes any formula depending only on those three numbers. The original question and the proposed complete counterexample were registered in issue #25997 before this native implementation.

The primary v5 text still poses 11.6. The inspected versioned texts 2608.19951v1 and 2508.16772v1 did not identify a prior answer to 11.6. Nakajima–Yamaguchi, 2608.19951v1, Example 6.8 and the following paragraph, instead refute Ta's 11.8 and 11.9. This is a bounded literature check on 2026-11-03, not exhaustive citation coverage, worldwide priority, or uniqueness. The existing repository 11.7 refutation does not settle this question.

Take native Mathlib semidirect products G_r=(Z/16) ⋊_r (Z/4), for r=5 and r=7, with multiplication (i,j)(k,l)=(i+r^j k,j+l). Both are nonabelian groups with invariant triple (64,4,22).

The central-label characterization is attributed to Ta Theorem 6.7. Our label z(x)=rho(x)x is the inverse of the source's zeta. Good maps correspond bijectively to class-invariant central labels satisfying z(z(x)x^-1)=z(x). A complete conjugacy partition and a complete enumeration of the actual center split this compatibility into independent blocks.

For r=5 the local factors are one 10, six 4s, and three 6s; hence the complete count is 10*4^6*6^3=8847360. For r=7 they are four 10s and three 6s; hence the count is 10^4*6^3=2160000. Each local block has at most four classes and four central labels. The local enumeration has at most 4^4 assignments, and the general bijection proves completeness for all maps; there is no enumeration of 64^64 functions or 4^22 global assignments.

The refutation would fail if either action did not define the native group, either group were abelian, any of the three invariants differed, the classifier omitted or conflated conjugacy classes, the center enumeration omitted central elements, the destination tables disagreed with native group operations, or the local counts did not count every compatible assignment. Those obligations occur on the live path to the full Lean result.

The mathematical source is D5/S3/FiniteGroups/Quandles/TaThreeInvariantRefutation.lean with literal claim : Prop and result : Not claim. The two complete counts are the conjunction good_counts, consumed by result. General classification and factorization live in D5/S3/FiniteGroups/Quandles/CentralLabels.lean and BlockCounting.lean, and the counts use the existing whole-carrier Good interface from D5/S3/FiniteGroups/Quandles/ConjugationSubquandle.lean, shared with the repository's 11.7 refutation.

This dossier does not certify independent review, admission, frozen status, required CI, merge, official acceptance, priority, or a solved-problem KPI. Those are distinct delivery obligations.

theorem; the designated resolution is refuted for the original three-invariant determination assertion. The two concrete groups, all three invariants, and both complete counts are required together. No claim about knot classification, all finite groups' counts, or counts modulo isomorphism is substituted for the original.

The mechanism explains the failure: the three invariant cardinalities do not specify how central twisted inversion links the conjugacy classes. The general block certificate retains that extra structure and yields a complete product formula. The central-label characterization itself is source-attributed, not claimed as new priority.

Source faithfulness remains an independent mathematical review judgment. The literature check is bounded as described above. Escape registration of the new public counting theorems remains unfinished under issue #25997: the full dependent group/partition/center telescope still needs a faithful realization bridge and variation, sensitivity and actual observational dependence proofs. No declared_validated registration is asserted.