[petalas]

Summary

Scoped proof with exact finite corroboration. Extend Ross Richardson's merged #3121 directed dirty-fanout tariff from XOR payloads to every nonzero unital ring, including noncommutative rings, with heterogeneous two-sided unit demands and coefficient-independent stationary complete prices.

A_i = a_i + lambda_i p, every lambda_i a unit
minimum price = min_F [sum_roots mu_i + 2 sum_edges c_ij]
attaining parent ratio = lambda_child * inverse(lambda_parent)

The new free-module cut argument handles zero divisors without field-rank assumptions. An exact Z/4 boundary has nonunit demands (2,1), source prices (1,9), ADD prices 1 and true optimum 4; applying the unit formula there would give 3. This is an algebraic interface extension, with no new kappa, physical supplier or selected-frontier change.

Only research/unit-ring-dirty-fanout/ is added. Credit Ross Richardson #3121/#3063, Cinna Davis #1944/#1968, Anthony #774, Oscar Fennell #759, Tekalign #3130 and their retained synthesis lineage. Edmonds supplies the external finite disjoint-branching theorem. Predecessor results are cited; no predecessor executable is replayed. This contribution is produced by Nick Petalas. Independent human and formal proof-kernel review are unclaimed.

Evidence

Before: #3121 explicitly restricts its tariff theorem to XOR and excludes other coefficient rings. After: the written proof gives the stated all-ring unit-demand implication and an ordered coefficient construction; a standalone exact checker corroborates it.

Local CPython 3.12.15, macOS 27.0.1 arm64, standard library:

~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/unit-ring-dirty-fanout
~/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/unit-ring-dirty-fanout/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/unit-ring-dirty-fanout/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/unit-ring-dirty-fanout --check-source-history

Normal and optimized direct checks and the registered package replay pass. Each registered command takes under one second locally. The checker covers 17,292 forest words, their literal inverses, 192 unrestricted optimal-cost comparisons over 48 two-row modular tariff instances, and 98,688 affine states across those instances. Five mathematical boundary groups include wrong noncommutative order and omitted dirt restoration. Generated RESULT.json binds proof, metadata and verifier hashes.

Git-free optimized replay passes with no Git on PATH. Actual proof-byte mutation and a Boolean receipt mutation fail, and an occupied receipt output is rejected without changing its bytes. Ruff 0.17.2 diagnostics E,F,B,I,UP,C4,SIM, formatting and whitespace checks pass. All six committed publication files and source-history ancestry pass at head 459a1d9984e8c5bf09257a409863e63282c282e6. The worktree is clean. Internal read-only proof review found no mathematical blocker; it is separate from human or formal-kernel acceptance.

The universal conclusion comes from the written argument and Edmonds' finite theorem, rather than the finite checks. General nonunit demands, coefficient-dependent prices, extra dirty workspace, intermediate observations, a physical tariff matrix, full supplier/caller pricing, all-size tape realization, Lean and selected scientific cold replay remain outside the checked scope.

Open next questions

• Determine a structural optimum for nonunit demands, beginning with all two-row Z/4 demands at one declared price matrix. Unit ratios no longer provide every required transport.
• Bind one actual caller to complete coefficient-independent tariffs, including source access, coefficient application, frames, lifetime, restoration, storage and precision. This missing physical bill blocks supplier admission.

Related open work

• Research: nonconsecutive commuting C3 fanout schedule #1865 studies commuting C3 fanout intervals in a selected scalar word. This theorem addresses unrestricted ring-valued output mixing under declared stationary tariffs, without repricing that word.

• Research: price the first profitable helper-star interface word (no κ) #3891 prices a concrete binary helper-star response. Its restored mutable helper is outside the disjoint read-only common-source model used here.

• Prove optimal weighted borrowed-source pair tours (no new κ) #4899 optimizes weighted pair-mask tours of a borrowed source. Here the common source is preserved and the dirty output rows may mix.

• Prove a Fibonacci paid-scan floor for two dense dirty outputs (no κ) #4891 gives a paid-scan support floor for dense dirty outputs. Its bank-support invoice differs from these abstract complete gate prices; no shared numerical comparison is claimed.

• Copied-information floor of the selected centre path #3120 studies copied-information rank for the selected centre consumer. Its source geometry and rank tariff are separate from this abstract ring-valued common-source contract.

• research: centre sink duality, dependent-triple braid sparsity (|B| <= 12), and circle-cancellation sharing ceilings #3147 studies centre-source incidence, dependent triples and sharing ceilings. No native route, centre schedule or shared numerical quantity from that work is computed here.

• Audit source-driven Gaussian halo copy enumeration (no new kappa) #4870 enumerates source-driven Gaussian halo copies. Its destination geometry and tape costs do not provide the coefficient-independent complete prices assumed here.

• Prove sharp routing costs for the displayed fourteen-point basis (no κ) #4911 proves routing costs for one six-source binary prefix set on a connector. This theorem preserves one common source and allows dirty outputs to mix; no fixed-set routing minimum is computed here.

• Prove optimal square-zero coupling compilation: four calls at rank three #4912 minimizes reflection calls for square-zero couplings with nonunit coefficient 1+[g]. The unit-demand tariff here does not price that compiler, and no common minimum is compared.

• Prove exact signed monotone unitriangular tariffs (no new κ) #4918 prices signed integer-unitriangular diagonal targets in a formal idempotent-call alphabet. Its targets and paid gates differ from scaled-ADD common-source fanout; no shared numerical quantity is computed.

The completion refresh inspected this account's ten open PRs created since the CLAIM at 2026-10-22T23:36:46Z, including this PR. The adjacent routing, coupling and signed-tariff scopes are distinguished above. The other new submissions concern subspace-map compatibility, full-mask and bipartite fitting ranks, idempotent orbit weights, a quadratic inverse bound and a tensor-error dimension threshold. None reports a competing ring-valued unit-demand fanout optimum. No additional scientific computations were run to audit those contributions.

Coordination and CI notes

Advance claim. Canonical main refreshed unchanged at 624de9d; priority revision 7 and selected kappa 0.00302015 are unchanged. The changed bit-helper-contract scope is recorded in TASK.json.

Live open-PR bodies and recent account submissions, latest handoffs and active Discussions, complete cached remote scans of 428 open PRs and 3,964 Discussion records, and distinctive main/merged searches were checked. Local materialized notes cover 131 of 6,319 tracked note paths; arbitrary diffs and unpublished work are not exhausted. No same-question result was found in this coverage. No novelty guarantee is inferred from that search.

Both hosted checks passed at exact contribution head 459a1d9984e8c5bf09257a409863e63282c282e6 in Registered research packages run 39873844613: planning in 3m48s and registered package replay in 12m16s. Run metadata confirms the matching head, completed status and successful conclusion. The checks were read once on completion.

The registered replay executes the warning-strict normal and optimized commands. Local direct/registered checks, committed-byte/source-history preflight, Git-free optimized replay and corruption controls also passed. Hosted CI certifies execution of these bounded checks; independent human and formal proof-kernel acceptance of the all-ring theorem remain unclaimed. No selected scientific cold replay, Lean or complete physical supplier validation was run.

Merge Danger

Door: two-way. Blast radius: isolated research package. Shared code, predecessor packages, frozen evidence and selected records are unchanged.