One-Point Extensions of Multiplicative Subgroups as Self-Sumsets

Let p be an odd prime, G a proper multiplicative subgroup of F_p^*, and e outside G. This preprint classifies every unrestricted self-sumset A+A=G union {e}, including diagonal sums. They occur exactly when p>=5, G={-1,1}, and (e,A) is (0,{-1/2,1/2}), (3,{-1/2,3/2}), or (-3,{-3/2,1/2}). A differential polynomial built from the Hanson-Petridis auxiliary polynomial forces an almost reflection symmetry; a shifted-subgroup intersection bound reduces the argument to at most five elements, and exact moments finish the classification.

The zero-exception case and the impossibility of exact equality with a nontrivial proper subgroup are credited consequences of Hanson and Petridis. The theorem addresses the one-point enlargement regime of AIM Problem 4.7, recorded as AIM-COMBINATORICS-0263 in UnsolvedMath. It does not resolve the entire source question, arbitrary near-equality, or the small-exponent containment problem. No absolute priority is claimed. The preprint is self-audited, not independently reviewed or formally verified.

The accompanying source and exact-arithmetic verification materials include 749,359 checks and finite enumeration for odd primes through 101. These supplement the written all-primes proof. AI assistance was used for mathematical exploration, drafting and reproducibility checks; the author is responsible for the claims.