Revision 1.1 internal proof review

Date: September 16, 2026. Review by the originating AI agent; no independent review is represented by this document.

Claim and derivation

Theorem 5.1 evaluates beta_r for every positive r. This answers the explicit question in Section 5 of version 1. The prior all-genus quasipolynomiality, parity oscillation, and fixed-r eventual growth proofs remain in place.

1. The full additive equality arrangement includes low-low-low, hole-low-hole, and low-low-hole equations. Distinct hyperplanes are independent in pairs. Fixing all but two coordinates bounds every intersection by O_r(g^(2r-2)); no periodicity of a lower coefficient is assumed away.
2. Outside the bulk, the low violation forces 2m in B. A further low sum forces a second equality unless it is m+m itself. This gives g/3<m<=g/2, other low elements above g-m, and all holes at most g+m. The upper-hole violation gives the second region, with forced pair a,a+m and the stated constraints on all other low elements and holes. Violating both bulk conditions costs two independent equalities.
3. Inside the bulk, only low-low-hole equalities invalidate a tuple. Their contributions add at leading order. The pair weights r, (r-1)/2, and (r-1)(r-2)/4 include the doubled minimum, doubled other elements, and unordered distinct pairs with the correct multiplicities.
4. Every forced coordinate is eliminated with coefficient one. The volume calculation therefore uses a lattice-coordinate measure, not Euclidean hypersurface area. Factorials account for unordered free sets.
5. The two beta integrals and their difference give the upper positive term explicitly. Expanding both falling factorials gives h_r and the lower mesh endpoint contributes -C_r/2 to the parity average. Their sum gives the displayed beta_r. The special r=1 case is evaluated directly and agrees with the same formula.
6. The asymptotic ratio beta_r/A_r follows from the central binomial coefficient estimate. It supplies no control of the genus remainder when r varies. No all-genus conjecture is declared solved.

Reproducible finite diagnostics

check_second.py uses exact integers and fractions. It checks the candidate sums against literal tuple enumeration on 17,568 tuples through genus 8. Every one of the 223 nonzero residuals has at least two distinct primitive normal vectors, hence two independent additive equalities. This directly tests the claimed support of the lower-order remainder.

For r=1,...,8, exact Newton interpolation on all twelve tested residue classes agrees with each candidate sum's proposed leading coefficient and passes 864 later evaluations. The beta_2 formula independently agrees with every residue of the source's degree-four formula. These tests corroborate the proof; their parameter ranges do not establish its universal claim.

The previous check.py and its saved diagnostic output are retained from version 1. No computational result is interpreted as peer review or a priority certificate.

Artifact checks

The final PDF has nine pages. Tectonic reported no overfull/underfull boxes or undefined references. The complete Markdown conversion uses the final TeX equation/theorem numbers and passes semantic round-trip verification. Final hashes and numerical artifact counts are recorded in artifact-qa.json and markdown-math-check.json. Every final PDF page was visually reviewed; the new proof pages were also inspected individually. Local MathJax checks do not constitute live GitHub-renderer verification.