# Verification of the one-point subgroup classification

The accepted result classifies every unrestricted self-sumset A+A=G union {e} over an odd prime field, with G a proper multiplicative subgroup and e outside G. Its only examples have G={-1,1} and e=0,3,-3, with the three two-element sets stated in the manuscript.

The broader AIM Problem 4.7 is not resolved. Exact equality and the zero-exception conclusion are prior consequences of Hanson and Petridis, not new discoveries claimed here. Novelty is undetermined after bounded primary-source review. This is an unrefereed preprint with originating-researcher self-audit, not independent review or formal verification.

## Mathematical checks

The written proof checks the characteristic restriction D=m+n-1<p; the nonzero leading coefficient of the differential polynomial; the precise exceptional derivatives; all root multiplicities; unordered representation counts; the reflection center; both saturation arguments; the parity-dependent intersection polynomial degree; and all six surviving cardinality pairs. The finite small-case exclusions are integer moment and greatest-common-divisor arguments, valid in every relevant characteristic. The final manuscript explicitly rules out A={0,b} in the zero-exception case: {b,2b}={-1,1} would require p=3, inconsistent with properness of the order-two subgroup.

The classification covers all prime fields by its written argument. No unbounded search or experimental extrapolation is part of that argument. The accepted research proof and detailed audit are preserved in reproducibility/complete_proof.md and reproducibility/proof_audit.md.

## Exact arithmetic and finite enumeration

The standard-library Python checker verifies 749,359 identities and finite conditions per execution. It includes subgroup intersections through prime 199, auxiliary-polynomial coefficients and derivatives, small-case rational identities, and all 6,634 one-point-enlargement systems for odd primes through 101. The clique search visits 1,361,347 nodes and is cross-checked against a separate powerset enumeration for graphs of at most ten vertices. It finds exactly 72 examples: three for each of the 24 primes from 5 through 101.

Ordinary and optimized executions agree. Checks do not rely on Python assert statements, so optimized execution cannot disable them. The recorded output files and both required scripts are included. These finite tests supplement the proof and do not certify untested prime fields.

## Typesetting and package scope

The English manuscript has six pages. Its source compiles successfully in the desktop LaTeX editor and exports with Tectonic without TeX warnings or unresolved references. All six rendered pages were inspected. The visible author line contains only Alper Ferudun; Mercury Software GmbH, email and GitHub are in the author footnote. AI assistance is disclosed in ordinary prose.

The source archive includes the manuscript, bibliography, metadata, license, this report, the bounded literature review and author-created reproducibility materials. It excludes third-party source PDFs, frozen dataset extracts and private coordination records. A SHA-256 manifest identifies each archived file. A prepared package is not evidence of a DOI or completed public release.