Symmetric 41-point kissing configurations in dimension 5: a computer-assisted partial classification

The kissing number in dimension five satisfies 40 ≤ τ5 ≤ 44. Let X be a hypothetical 41-point kissing configuration in R^5 (41 unit vectors with pairwise inner products at most 1/2) and G = Aut(X), a finite subgroup of O(5). We prove that −I is not in G and G is reducible; that every element of G has order 5, 4, 3^b or 2·3^b (b ≥ 0); that G is not A5, Z5², Z2⁴:Z5, Z2⁴, Z2⁴:Z3, D5, Z4×Z4, Z4:Z4, Z4×Z2×Z2, SmallGroup(16,3), Dic3 or S4×Z2, and contains neither Q8 nor Z3²; and that if 5 divides |G| then G = Z5. Consequently G = Z5, or G embeds in S4×Z2 (G ≠ S4×Z2, and S4 can act only as 1+1+T1), or G contains an element of order 9 or a subgroup Z6×Z2. This is a partial result. Open: the order-9 case; the Z6×Z2 case (11 of 51 sub-cases open). The exclusion of order 8 rests on a floating-point interval standard; an exact (directed-rounding) re-check is open. The theorem says nothing about configurations with trivial symmetry and does not improve τ5 ≤ 44. Proof standard: some parts are by hand; some use interval arithmetic with outward rounding; the rest use floating-point interval branch and bound with stated slack (not directed rounding). The note states the standard of each part. The supplement has the checker sources, result and red-team reports, small logs, and SHA-256 checksums; files over 4 MB are listed with checksums. AI use: the case split, checkers, computations and proofs were produced with Claude (Anthropic) under the author's direction, and checked by separate sessions of the same AI system. This is not independent human review. Nothing here has been peer reviewed. Effort: about one week. Three cited results were accepted with fixes after review; checker bugs in the Q8 and order-8 runs and one invalid batch of order-16 runs were caught, fixed and re-run before acceptance. To our knowledge no earlier result restricting the symmetry group of a 41-point kissing configuration in R^5 has been published; we would be grateful for pointers. Companion note: doi:10.5281/zenodo.23784232.