Research: local block splitting excludes 49 mixed-rank patterns

[aliromman]

Uniform mode 3 permits at most one regular anchor block. This closes all 28 four-exception and all 21 five-exception patterns left open by research/kappa-nine/MIXED-RANK-TEST.md. These are regularity patterns inside one remainder-mode branch. The seven six-exception patterns, the all-exception pattern, and all 103 complete remainder-mode branch representatives remain open. No new kappa is claimed.

Review files: Proof and scope · Evidence · Source pins and credit.

The new algebraic result is a forced local decomposition on every anchor block: the distinguished Y restrictions for outside points 7 and 8 give complementary idempotents. Their paired A ranks and paired B ranks each sum to 2. Two regular blocks force both paired restrictions on a third block to have rank 2, by Douglas Colkitt (@Mister-Meeseeks)'s existing complementary-block lemma. This contradicts the new rank sum. No covariance, common form, positivity, finite-field inference, or nonlinear solver is used.

The six-file package adds the proof, source pins, a graph-certificate builder, an independent replay, a frozen copy of the experimental selection, and the local scope claim. The exact base is abdc6a7002abdbcd9e21bfc5e1a795784deacfdf. A separate live-main note records PR #1042 at ba82555c64e1879e4b7decce28186f208fcb8262; the executable experiment keeps its original pins and does not read the mutable selected record.

python3 -B -O research/kappa-nine/verify_mixed_geometry.py

Expected output: 7 local block certificates, 378 sole-common-block edges, 21 excluded regular pairs, 49 newly excluded exception patterns, 8 remaining patterns, 3 rational controls, and 7 rejected mutations. The replay reconstructs the graph without importing the producer's graph helpers. It also checks all 128 exception sets, a full 42-label Y coordinate control, and exact root enclosures for the inherited optimistic cost profile.

Validation completed:

• Saved and rebuilt certificates are byte-identical; optimized replay passes.
• Twelve focused predecessor tests pass.
• Replay passes on the pinned source files from live main, with its live selection file absent.
• The coordinating AI agent independently replayed the certificate and reviewed the algebraic proof and inherited block-inverse obligation. This is not external human review.
• git diff --check passes. Repository-wide make verify, full supplier replay, hosted CI, and Lean were not run; no supplier or selected result changes.

The historical dimension-15/five-role relaxed profile passes a=1/511 but fails a=1/99. It is not an implemented producer. A bit-only improvement cannot advance the outer grid while either recorded complex endpoint stays fixed. A useful full construction still needs the remaining geometry, extension to 16 points, a compact charged producer, the copied/ordered transfer interfaces, and a complex counterpart.

Credit: Douglas Colkitt (@Mister-Meeseeks)'s original kappa-nine work and complementary-block proof; PR #1066 by @woklerx at 2e72784a7908d45511e953c182d42cdcb911e6cb independently corroborates that inherited lemma. This contribution adds the forced local decomposition and 49 exclusions. The final overlap check at 2026-10-13 18:28 UTC found PR #1066 unchanged and no competing extension. Ali Romman (@aliromman) prepared this package with substantial OpenAI Codex assistance. New work is Apache-2.0.

[Mister-Meeseeks]

100-PR batch review at b084bc5bc7aaa4ee51aba7e3d668ea3da9af4934.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Complementary block-idempotent proof excludes 49 regularity patterns inside uniform mode 3; eight exception patterns and other branch modes remain open.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.