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A kissing configuration in dimension five with at least 43 points has at most seven antipodal pairs

Year: 2026
Type: preprint
Abstract: A kissing configuration in R^5 is a set of unit vectors with pairwise inner products at most 1/2; an antipodal pair is a pair {x, −x} of points of the configuration. We prove that every kissing configuration in R^5 with at least 8 antipodal pairs has at most 42 points. Equivalently, a kissing configuration in R^5 with at least 43 points has at most 7 antipodal pairs. The proof does not use any upper bound on the kissing number τ5. For comparison, every antipodal kissing configuration in R^5 has at most 40 points (Anstreicher 2002), and the 40-point configurations D5, L5, Q5, R5 (Cohn–Rajagopal, Szöllősi) have 20, 12, 10 and 6 antipodal pairs. This does not change the known bounds on τ5: τ5 ≥ 40, the best published upper bound is τ5 ≤ 44 (Mittelmann and Vallentin 2010), and a 2026 preprint by T. S. Tomar (doi:10.5281/zenodo.23638109) gives τ5 ≤ 43 with an exact computer certificate (not yet refereed; not used here). We claim no priority on τ5 ≤ 43. The proof is an exact rational certificate for a Bachoc–Vallentin three-point bound in which the points carry two types, paired and unpaired (degree 8). Its conditions are checked in exact arithmetic (LDL factorisation, Sturm sequences, Bernstein branch-and-bound on the Gram region) by two separately written programs, which are in the supplement together with an exact bookkeeping test on the D5 configuration, tamper tests and a 40-point control. Related work: Anstreicher's antipodal linear programming bound; A. Kriebel's public research dossier on hypothetical 41-point configurations with r antipodal pairs (not peer reviewed); Musin's interval pair counts in dimension four. As far as we checked, a bound of the form "k antipodal pairs imply at most M points" in dimension five is new; pointers to earlier work are welcome. Version 2 (11 Nov 2026) is a display fix: the count line now prints D + k·H_PP(−1) = −48530.6955912… + 8·6066.2129116… = −0.9922987…; version 1 printed two-decimal roundings that do not reproduce the value. The certificate and the result are unchanged. AI use: the computations, the certificate, the checkers and the text were produced with Claude (Anthropic) under the author's direction; the second checker was written by a separate session of the same AI system, which is not independent human review. Not yet peer reviewed. Contact: [email protected].
DOI: 10.5281/zenodo.24132849
Language: English
Open Access status: green
OpenAlex ID: W7225789544
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