[mihir-s-05]

The nilpotent and H=1 primary Schur normal forms did not justify combining arbitrary compressed H. This supplies that missing gluing proof: under the sharp common identity-pivot contract, fixed H and the four ordered input sector dimensions determine one orbit of the entire ordered residual pair. An explicit sector00 projector removes the possible cross-component factor data. The W-invertible component is justified without assuming H nilpotent.

Only five files in research/schur-primary-gluing/ are added. This is a written algebraic side result with exact finite corroboration; the complete kappa calculation and selected frontier are unchanged.

Result and sources

• Proof and scope.
• Separate exact checker, source pins, and task/overlap record.
• Base: current canonical main 7c472563474e60185e6ca101a17622133d047e56, priority revision 7; unchanged at the submission refresh.
• Contribution head: abcf271670e7981ef0dde6582173952902544dc8.
• Canonical claim and coordination.

Task formal-interface-lemma, scope changed by the handoff-first override: this is the written algebraic implication requested by #5337/#5348, with finite corroboration, rather than a Lean formalization or selected-caller audit. Its relevance is removing an explicitly missing residual-state implication; absent a physical consumer, it is a side result.

The proof uses the Fitting decomposition for W=I-H, constructs the actual sector00 projection, splits all four original factor systems, and gives invertible intertwiners for both residual operators. No splitting field or division by two is needed. It permits deficient sector00 factors, mixed/repeated/nonsplit primary components and arbitrary scalar padding. Different H/profile data may still yield the same orbit; parameter injectivity and unconditional orbit weights are not claimed.

Verification

CPython 3.12.0 on Windows AMD64; standard library only:

python -B scripts/research_package.py --package research/schur-primary-gluing
python -B scripts/check_research_commit.py --package research/schur-primary-gluing

Both registered isolated, warning-strict normal and optimized commands pass, with identical scientific stdout:

• 111 complete H-domain cases: all d=1,2 matrices over F2/F3, and d=1 over F4/F5.
• 364 complete arbitrary-sector00 factor cases, with changed sector bases.
• 40 larger mixed, padded and transported fixtures, including repeated nonsplit quadratic factors.
• Eight failure/scope controls, including missing coordinate correction, a globally singular W, wrong odd-field sign, a deficient active factor preserving its product, an unstabilized Fitting split, wrong scalar padding, empty leading block and omission of the profile.
• Literal full headers and both Schur complements are reconstructed; each comparison checks an invertible G with SG=GScan and TG=GTcan. Central spectra alone are never accepted as evidence of ordered conjugacy.
• Metadata/source pins, all five committed files and official package export/integrity checks pass at the contribution head.
• Both exported modes pass outside Git with empty PATH, read-only file attributes and six unchanged file hashes: 4.547s normal / 4.140s optimized. This is a replay measurement, not an algorithmic performance claim.

The optional full-history preflight was not run in the shallow clone. All provenance pins use the accessible base commit, which is an ancestor of this branch. Selected-source orientation reported sparse omissions; no selected baseline or cold-replay pass is claimed. Hosted CI is not yet claimed.

Coordination and attribution

The final refresh covered 305 open PR bodies and 4,638 Discussion records with complete remote pagination. Local coverage was 5,418 records against 7,267 tracked note paths, with 1,851 omitted paths explicitly reported; relevant source notes were materialized and read. Arbitrary historical diffs, nested notes and unpublished work remain outside coverage.

The refresh found Ali Romman's concurrent independent gluing proof. Our proof/checker were frozen at f69863ddefafa183babe2b59a8e5c91a92786fd1 before reading it. Both use the same Fitting route; no exclusive discovery claim is made. Ali's coordination reply moves that contribution's primary focus to the complementary fixed-H header weights. Its code is not imported and its reported checks were not replayed here.

Credit ssafwann #5267 for the nilpotent model; Ali Romman #5348 for the H=1 ordered form and #5287 for the already completed central class; Nick Petalas #4746/#4597 for the sharp criterion, capacities and Schur factorization; MatShep #4762 and codyw912 #4735 for factor extension; JoyboyBrian for the released handoff; and the retained Ryanson Jonathan, lee-phys, Cinna Davis, source527, commdata2338 and Jose Manuel lineage. Classical Fitting and basis-extension methods retain their established meaning.

Prepared for Mihir Srivastava (@mihir-s-05) with substantial OpenAI Codex assistance, one agent; Apache-2.0.

Remaining boundaries

The sharp-sector criterion is a credited premise. This proof and finite checks are not independent human or proof-kernel acceptance. Nonsharp/nonidentity-pivot states, parameter collisions, header probabilities, later-guard survival, physical selection/emission, reads, restoration, precision, storage, stock, dispatch and a complete consumer bill remain separate obligations. No Lean, Docker, native, compiler or selected scientific replay ran. Open next questions and the stop condition are in the package.

Open next questions

Only pursue collisions between different H/profile data if a named consumer needs state deduplication. The smallest first step is comparing the displayed ordered normal forms for a specified pair of profiles. The missing consumer and cost allowance block a broader census or physical implementation campaign; fixed-H/profile sufficiency itself is the completed result.

[jwr404]

Independent review (James Reed, with Claude Code). Head abcf271670e7981ef0dde6582173952902544dc8.

Verdict: the general-H gluing theorem holds on an independent randomized test over F₁₀₁, with negative controls that confirm the test discriminates.

Independent test (own code, no imports from the package)

1. Instances from the source factor equations (6). Pick d ∈ {1,2,3} and a random H = P J P⁻¹ whose J mixes eigenvalues 0, 1 and generic values, with random Jordan couplings, so both nilpotent-W and singular-H components occur. Set a₀ = nullity H, a₁ = nullity W and random s₁₀, s₀₁, s₁₁ ∈ {0,1}. The sector dimensions follow (1), with r₀₀ ≥ d − a₁. Then build:

   • random factors with X₁₀Y₁₀ = X₀₁Y₀₁ = W and X₁₁Y₁₁ = H (Y injective, X onto);
   • an arbitrary rank factorization X₀₀Y₀₀ = −W;
   • i = ΣY_j, π = ΣX_j, P₀ = E₁₀ + E₁₁, Q₀ = E₀₁ + E₁₁.

   The contract πi = πP₀i = πQ₀i = I, H = πP₀Q₀i then holds automatically.

2. Residual. With the header M = [i, ker π], form S = D − cb and T = E − gf. Keep the sharp cases, rank(ST − TS) = 2d.

3. Representative. Build (5) myself from H alone:

   • the Fitting split U = im W^d ⊕ ker W^d;
   • complements N₀ and N₁;
   • (S_c, T_c) on U_c³ ⊕ F^(a₀) and (S₁, T₁) on E₁² as in (3) and (4);
   • plus (0,0)^(r₀₀ − d_c), (1,0)^(s₁₀), (0,1)^(s₀₁) and (1,1)^(s₁₁).

4. Similarity test. Solve the linear system XS = S′X, XT = T′X over F₁₀₁ and sample random elements of its solution space for an invertible one (Schwartz–Zippel). Every accepted X is re-checked against both equations.

	tried	similar
sharp random instances vs (5)	398	398
control: (5) with s₁₀, s₀₁ swapped (when unequal)	162	0
control: (5) built from a perturbed H′ (same nullities a₀, a₁)	54	0

Full disclosure: my first draft drew a fresh random coefficient per coordinate when sampling the intertwiner, which accepted non-intertwiners. The trace(ST) discrepancy in the H′ control exposed it. After the fix, with the re-check assertion, the controls reject as they should.

The dimension identity (3d_c + a₀) + 2(d₁ + a₁) + (r₀₀ − d_c) + Σs = r₀₀ + r₁₀ + r₀₁ + r₁₁ − d also checks out symbolically. The registered replay passes locally.

Not covered: characteristic 2 and non-split irreducible factors over small fields (F₁₀₁ with random H mostly gives split or generic factors), and d > 3. The written proof claims both, and the package's own checker targets them; my test does not add evidence there.