[petalas]

Summary

Prove that PR5456's linear-support BIT obstruction survives arbitrary finite field ownership. This is a scoped written algebraic proof with exact finite corroboration, produced by Nick Petalas. No new kappa is claimed.

final response restricted to its support
  -> blocks with the same latest earlier owner
  -> singleton blocks when supports meet pairwise in at most one field
  -> no nonzero native main Jordan chain

The terminal-block criterion is also sufficient in the unrestricted linear endpoint relaxation, with native-chain requirements explicitly excluded from that converse. A full three-call, three-owner boundary control has all native chains and closes outside the linear-intersection hypothesis.

Credit Ali Romman / PR5456 at 0d64d4d5d73f370182ada9825a611c86cfbf0049 for the model, coefficient equation, two-owner theorem and open ownership question; Dillon Amburgey / dillona's PR5388 early-helper interface and the retained BIT/native/owner/compiler lineage; Brian / PR5507 for the complementary two-owner multigraph funding theorem. These are cited inputs, not audit or replay deliverables. The new proof and checker import no predecessor executable. Advance CLAIM.

Evidence

Before: PR5456 excludes fields with more than two owners from its last-call theorem. After: the written proof removes that cap, and gives the exact terminal response blocks for arbitrary coordinate-support histories.

Local checks pass on Darwin 27.0.0 arm64, CPython 3.12.15, standard library only:

• All 18,795 ordered nonempty-support histories on three coordinates at lengths three through five with at least one field owned by three or more calls. Exactly 11,106 of 33,048 terminal off-diagonal slots are feasible in the linear endpoint relaxation.
• 2,943 independently interpreted literal formal-product impulses, 781 attaining terminal matrices, all 5,040 four-support Fano orders, and eight boundary control groups.
• Warning-strict normal and optimized direct checks, both registered commands, metadata/source pins, six committed files and full declared source-history ancestry.
• Read-only Git-free copied-package normal and optimized checks with empty PATH. Their generated hash-bound reports equal the local report.

Each registered command was timed directly at 0.27 and 0.28 seconds; the complete registered launcher took 0.77 seconds. These are check durations, not a performance-improvement claim. The focused staged diff passed whitespace checks. Static parsing/compilation passed; the largest function has 31 lines.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/bit-linear-support-ownership/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/bit-linear-support-ownership/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/bit-linear-support-ownership
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/bit-linear-support-ownership --check-source-history

Substitute another CPython 3.12 executable when needed. Expected markers are PASS_BIT_LINEAR_SUPPORT_OWNERSHIP, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Receipts are generated from actual enumeration and bind all package files by SHA256. No receipt is handwritten.

Limitations and unrun obligations

Formal coefficientwise equality over F2 with independent commuting stage involutions and q squared zero is essential. Supports are coordinate supports, every response column belongs to its call, and outside-support response rows are permitted. The obstruction does not cover arbitrary many-owner intersections, dependent stage symbols, non-coordinate stage projectors, extra thin calls or free interleaved address maps. Relaxation attainment supplies no earlier native chains or funded caller.

The universal statement relies on the written proof; bounded checks corroborate it. No physical implementation, paid/productive invoice, uniform compiler, precision/tape, stock, analytic recovery, all-size multiplication theorem or selected cold replay is supplied. The selected frontier is unchanged and all five framework assumptions remain. Independent human and formal proof-kernel review are unclaimed. Every registered command ran locally; no proof check is deferred to CI.

Base: canonical main 7c49fe32c55de02c2c873655c804b16e4c3fef0a, refreshed unchanged before submission, priority revision 8 and selected PR6030. The bit-helper-contract card explicitly changes to this algebraic interface obstruction. The selected binding-only positive control passed. Live PRs, current claims, local reservations, distinctive main and merged-object/general-form searches, and the complete cached overlap inventory were checked. Arbitrary code diffs and unpublished work remain coverage limits.

Open next questions

Can a source-compatible many-owner caller with a repeated terminal pair improve the complete signed bill? First give a completed native word whose final chain lies in a nontrivial latest-owner block, then charge all distinct productive helpers, paid calls, restoration and consumer costs. Passing this necessary test alone is insufficient; no funded candidate is supplied.

Related open work

The live exact-source and object searches found no open PR studying this many-owner independent-stage coordinate-support endpoint. PR5456 and PR5507 are merged predecessors. PR5527 has one shared D and a different rank/invoice model. PR5928's local two-matrix spectral envelopes concern a different predicate, without chronological support histories. Those results are cited context, with no additional computations to compare them. Completion refresh checked all 12 account open PRs created since the CLAIM at 2026-11-03T11:07:16Z, including this PR. No other entry studies this many-owner independent-stage support endpoint. Nearby work has different hypotheses and quantities, with no numerical comparison asserted:

• PR6601 bounds one-query width for a triangular carrier; it does not address this coordinate-support endpoint.
• PR6597 extends an abelian one-relation paid-rank bound to nonfield rings; it does not address independent-stage native chains.

CI notes

Both hosted checks passed at head 416978925318bf73554ff5ec1014df8b71b3d504 in Registered research packages run 41625860675: plan in 4m12s and replay (research/bit-linear-support-ownership) in 1m21s. The completed checks were read once with gh pr checks. Hosted replay runs both registered normal and optimized commands. All local checks listed above also passed; no registered check was deferred to CI. Passing replay confirms these registered checks, without claiming independent human or formal proof-kernel acceptance.

Merge danger

Door: two-way. Blast radius: isolated. Six new files in one research package; no shared code, frozen evidence, predecessor credit or selected-frontier records change.