[petalas]

Summary

Scoped all-field proof: for any finite bipartite permitted-read mask with nonnegative integer payload widths, the exact minimum block-fitting rank is the maximum independent-set payload weight. Arbitrary rectangular, asymmetric and noncommuting permitted blocks cannot improve on integral coordinate sharing.

minimum query dimension
  = total payload - maximum integral edge sharing
  = maximum independent-set payload weight

For the path with widths (4,1,1,4), the exact answer is 8 although each bipartition side has width 5. This is a query-capacity obstruction, with an attaining construction. It supplies no physical supplier or new kappa.

This contribution is produced by Nick Petalas. Credit Tekalign's merged #3362 for the unequal-payload block-fitting model, Sagar Thakrar/#2484 and ywz65/#2471 for the P112 cycle setting, and classical weighted matching/vertex-cover duality. Cinna Davis, Adam Gall, ctoth, Philippe Van Velzen and the retained P112/E8/native/compiler lineage retain their credit. The predecessor's C5 result and geometry are cited without replay or audit. No literature priority is claimed for the classical graph-theoretic ingredients.

Evidence

• Before: Research: sharp query dimension for unequal C5 payloads #3362 supplies an exact unequal-width five-cycle model. This package changes to the bipartite graph class and proves its complete all-field, all-dimension formula.
• After: PROOF.md combines an identity principal-submatrix lower bound, an elementary integral flow/cut argument and a literal rank-one pair construction. There are no external scientific runtime inputs or predecessor imports.
• Every one of 64 masks on parts of sizes 2 and 3 and every width vector in {0,1,2}^5 passes: 15,552 recipes, independently enumerated independent-set optima and 46,656 actual constructed ranks over GF(2), GF(3), GF(101).
• Exhaustive directed binary completion gives 1,297 mask/width/matrix cases on parts of sizes 2 and 2, including nonsymmetric blocks and repetitions across zero-width masks. Thirteen fixtures and rejection controls pass, including a 4096-bit compressed capacity case.
• Warning-strict normal and optimized registered commands pass locally on CPython 3.12.15, macOS arm64. Timed direct checks took 0.63 and 0.62 seconds respectively, well below their 240-second limits. Lint and formatting checks pass without suppressions. The two modes emit identical receipts.
• Committed-byte/source-history preflight and official export integrity pass. An isolated optimized Git-free replay of final committed bytes passes and leaves all exported files unchanged. Generated receipts bind all six package files by SHA256.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/bipartite-query-capacity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/bipartite-query-capacity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/bipartite-query-capacity
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/bipartite-query-capacity --check-source-history

Expected scientific marker: PASS_BIPARTITE_QUERY_CAPACITY. Substitute an installed Python 3.12 path elsewhere. Standard library only. Optional --receipt /absolute/external/new-file.json generates a new source-bound receipt; default checks write nothing.

Limitations

The main claim is a scoped written proof with finite corroboration. Independent human and formal proof-kernel review are unclaimed. It is not a complete multiplication theorem. Native COPY tariffs, neighbor access, frames, arbitrary dirty-state restoration, ownership, hosting, stock, precision/tape, all-size compilation, supplier admission and outer transfer remain unproved. Nonlinear encoders and nonbipartite masks are outside the theorem. The selected frontier is unchanged.

No full selected cold replay, Lean or Docker suite was run. The current card's historical first ceiling command was attempted and fails with KeyError: 'complex_profile' because it reads an obsolete selected-certificate schema. Its separate lever command runs. The current selected-baseline binding command passes, as does the field-extension positive control. These are orientation evidence, not proof premises or predecessor audit deliverables.

Open next questions

1. Can a named triangle-free nonbipartite mask have a block query rate below integral edge sharing? Begin with an exact fitting identity and its edge-sharing optimum. No candidate is supplied here.
2. Can an attaining pair-query factorization fit a fully charged native consumer? Specify one caller and price every neighbor read, frame conversion, helper restoration and target write. Physical acquisition and all-size admission remain the blockers.

Related open work

• Prove exact full-mask rank with separated-port excess (no new κ) #4917 proves exact full-mask fitting rank with separated-port excess on the fixed P112 mask. That mask contains a C5 and is outside the present bipartite theorem; no shared new numerical value is computed or compared.
• Research: actual Turyn profiles have asymmetric block rank 9d (no new kappa) #1037 studies asymmetric block-fitting rank for actual Turyn profiles. Its mask and geometric domain differ; this theorem does not calculate or recheck its profile value.
• Non-dense fitting B: every E8 term with x·y ≠ 0 has rank ≥ 9; square root-code floors E8 ≥ 72, P112 ≥ 64, P104 ≥ 56, D4 ≥ 12 for any fitting B (no κ claim) #3045 studies sparse fitting pairings and square root-code decoder COPY floors. It has a different mask and tariff; no shared numerical claim is computed here.
• Prove the sharp partial-axis root-code floor (no new κ) #4863 studies a dense root-code partial-axis COPY floor. It does not classify these unequal-payload bipartite query masks, and no numerical comparison is asserted.

Targeted main README, merged-PR and live open-PR/Discussion searches found no competing bipartite-capacity result within their coverage. The cached card scan covered 426 open PRs and 3,958 Discussion records, but only 106 local notes of 6,319 inventoried notes. Sparse omissions, arbitrary diffs and unpublished work remain coverage limits; search results are not a novelty certificate.

The completion refresh read this account's seven other open PRs created since the CLAIM at 2026-10-22 23:34:43 UTC: #4910, #4911, #4912, #4914, #4915, #4916 and #4917. Only #4917 studies the same general fitting-rank quantity, on its different nonbipartite mask as described above. The others concern tensor-error dimension, a fixed source-basis walk, square-zero call compilation, a quadratic inverse bound, idempotent orbit counts and ring-valued fanout tariffs. No additional computations were run to compare predecessor numbers.

CI notes

Both hosted checks passed for head 1244701f6f868f690104fd64d65013d822e59bdc in run 39873348102: planning in 3m5s and registered package replay in 2m14s. Run metadata confirms the exact head, completed status and successful conclusion. The checks were read once at completion. Registered replay executes both complete warning-strict normal and optimized commands.

Local registered replay, committed-byte/source-history checks, export integrity and isolated Git-free replay also pass. Hosted CI certifies execution of these bounded checks, not independent human or formal proof-kernel acceptance of the universal theorem. No full selected cold replay, Lean or Docker run was performed.

Merge danger

Reversible. Adds only research/bipartite-query-capacity/. Shared code, frozen sources, selected records and other research packages are unchanged.