[ssafwann]

Change and mathematical scope

For nilpotent H, this closes the whole-residual extension question left by #5256. Fix a commuting idempotent pair, its four joint-sector dimensions, a common identity-block deletion and the exact compressed product H. If the Schur commutator has maximal rank 2d, the entire ordered residual pair has one simultaneous-conjugacy orbit. The package gives its all-field representative and its exact fixed-H split/header weight.

In odd characteristic, every H Jordan block of size k produces central blocks of sizes k and k+1, together with explicitly specified scalar summands. Thus the motivating ternary H=J3 example has whole central blocks (4,3,1,1,1): the extra length-four attachment is determined, rather than merely bounded by an annihilator. An explicit coordinate isomorphism proves the full ordered-pair statement, including the complement of the commutator image.

For profile r=(d+s00,d+s10,d+s01,d+a+s11), a=nullity H, the number of sharp splits with this exact H is

N_H = product_j |GL_(r_j)(F_q)| /
      (product_j |GL_(s_j)(F_q)| q^(a²+2a s11)).

There are N_H |GL_(n-d)(F_q)| complete ordered headers. These all reach the same residual orbit, but other H may reach it too: this is fixed-H event mass, not unconditional state mass. The Jordan formula requires odd characteristic; the all-field representative, rigidity and count include characteristic two.

Only the five files in research/schur-nilpotent-extension/ are added. Claim type: scoped written proof with exact finite corroboration. The generic factor-equivalence and stabilizer methods are credited premises; the new work is their Schur specialization, explicit complementary attachment and full central Jordan law. No new dimension minimum, general-H classifier, physical supplier, κ or selected-frontier change.

Task and overlap

Base: freshly fetched canonical main ff3cc58e370d0f678e529246f8c73e8094e18fcf, unchanged at the submission checkpoint. Priority revision 7; selected conditional κ=0.00302015. Contribution head: 8ab280c8a3ef0544f13ff23a1c69df4757cd4900.

Canonical claim / investigation, with the #1166 handoff pointer. formal-interface-lemma is explicitly changed to this written algebraic side result, not formalization or a selected-caller audit. The missing obligation was the whole extension beyond im K, rather than another seed of the completed dimension/count/census work.

Exact immediate sources:

• Prove the exact identity-pivot Schur minimum over every field (no κ) #4746, 5f014560d6225cae17598ac479fb29d2947d8e6b: all-field sharp sector criterion and construction.
• Prove a complete factor invariant for ordered idempotents (no new kappa) #4762, 3b4e0449262b0b1e07b80e009ef71cc0ca5bf816, following Derive invariant Schur guard transitions without GL4/GL5 enumeration (no new κ) #4735, 4afbea22e79e19d1d65702eb4090d454ee39d967: factor-system extension principle.
• Prove exact orbit weights for ordered idempotents (no new κ) #4915, 906be31cc1337f599dae85718fbe8187a97453f2: extension-stabilizer counting method.
• Pending Prove block Schur central action and the exact sharp restriction (no κ) #5256, c72d6ff431cf5c979f5010018d8eba0c15577bce: motivating handoff and adapted matrix helpers. Its scientific claims are not theorem premises or executable inputs; this PR starts from main.

The archived source bytes are pinned in SOURCES.json; the pending helper's exact blob/hash and further lineage heads are in TASK.json. A merge into the archive is not scientific acceptance.

Final overlap covers all 7,091 tracked in-scope notes, 327 open PR bodies (2026-10-23 14:49 UTC) and 4,530 Discussion records (14:52 UTC), with complete pagination and no coverage warnings. The other open Schur matches concern P112 diagonal completion (#5249) and downset inertia (#5247). No competing extension result was found within coverage. Arbitrary diffs, nested notes, exhaustive closed-PR bodies and unpublished work remain outside coverage; targeted all-state searches supplement the initial snapshot. No literature-wide novelty claim.

Validation

CPython 3.12.11 / macOS arm64; standard library only:

python3.12 -B scripts/research_package.py --package research/schur-nilpotent-extension
python3.12 -B scripts/check_research_commit.py --package research/schur-nilpotent-extension --check-source-history

• Both registered isolated, warning-strict normal and optimized replays pass, with identical scientific stdout and explicit exception-based checks.
• All 834 nilpotent H in complete domains d=1,2,3 over F2/F3 and d=1,2 over F5 pass literal full-matrix Schur/coordinate checks. All 765 odd-field cases also verify the core and full central rank sequence.
• 45 padded fixtures check the ordered scalar decomposition and construct actual factor/header/residual intertwiners, including multiple nilpotent Jordan blocks.
• Five exhaustive split inventories test the sector criterion against literal Schur ranks and transport every one of the 188 sharp outcomes to its canonical ordered pair. The ternary scalar case has 128 passing splits and 3,105,423,360 complete headers.
• Six independent sector-factor censuses check the stabilizer denominator; nine rejection controls cover the characteristic-two overclaim, omitted extended chain/exponents, nonsquare/nonnilpotent H, negative padding and deficient factor rank.
• All five committed files and source-history pins pass. Official export/integrity checks pass at the exact head above. The affected-package planner selects only this package.
• Both isolated export replays pass outside Git with PATH=/nonexistent, read-only files/directories and unchanged hashes for all six exported files (including the receipt): 3.26s normal / 3.27s optimized. These are replay durations, not algorithmic performance claims.

Export SHA256: 7fe8a9bff73c8eb41f6ae2a8d829e5e9608ecfabf3d06bb497b565e39e8b1f4f.
Scientific stdout SHA256: ba358675b3031aef57895701547380f864223c83ef89a9f44f039714f1a97fa0.
Hosted CI at the exact contribution head is in_progress at handoff; no hosted pass is claimed. This is separate from the completed local checks.

The all-field/prime-power theorem rests on the written argument; the executable checks are finite prime-field corroboration. Independent human/proof-kernel acceptance, Lean, Docker and selected scientific cold replay are unclaimed. Prior source-binding orientation stopped on a sparse-omitted checkpoint; it was not a baseline pass.

Credit

Prepared for ssafwann with substantial OpenAI Codex (gpt-6-astra) assistance, one agent; Apache-2.0. Credit Nick Petalas #4746/#4915/#4597/#4801, MatShep #4762, codyw912 #4735, the earlier ssafwann #4657/#4671/#4678/#5256 lineage, Ryanson Jonathan #4609/#4617, lee-phys #4634, and distinct successor work by Mihir Srivastava #4689, JoyboyBrian #5072, and Krzysztof Wojt #5128. Cinna Davis #4100 and source527/commdata2338/Jose Manuel retain the BIT/header and normalizer context. Predecessor packages and their disclosures remain unchanged.

Open next questions

The README gives a bounded characteristic-two question: start with its all-field model for H=J2/J3 and derive the full central partition without division by two. No claim is held. Unconditional or later-guard state weights should wait for a named consumer/header distribution and cost allowance. Physical cover, selection/emission, reads, owned/borrowed state, dirty restoration, precision, storage, stock, dispatch and the complete consumer bill remain unknown, not zero. Stop this branch at the present result without that contract.