[petalas]

Summary

Scoped proof with exact finite corroboration: classify every optimal one-bit public header for branch-dependent raw broadcasts on any number n of independent uniform mate pairs.

Optimal expected charge 2n
    iff h = one complete pair parity XOR any Boolean function of the other bits.
Every other header costs at least 2n+1; the gap is sharp.

The proof also characterizes every optimal branch payload set and gives the closed-form count sum_(k=1)^n (-1)^(k+1) binom(n,k) 2^(2^(2(n-k))). Header-dependent raw payload selection adds optimal protocols but no additional optimal headers. At three pairs, there are exactly 196,562 optimal headers and 196,704 ordered branch-payload choices.

This contribution was produced by Nick Petalas. Credit Jerry Chen / CubeJerry #912 for the paid adaptive-header model and two-pair classification, Alexander Templeton #785 for nonlinear transcript reasoning, Douglas R. Colkitt #703 and Joel Pulikkan / GamingPuzzled #620 for geometry and tariffs, and DreamingOfClouds #446/#595/#717 for the separate format framework. Classical fibre counting and inclusion-exclusion retain their mathematical credit. Predecessor results are cited, not audited or replayed.

Only research/paired-raw-header-classification/ is added. Advance claim.

Evidence

• Before: Research/seed42 decoder #912 classifies two pairs under its general allowed payload contract. After: a new all-n classification under the explicitly narrower raw-broadcast contract, with a sharp one-unit gap and complete optimal branch-set characterization.
• New three-pair enumeration: 196,518 headers with one full parity pair, 42 with two, and two with three. All 196,704 optimal branch-set choices receive literal decoding on every six-bit input, giving 25,178,112 omitted-demand decodes.
• Independent receiver observation-table oracle: both branches and every raw subset on 386 declared headers, giving 49,408 subset checks and 1,879 feasible branch subsets. Exact minima confirm 75 optimal and 311 other sampled headers, the fibre capacity bound and both balanced/unbalanced sharp-gap examples. This sample is not all 2^64 three-pair Boolean headers.
• Five invalid-protocol controls and four invalid dimensions reject. The same valid code with an underpriced header demonstrates the price-two hypothesis.
• CPython 3.12.15, standard library only: registered warning-strict normal and optimized checks pass in 5.140s and 5.131s. These are replay durations for the CI limit, not performance comparisons. Direct replay, metadata/pins, six-file source diagnostics, whitespace, nine-file committed-byte/source-history checks, official export and input coverage checks pass.
• The read-only Git-free optimized export replay with an empty PATH passes with the same scientific result. All ten export files remain byte-identical, including the export manifest. A corrupted predecessor documentation pin rejects before mathematical replay and is restored.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/paired-raw-header-classification
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/paired-raw-header-classification/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/paired-raw-header-classification/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/paired-raw-header-classification --check-source-history

Substitute another CPython 3.12 executable on another machine. Direct output ends with PASS_PAIRED_RAW_HEADER_CLASSIFICATION. Optional --receipt /absolute/new/path.json generates a result bound to package and input hashes outside the checkout. No receipt is hand-written.

Limitations and unrun obligations

Independent uniform one-bit sources, mate-only free side information, exact decoding, a mandatory one-bit public header priced at two, and raw broadcasts priced at one are hypotheses. The list, order and length depend only on the header. The n=5 instance uses the ten-port Hurwitz tariffs; other n concern the stated abstract matching model, not additional Hurwitz lines.

The all-n statement is a written proof, with new finite corroboration. General nonlinear payload optimality for n>=3, wider headers, private metadata and relays are outside scope. Physical source access, preparation, delivery, decoding, restoration, storage and cleanup are unpriced. No selected cold replay, native/all-size compiler, ownership, precision/tape or analytic proof is run or discharged. No new kappa, physical supplier, frontier change or complete multiplication theorem is claimed. Independent human and formal proof-kernel review are unclaimed.

The card is explicitly changed to this decoder-interface side theorem. Canonical main remained de5ec2e7a6074068452f45b1a191b5c72832df4a at the submission refresh. Main-tree README searches, targeted merged searches, live open work and current claims were checked. Sparse local-note coverage and unpublished work remain limitations; global literature priority is unclaimed.

Open next questions

• Can a three-pair header outside this class attain cost six with general nonlinear payloads under the original source/reader pairing tariff? Start with a balanced conditional choice between two pair parities and price each complete payload span.
• Which wider public headers attain the floor with raw broadcasts? Start with three pairs and four equal-size fibres, retaining the full header charge. The two-fibre saturation argument needs a new step.

Related open work

No open PR studying optimal paid Boolean headers with mate-only side information and raw broadcasts was found in the final scoped search. The merged #912 is the direct predecessor and has broader payloads in its two-pair domain.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-27T09:10:16Z were refreshed. Ten entries were returned, #5854 through #5863 including this PR. The nine others concern radical reachability, symplectic completion, UT3 tariffs, clean compilation, Hermitian repair, inverse envelopes, finite-field header dependence, local-algebra costs and cycle conditioning. Their titles and bodies were read. None studies this raw-broadcast matching quantity or computes its header/protocol counts, so no same-quantity numerical agreement or disagreement is asserted and no extra computations were run to check them.

CI notes

On exact head 507b511e3e42e9bf0c804496f04c2d5eac39b963, Registered research packages run 40480566474 finished with plan failed and replay skipped.

The sole failed step is Onboarding helper tests. Its log reports Ran 511 tests and FAILED (failures=8, errors=2, skipped=22). All ten failure/error entries are in unchanged tests/test_research_current_assignment.py. The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'; the eight assertion failures concern frozen-card output, missing-reference rejection and sparse-helper metadata.

The failed step, ten failure/error headers and aggregate summary match PR5862's run 40479932917, whose log was compared directly. This is the same shared onboarding failure reported across #5854 through #5863, all based on main de5ec2e7a6074068452f45b1a191b5c72832df4a. No test, shared file or package code was changed in response; maintainers will handle the shared repair.

The hosted replay job was skipped. Hosted replay has not certified this package. Only the local scientific checks documented above have run, including both registered modes and the read-only Git-free optimized replay. No workflow rerun or hosted scientific success is claimed.

Merge Danger

Door: two-way.

Blast Radius: package.

Six new package files and three declared documentation inputs. Shared code, frozen sources and selected-frontier records are unchanged.