[aliromman]

This scoped classification gives every nonadditive rank-59 complex diagonal completion of the doubled P112 Gram matrix. Exactly 112 supports form one orbit under the stated frame-preserving E8 group; one explicit reciprocal-coefficient equation is necessary and sufficient on each support. The inherited root-flat premises remain explicit. The complete kappa calculation is unchanged.

Proof and scope, exact checker, replay declaration, and source provenance are pinned to the reviewed commit. Base: canonical main ff3cc58e370d0f678e529246f8c73e8094e18fcf, priority revision 7.

CPython 3.12.14 registered normal and optimized replays passed: 120 root normals, 112 equality supports, 1,344 affine maps, 18,816 plane actions, two literal 112-row ranks and six 53-row Gaussian-rational rank checks. Both print PASS_P112_RANK59_CLASSIFICATION; the launcher prints PASS_REGISTERED_RESEARCH_PACKAGE. All five package files match their committed bytes, and the base is an ancestor of the package commit.

python3 -B scripts/research_package.py --package research/p112-rank59-equality

Use Python 3.12 and the standard library. No external runtime inputs or third-party Python packages are required. Hosted planning and the registered package replay passed on this exact head in run 40000608957. No predecessor scientific suite, selected cold replay, native replay, compiler, Lean or Docker build was run. The optional full-history preflight is unavailable in this shallow checkout. Finite checks corroborate the written proof; they do not constitute independent human or proof-kernel acceptance.

Test a specified diagonal-repair consumer and charge its full decoder bill before using this algebraic rank reduction. Source preparation, native operations, precision, movement, and supplier/outer admission remain separate obligations.

Credit @petalas's PR #5249 supplies the exact model, rank stratification, rank-59 witness and equality question in its pinned proof. It retains @Arham-Tahir64's PR #1313 diagonal-mask question and pivot method, @woklerx's PR #1447 code-distance premise, and @themilkman216's PR #968 root-flat classification. @DreamingOfClouds and the retained E8/Hamming lineage keep their geometric credit. Rank-one determinant identities and the affine-plane model are classical. The new result is the complete first nonadditive coefficient/support class and its specified symmetry quotient.

Prepared by @aliromman with substantial OpenAI Codex assistance; Apache-2.0.

[jwr404]

Independent review (James Reed, with Claude Code). Head f8cee42b6aa2d650de8b359061a2660031506fe8.

Verdict: the support count, the coefficient equation and the symmetry group all check out under independent exact computation. Completeness rests on the inherited premises, as the PR states.

Independent exact checks

My own code (exact Fraction elimination, no shared code) builds the 120 E8 root lines in doubled coordinates, removes 2e_(2j) ± 2e_(2j+1), and forms G = XXᵀ on the 112 live lines.

• All 120 normals.

  normal	count	|S| = nonzero entries of Xα	rank of the unchanged rows
  live	112	53	7
  excluded	8	56	7

  This matches the "59 vs 56 live zero rows" argument.

• Squares on a live support. One entry with c_i² = 64 (α's own row) and 52 with c_i² = 16, giving (2) from (1). ✓

• Rank dichotomy, exact.

  • With random rational t_i summing to −1 and δ_i = c_i²/(8t_i) on a live support, rank(G + Δ) = 59.
  • Doubling one coefficient gives rank 60.
  • Additive control: a random 51-row support whose complement has rank 8 gives rank 59.

• Group order. 2⁸ · |AGL₃(2)| = 256 · 1344 = 344,064 ✓. Transitivity on the 14 planes and the 8 sign classes per plane gives the single 112-orbit. ✓

Proof read

The Schur reduction (pivot on 7 independent unchanged rows, leaving diag(δ_S) + c_S c_Sᵀ/8) and the nullity-one criterion via the candidate kernel diag(δ)⁻¹c_S are correct for arbitrary nonzero complex δ, with ordinary transposes. The frame-preserving group argument (signed permutations preserving the 14 affine planes, translation then additivity) is standard and reads correctly.

Scope note

Completeness ("there are no other cases") uses #5249's rank stratification (r = 7 ⇒ |S| ≥ 53, r ≤ 6 ⇒ rank ≥ 62) and #968's root-flat classification. #5249 is still an open PR. The README says the premises are inherited, but it would help to say explicitly that the classification is conditional on an unmerged predecessor, so a maintainer merging this one doesn't take it as standalone.

The registered replay passes locally (PASS_REGISTERED_RESEARCH_PACKAGE).