[petalas]

Summary

Make the supplied-binary-block step in Charan/theon199 #2330, Open next question 1, constructive. The new extractor returns maximum disjoint nonsingular minors and an equal-valued original-index cut in deterministic polynomial time. This is a scoped constructive algorithm and written proof with finite certificates; no new kappa or complete multiplication theorem is claimed.

binary color blocks
  outer row-partition / induced-matroid intersection
    inner column-partition / linear-matroid intersection
  maximum split + equal-valued rank cut

The implementation specializes classical matroid intersection and Schrijver's induced-matroid theorem. The new contribution is this decoder extraction/cut-recovery implementation and its independently checkable interface, not a new general matroid theorem. Credit Charan/theon199 #2330 for the explicit question and cut contract, DreamingOfClouds #2162 and the #1590/#2111 decoder model, and the classical Schrijver/Edmonds/Goemans foundations cited in the proof. Produced by Nick Petalas.

Evidence

• Before: Research note: justify decoder split min-max via linking systems #2330 explicitly left polynomial split/cut extraction open, with a deliberately small exhaustive checker. That predecessor is cited; its verifier was not replayed.
• After: 4,368 new exhaustive width-two inputs, 30 specified larger sampled inputs, nine structured inputs through width 32, a greedy-trap fixture and 20 malformed-input/certificate rejection controls pass. Every returned split and cut receives an independent check. Exhaustive small checks use permutation determinants and exhaustive cut minimization; larger checks use a separate dense-rank implementation.
• Source-bound metadata, direct warning-strict replay, all registered normal/optimized checks, committed-byte/source-history/mode preflight and portable export checks are recorded below. Generated receipts bind the package hashes; no receipt is hand-written.

From the repository root, with CPython 3.12 and the standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -I -B research/polynomial-decoder-split/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/polynomial-decoder-split
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/polynomial-decoder-split --check-source-history --check-modes

The direct checker ends PASS_POLYNOMIAL_DECODER_SPLIT; the registered runner ends PASS_REGISTERED_RESEARCH_PACKAGE research/polynomial-decoder-split outcome=completed. Any CPython 3.12 path can replace the shown interpreter path.

Limitations

The written construction concerns arbitrary finite binary blocks. The executable accepts 1–8 colors, width 1–32, and at most 96 color copies per side. Larger unstructured verification inputs are sampled. The actual tied_min pairing/support adapter, nonbinary arithmetic, physical decoder, complete supplier invoice and selected-frontier admission remain open. No Lean or production supplier replay was run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Source-bind one actual tied_min pairing and its support masks to this supplied-block interface, then export its split or obstruction. The input adapter is still needed.
2. Extend executable rank and certificate arithmetic to prime and extension fields, preserving the same oracle/cut contract. Only binary arithmetic is implemented here.

Related open work

• Decoder floors: universality, projection transfer and batching reductions; C(P112) ≥ 64 reduced to named conjectures, C(P120) ≥ 72 restated on N₀ (no κ claim) #2832 studies rational decoder cost floors and batching reductions. This binary split extractor does not establish those cost conjectures or replace their hypotheses.

No open PR found in the live split/min-max overlap search delivers the same extractor. Main remains de5ec2e, priority revision 8; the selected frontier is unchanged.

Related open work was refreshed against this account's open PRs created since the advance CLAIM. No additional PR studying the same binary split-packing and equal-cut quantity was found. The partial-query acquisition result #5878 and dyadic tensor-capacity result #5882 use different objects and optimization quantities; no common numerical value is compared.

CI notes

At commit d20ba5c5883035557987259e515dc43b43e9c02b, the registered-package run failed in plan, solely at Onboarding helper tests. All ten failure/error entries are in unchanged tests/test_research_current_assignment.py; the log ends FAILED (failures=8, errors=2, skipped=22). The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'; the assertions include zero resolved cards instead of 16, missing expected ValueErrors and empty sparse-reference results. These are the shared onboarding failures on base de5ec2e7a6074068452f45b1a191b5c72832df4a, outside this package. The ten failure/error labels and final summary match PR #5874's failed run exactly; both logs were inspected.

The package replay was skipped. Hosted replay has not certified this package; only local verification has run. The shared failure is left for maintainers; no onboarding test or shared file was changed.

Local direct replay passed in 26.33 s. Registered normal phases took 1.017 s, 24.663 s and 0.157 s, and the complete optimized replay took about 26.6 s. All commands stayed below the 300-second bound. Committed-byte/source-history/owner-execute checks passed for all nine package/input files; official export integrity and both declared input checks passed. Every registered command also passed from the extracted read-only Git-free export, with all ten exported files unchanged.

Merge Danger

Two-way door. Blast radius: the new isolated research package. Shared code and selected certificates are untouched.