[JoyboyBrian]

Change and mathematical scope

Derive the first-pivot input multiplicities left open by the complementary-plane continuation handoff, using scalar sector incidence rather than the previous 496-split/GL4 census.

h=0: four 11-sector incidence types -> 1,3,3,3
h=1: 27 one-dimensional type tuples x 6 realizations -> 126,18,18
published state order -> (1,3,126,18,3,3,18)

The binary scalar incidence-sum formula is proved for arbitrary sector dimensions; sufficient residual normal forms are proved only for profile (1,1,1,2). Input incidence, complete-header fibres, and centralizer multiplicities are kept distinct. Exact retained successor README bytes preserve Sagar Thakrar's continuation proof/table and credit; composing that credited premise recovers 93564 headers without claiming another continuation theorem or a new kappa.

Validation

• Before: PR4689 enumerated the seven input weights; successor 93bebe5e1ad3dd17a97ae79f27e1a35edd3630aa imported them explicitly. After: image/kernel incidence derives all seven weights, including the first weights 1 and 3 before centralizers enter.
• Registered normal and -O checks pass: 31 canonical tuples with actual invertible intertwiners, 30 dimension-two/three direct split controls, five semantic rejections, retained-byte authentication and exact output replay. No GL4 or full 496-split enumeration. Commit-byte/source-history and archive integrity checks pass.
• Credits retained: Mihir Srivastava Classify seven residual Schur-pair states and full-guard counts (no new κ) #4689, ssafwann Count complete headers with sharp identity-pivot Schur blocks (no κ) #4678/Prove exact identity-pivot Schur dimension for every prescribed H (no κ) #4671, Sagar Thakrar, Nick Petalas Prove exact orbit weights for ordered idempotents (no new κ) #4915/Prove the sharp 2d block Schur commutator bound (no new kappa) #4597, MatShep, codyw912, Ryanson Jonathan, lee-phys, Cinna Davis and their cited lineage. Prepared by JoyboyBrian with substantial OpenAI Codex assistance.

Merge Danger

Door: two-way. Isolated research package; no selected result or existing code changes.

Blast Radius: research. The continuation numerical table is source-bound but not independently replayed here. General residual classification, physical header costs and supplier admission remain open. No Lean, Docker, production build or selected cold replay was run.

Task and overlap

Refined task b2_math_schur_split_incidence on card formal-interface-lemma; canonical investigation. Scientific base 4c0a359d88f22f05189d51f48387750b801a45d4, priority revision7; refreshed canonical main is unchanged at the submission checkpoint. Exact dependency heads and retained source/path/hash records are in the scoped package. The claim states predecessor coverage, changed premise, live parallel exclusions and bounded stop rule. Arbitrary diffs and unpublished work are outside overlap coverage.

Credit

Mihir Srivastava #4689; ssafwann #4678/#4671; thakrarsagar complementary-plane successor; Nick Petalas #4915/#4597; MatShep #4762; codyw912 #4735; Ryanson Jonathan, lee-phys and Cinna Davis. Substantial OpenAI Codex assistance is disclosed; retained source-specific attribution and licenses are preserved.

Open next questions

See the package questions for the smallest remaining caller/interface obligation and its blocker. This contribution stops at the stated scoped result.

Reproduce with CPython3.12 on macOS arm64: python3.12 scripts/research_package.py --package research/schur-split-incidence. A separate Astra review of this exact HEAD found no actionable issue and reran normal/optimized targeted checks. Scientific review remains AI-assisted, not human or proof-kernel acceptance.