[petalas]

Summary

Prove a scoped extension of ywz65's merged #2641: localizing a Gaussian public routing header's source support lowers its rank tariff but preserves a strict expected copy-cost surcharge.

M = h(X_A), A nonempty; header price r = rank_Gaussian span(A)
E[C] >= (2/3)B + rL - lambda(A)H(M) >= (2/3)B + L/3

B is total independent uniform demand, L expected prefix-coded header length, and lambda(A) the retained weighted-clique incidence of A. Equal b-bit demands retain the bound 40b + L/3. The support-sensitive entropy estimate is attained by parity headers; attaining the complete communication lower bound is not claimed. The new step retains the header's source support in the conditional entropy argument and obtains its positive gap from the credited flat census.

Claim type: scoped proof with exact finite corroboration, conditional on the cited Gaussian geometry. This answers the localized-source/public-reader part of #2641's open question. It is an original extension, not a predecessor audit. No new kappa or selected-frontier change.

Evidence

• Before: Bound Gaussian adaptive routing with a paid public header #2641 prices public header bits at full-space rank four and leaves localized sources/readers open.
• After: the source-sensitive bound covers every nonempty declared source support, including ranks one through three; their uniform geometric surcharge is at least 1/3 per expected header bit. Eight inherited flat types are transformed without replaying the predecessor's geometry census.
• New exact checks cover all 65,536 four-bit Boolean headers (32 affine equality cases), 63 parity cuts / 4,032 input-cut pairs, three 64-input adaptive protocols with 1,152 receiver observations, and eight rejection controls. Protocol expected charges are 7, 7 and 8; these are examples, not communication optima.
• On CPython 3.12.15, registered ordinary/optimized replay passes. Each registered command took approximately 0.3 seconds locally; the whole registered invocation took 0.78 seconds. These times document timeout margin, not a performance comparison.
• Committed bytes/source history pass for all eight package/live-input files. Official committed export integrity, optimized Git-free replay, source and receipt corruption rejection, and exclusive-output rejection pass. The selected metadata orientation command also passed; no supplier replay was run.

Exact reproduction from the repository root:

~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -I -B research/gaussian-localized-header-tax/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -I -O -B research/gaussian-localized-header-tax/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/gaussian-localized-header-tax
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/gaussian-localized-header-tax --check-source-history

Expected markers: PASS_GAUSSIAN_LOCALIZED_HEADER_TAX surcharge=1/3 boolean_headers=65536 protocols=3 controls=8 no_new_kappa, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES. The generated receipt binds the package and both live predecessor inputs by SHA256.

Limitations

The header is public to every receiver and payload encoder; every payload schedule/codebook is determined by that paid header. Pairing-rank pricing is an abstract premise. Private readers, hidden routing metadata, interactive payload relays, correlated demands, errors, non-prefix termination and cheaper physical sharing are excluded. The estimate bounds expected and hence worst-case charge, not each execution. Geometry is cited and source-pinned, not independently replayed. Physical supplier realization, dirty restoration, native/compiler integration, precision/tape transfer, formal proof-kernel acceptance and the complete multiplication theorem remain unrun/unclaimed.

Open next questions

1. Specify a localized private-reader header with every recipient's observations and all schedule information charged, then test whether a comparable entropy bound survives.
2. Formulate two sequential public headers whose later source support and rank price depend on earlier paid information; determine whether an L_total/3 surcharge survives. This PR claims only one header.

Related open work

No open PR studying this Gaussian localized public-header contract was found in the complete 412-PR cached overlap snapshot and the publication-time live refresh within their stated coverage. No agreement or disagreement with unrelated Gaussian numerical/adaptor work is claimed. Merged #2641 is the direct predecessor; #738 provides the Gaussian geometry and flat certificate, and #888 provides nonlinear transcript composition. The concurrent Eisenstein nonlinear-restriction and Hurwitz decoding-error claims in #1166 study different geometries/contracts and are not numerical comparisons.

The pre-DONE refresh examined all 71 open PRs on this account, including ten created since the CLAIM and nine other submissions. None of those nine studies this Gaussian localized public-header contract or computes a competing value for its fixtures. No additional predecessor computation was run for comparison.

• #4895 studies nonlinear coordinate restrictions in the Eisenstein geometry, with fixed-support messages. It does not compute this Gaussian public-header surcharge.
• #4900 studies budget versus simultaneous decoding success in the Hurwitz matching model. Its error/budget tradeoff is distinct from this zero-error Gaussian expected-charge bound.

CI notes

Both hosted checks passed at b36e5f86954dca9a5b1225cd7878f2eca23fa820 in Registered research packages run 39870352751: planning in 3m07s and registered package replay in 1m12s. Run metadata confirms the matching head and successful completion. The replay executes both registered warning-strict normal and optimized commands. Local registered/direct replay, committed-byte/source-history preflight, export integrity and Git-free replay also passed. Hosted success certifies execution of the scoped automated checks, not independent mathematical or formal proof-kernel acceptance.

Initial local metadata validation rejected the outcome label scoped-proof; changing it to the schema's completed value resolved that packaging issue before publication. The mathematical claim and check domain were unchanged.

Merge Danger

Door: two-way. Blast Radius: package.

Adds six files under research/gaussian-localized-header-tax/; shared code and selected records are unchanged.

This contribution is produced by Nick Petalas. Credit ywz65 #2641, Sagar Thakrar #738, ToikY #888, Alexander Templeton #785, Michael Fischler #814 and Jerry Chen #912, retaining Joel Pulikkan, Douglas Colkitt and DreamingOfClouds's predecessor credit. Entropy, Kraft's inequality and Boolean influence are standard methods. Independent human and formal proof-kernel review are unclaimed. Apache-2.0.

Coordination: #1166 (comment)