[petalas]

Summary

Classify the saturated cubic equality case left open by CubeJerry's PR #6768, Open next question 2. Claim type: scoped written proof with exact finite corroboration. This contribution was produced by Nick Petalas.

characteristic two, dim V=3s, rank D=2s
A+B=D, D^2=D, A^3=B^3=0
    => one simultaneous similarity class
    => exactly |GL_(2s)(F_q)| ordered pairs for each fixed D over F_q

For dimension six over F2, the exact count is 20,160. The equality observation map v -> (Ev,EAv,EA^2v), E=I+D, is an isomorphism. In its coordinates the first block row of B^3 forces both free blocks to be identities. The joint centralizer then gives the count. This classifies every saturated maximizer, rather than giving another attaining example.

Credit CubeJerry for the sharp capped spectrum, surviving-kernel idea, attaining family and explicit equality question. Nick Petalas's earlier #5928 spectral envelope, JoyboyBrian's #5507 two-owner model, Ali Romman's #5456 early-helper model and the retained #5452/#5388/#589/#555 lineage keep their credit. The triangular construction of Ferrero, Puczylowski and Sidki, Jordan theory and orbit-stabilizer are classical. Predecessor results are cited, not audited or replayed as this contribution.

Evidence

• Before: the capped-spectrum package gives the maximum and witnesses, leaving equality classification open. After: PROOF.md gives the saturated all-field normal form, stabilizer and fixed-endpoint count.
• Literal three-dimensional enumeration checks all 512 F2 and 262,144 F4 matrices A. Exactly six and 180 pairs pass, and every passing pair has the predicted observation normal form.
• The complete binary GL4 orbit enumerates 20,160 distinct six-dimensional pairs. Each satisfies both cubic equations and the normal form. This is an orbit enumeration; completeness in the ambient six-dimensional domain follows from the written classification, not an ambient 2^36 census.
• All 276 declared block forms, four joint-commutant systems, twelve larger conjugates through dimension 24 and six hypothesis-rejection controls pass. Explicit checks remain active under optimization. Both modes emit identical generated receipts bound to all six package-file hashes.
• Both registered commands ran locally. The matching direct commands took 2.97 s and 3.09 s on CPython 3.12.15. These are replay-deadline measurements, not speedup claims. Metadata, exact committed bytes and declared source-history checks pass at 27aa93132008b940d392c02c8d9230291759c776.
• Official export integrity and declared-input coverage pass. Both direct modes pass from a read-only, Git-free export with empty PATH; their complete receipts equal the local receipts. No registered check is deferred to CI.

Exact reproduction from the repository root, using CPython 3.12 and its standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/saturated-cubic-pair-rigidity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/saturated-cubic-pair-rigidity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/saturated-cubic-pair-rigidity
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/saturated-cubic-pair-rigidity --check-source-history

Any CPython 3.12 path can replace the displayed interpreter. Expected markers: PASS_SATURATED_CUBIC_PAIR_RIGIDITY, PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES.

The source base is canonical main 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, refreshed unchanged before committing. The selected JSON gives conditional kappa 0.003125804. The bit-helper-contract card is explicitly changed to this algebraic side result. The source question is pinned to #6768 head 48233eac64d0645e385407bee33195d78deb2e52. Its proof and question hashes are citations, not unavailable runtime dependencies; the new proof and verifier are self-contained. Advance CLAIM.

Limitations and unrun obligations

The all-dimension and all-characteristic-two-field statement rests on the written proof. Finite corroboration is not formal proof-kernel acceptance. Independent human and formal proof-kernel review are unclaimed. Nonsaturated dimensions, higher caps, nonzero-primary parts and other characteristics are outside this classification.

No physical realization, native chronology, ownership, dirty restoration, complete paid/target invoice, precision/tape, compiler closure, supplier admission or complete multiplication theorem follows. All five selected assumptions remain. No new kappa or selected-frontier change is claimed. No predecessor scientific replay, selected cold replay, Lean or Docker run was needed or performed.

Open next questions

• Classify nonsaturated cap-three maximizers at corank two. Start with the invariant observation-image subspaces for globally cubic-nilpotent pairs at r=4,k=2; handle the possible nonzero-primary part at r=5,k=3 separately.
• Determine whether saturated odd-cap-five pairs are rigid. Start with the ordered first block row of the fifth power in observation coordinates. The cubic argument makes no higher-cap claim.

Related open work

• #6768 proves the sharp capped zero-primary maximum and asks for equality classification. This adds the saturated cap-three classification and fixed-endpoint count. Its scientific engine was not replayed.

• #6560 compiles a single nilpotent-chain operator in a priced reflection model. This pair classification imposes no chronological call word or reflection tariff and does not extend that compilation model.

• #6800 proves a restored-dirty-workspace capacity theorem for three or more square-zero query terms. It studies a different nilpotency and chronological contract and computes no competing cubic equality count.

The completion refresh inspected the eight other open PRs from this account created since the advance CLAIM: #6795, #6796, #6798, #6799, #6800, #6801, #6802 and #6803. Apart from the adjacent square-zero contract above, their tape, dirty-word, cube-lattice, temporal-cut, arithmetic-refinement and singleton-return results concern different objects and quantities. No predecessor or peer scientific replay was added for this check.

CI notes

Both hosted checks passed at 27aa93132008b940d392c02c8d9230291759c776: plan in 4m25s and replay (research/saturated-cubic-pair-rigidity) in 1m37s, in run 42228457634. The completed PR checks were read once; the run reports completed success and the matching head. Hosted replay executes both registered normal and optimized finite checks. Local registered/direct checks, committed bytes/source history and read-only export checks also passed. No registered check remains pending. Hosted execution certifies the declared finite replay, not independent human or formal proof-kernel acceptance of the universal proof.

Merge danger

Door: two-way. Blast radius: isolated. Six files are added in one new research package. Shared code, frozen evidence, predecessor files and selected-frontier records are unchanged.