[petalas]

Summary

Scoped proof with exact finite corroboration: determine the exact maximum probability that all ten independent uniform b-bit Hurwitz receivers decode correctly under fixed integer copy-charge budget C.

P_max = 2^(-max(0, 5b - floor(C/2)))

Saving even one unit below the zero-error floor 10b forces simultaneous failure probability at least 1/2. Successful-fiber counting proves the upper bound; partial mate-parity delivery attains every budget, including the odd-budget plateaus. The proof also covers any abstract matching with the stated tariff and independent random coins under the same per-realization budget.

The contribution was produced by Nick Petalas. Alexander Templeton #785 retains the fixed-support nonlinear model and zero-error transcript argument; Douglas R. Colkitt #703 retains the geometry, tariff and transversal method, following Joel Pulikkan #620. Jerry Chen #912 treats a separate adaptive-header expected-cost boundary. These are cited premises; no predecessor result is audited or replayed.

Evidence

• Before: Scoped negative result: Hurwitz copy floor for fixed-support nonlinear messages #785 establishes the zero-error floor and excludes decoding errors.
• After: an all-width written sharp tradeoff, a matching code, and a new exact finite decoder census.
• All 65,536 Boolean one-bit broadcasts on two mate pairs receive exact decoder optimization by conflict-graph independent sets, then literal optimal decoding on 1,048,576 input tuples. The optimal-success histogram, counted over encoders, is 4:1810, 5:17632, 6:28256, 7:15136, 8:2702 out of 16 inputs.
• 117 partial-parity code cases pass over 143,235 literal input tuples. Five invalid cases reject; odd-budget rounding, omitted parity, correlated-input boundaries and conflicting decodes are distinguished.
• Both registered warning-strict checks pass locally on CPython 3.12.15: normal 0.822 seconds, optimized 0.793 seconds. These are replay-duration measurements for the CI limit, not performance comparisons.
• Metadata replay, seven-file committed-byte/source-history preflight, official committed export integrity, Git-free normal/optimized replay, and proof/receipt corruption controls pass. Every exported byte is restored. Local replay planning selects only this package.

Exact reproduction from the repository root:

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

The direct marker is PASS_HURWITZ_BUDGET_SUCCESS. The generated receipt binds the proof, verifier and metadata hashes. On another machine, substitute its CPython 3.12 executable for the absolute interpreter path.

Limitations and unrun obligations

Fixed source/reader supports and widths, independent uniform inputs, mate-only free side information and additive pairing-rank pricing remain premises. The result measures simultaneous correctness, not marginal errors. It does not cover an expected-charge budget, adaptive/private routing, variable lengths or interactive relays. Physical encoding, decoding, source access, hosting, cleanup and tape costs are unknown. No physical supplier, new kappa, selected-frontier change or complete multiplication theorem is claimed. No native/Lean/full-selected replay or independent human/formal-kernel proof review was run.

Open next questions

• Determine the achievable individual-receiver error vectors, starting with two mate pairs and one broadcast bit.
• Determine the sharp unequal-width counterpart, starting with two mate pairs of widths (1,2,1,1). Transversal dimensions then differ.

Related open work

The pre-DONE refresh covered all 71 open PRs on this account, including ten created since the advance CLAIM and nine other submissions. None of those nine computes this Hurwitz budget/simultaneous-success tradeoff. The adjacent nonlinear-message contributions are:

• #4895 bounds fixed-support zero-error Eisenstein copy charge through coordinate restrictions. It uses a different geometry and does not compute this Hurwitz decoding-success probability.
• #4897 proves an expected-charge surcharge for localized public Gaussian headers. Its zero-error adaptive-header contract differs from this fixed-width Hurwitz error/budget model.

No competing value for this Hurwitz quantity was found, and no additional predecessor computations were run for comparison. #703, #785 and #912 are merged predecessors.

CI notes

Both hosted checks passed at c12dddd5765150fe5c37a7e9e2a83d58f6c44ccd in Registered research packages run 39870488514: plan and replay (research/hurwitz-budget-success) / replay (research/hurwitz-budget-success). Run metadata confirms the exact head, completed status and successful conclusion. The registered replay executes both warning-strict normal and optimized commands. Local registered/direct replay, committed-byte/source-history preflight, official export integrity, Git-free replay and corruption controls also passed. These automated checks do not establish independent human or formal proof-kernel acceptance or physical supplier admission.

Merge Danger

Door: two-way.

Blast Radius: package.

Adds only the seven files in research/hurwitz-budget-success/; no shared runtime or selected source changes.