Exact local exclusions and structure lemmas for 41-point kissing configurations in dimension five

The kissing number in dimension five satisfies 40 ≤ τ5 ≤ 44. This note collects small, exact, computer-assisted results about hypothetical 41-point kissing configurations in R^5, that is, 41 unit vectors with pairwise inner products at most 1/2. (A) No such configuration has 40 points each within 10.25° of the 40 points of D5, after one rotation or reflection and a matching. The same holds near L5 (10.25°), Q5 (9.80°) and R5 (10.1°). (B) Around the best known 41-point near-misses, every 41-point family with pairwise inner products at most 1/2, in any dimension, has a point that has moved more than 6.69° to 7.47°. (C) Every 41-point configuration contains three points whose pairwise angles all lie in [60°, 66.11°). (D) A configuration containing the 16-point half-cube has at most 26 points. New in version 3: (E) In a 41-point configuration every point has at least 26 others within 120°. (F) Unit vectors in R^5 whose pairwise inner products lie in [−1, 1/2] and in (1/8)Z, in (1/12)Z, or in (1/4)Z ∪ (1/10)Z number at most 40. The last of these alphabets contains every inner product of the four known 40-point codes. (G) Explicit 16- and 20-point blocks of each known code, even when moved by up to 1.89°, cannot be part of a 41-point configuration. The same holds for the cross-polytope ±e1,…,±e5: a configuration containing it has at most 38 points. (H) The symmetry group G of a 41-point configuration acts reducibly. It contains neither −I nor A5. Every element of prime order has order 2, 3, 5 or 13, and no element has order 20, 25 or 40. If every nonidentity element of G has prime order, G is one of 16 explicit small groups. The supplement contains all certificates and standalone exact checkers. None of these results changes the known bounds on τ5. Our kill-test queue for 19–26 October 2026 has about 300 entries, most of them for this problem; these results are the few that survived. We found no earlier record of these statements, and corrections and pointers are welcome. The computations, certificates, checkers and proofs were produced with the AI system Claude (Anthropic) under the author's direction. Each result was re-checked by a separate session of the same AI system with its own code, which is not independent human review. Not yet peer reviewed. Contact: gameruchral71@gmail.com.