[krzysztofwojt]

Change and mathematical scope

PR #4916 proves the dirty-fanout tariff for unit demands and leaves the structural nonunit optimum open. This proves an exact five-term price envelope for two dirty output rows whose demand coefficients generate comparable principal left ideals, including every pair over Z/p^k, zero demands and suitable noncommutative-ring pairs. Prices are arbitrary positive, stationary and coefficient-independent.

The proof classifies arbitrary words by source-insertion counts and proves that a nonzero ADD direction cannot occur exactly once while returning the original dirt. Explicit words attain the bound. The new distinction is between paying a second source insertion and paying reversible mixing in both directions: for Z/4 demands (2,1), source prices (100,1000) and ADD prices (1,1), the optimum is 104, while repeated insertion costs 202. The predecessor's known optimum-four example is credited calibration.

Only research/two-row-nonunit-fanout/ is added. This is a scoped written algebraic proof with exact finite corroboration, no new κ. More rows, incomparable demand ideals, coefficient-dependent prices, a physical caller/tariff matrix, native/hosting/precision/tape implementation and compiler/outer admission remain outside scope. No independent human or proof-kernel acceptance is claimed.

Task and overlap

• Pinned scientific base: f259dabc7d70e0db8d6d36bfda888f008bbfecc1, priority revision 7, freshly cloned and re-fetched unchanged before submission. During publication canonical main advanced to 4c0a359d88f22f05189d51f48387750b801a45d4; the pin remains its ancestor, and a targeted diff found no changes to root guidance, current priorities, task catalog, selected record, predecessor package, this package path, package verification/export scripts or registered workflows.
• Predecessor: Nick Petalas's Prove exact dirty-fanout tariffs over unital rings (no new κ) #4916 at 459a1d9984e8c5bf09257a409863e63282c282e6; its retained README is byte-pinned. Its unit theorem and single nonunit example are not presented as new.
• Canonical investigation #5030, taking the explicit #1166 handoff.
• bit-helper-contract scope is changed to this handoff's algebraic interface question; TASK.json preserves the original card and explains the side-result boundary. Abstract savings are not promoted to a supplier gain.
• The submission refresh covered 326 open PRs and 4,148 Discussion records (07:10–07:11 UTC), with complete reported pagination and all 6,710 base README/TASK notes. Targeted live fanout PRs, latest submissions and DONE: exact two-row nonunit dirty-fanout tariff (#5045) #5030 replies were checked again. No competing structural two-row result was identified within this coverage; arbitrary diffs and unpublished work remain outside it.

Concurrent #5043, inspected at publication, prices a fixed native common-source fan rewrite on the round-three contact graph. Its Gaussian transition invoice and bounded native occurrences differ from this abstract ring-valued two-row all-price optimum. No shared numerical quantity or native-profile improvement is claimed here.

Validation

At head 8639d7b6348d9b09683e71d74afcceedf2d4bdc0, using Python 3.12 on macOS arm64 and only the standard library:

• 17,313 optimum comparisons across 522 independently implemented exact Dijkstra runs: every demand pair over Z/2, Z/3, Z/4, Z/5, Z/8 and Z/9 for 87 price vectors. No depth cap or probabilistic state keys. The oracle imports no tariff code.
• 2,591 attaining words, 7,773 coefficient-column/inverse checks and 4,800 literal dirty/source-state/inverse checks, including F2[e]/e² and M2(F2). Incomparable pairs and malformed domains reject.
• Registered normal/optimized replay, source pins, exact committed-byte/source-history checks and official export integrity pass. Both normal and optimized direct replay pass from a read-only Git-free export with PATH=/nonexistent.
• Seven local disposable-export corruption controls reject changed predecessor/proof bytes, a bool-to-int receipt change, occupied receipt output, and three semantic mutants even after rebinding their source hashes: a false unit-demand floor, a wrong oracle price and an unreturned mixing gate. All eleven original export files remain unchanged.

python3.12 -B scripts/research_package.py --package research/two-row-nonunit-fanout
python3.12 -B scripts/check_research_commit.py --package research/two-row-nonunit-fanout --check-source-history

The package's generated RESULT.json records deterministic evidence and source hashes. The universal statement rests on PROOF.md, not on extrapolating the finite checks. Selected-source metadata orientation passed; no selected scientific cold replay, Lean or Docker run was needed. Hosted CI run 39917159334 passed both plan and this package's registered replay at the exact head above.

Credit

Krzysztof Wójt (@krzysztofwojt), with substantial OpenAI Codex assistance in derivation, implementation, verification and writing. A separate supporting agent implemented the finite graph oracle; the primary agent reviewed and integrated it. This is not independent human review.

Credit Nick Petalas #4916; Ross Richardson #3121/#3063; Cinna Davis #1944/#1968; Anthony #774, Oscar Fennell #759, Tekalign #3130 and the retained fanout lineage. Apache-2.0. Selected κ remains 0.00302015.

Open next questions

README.md and TASK.json give explicit first steps for incomparable demands (begin with (2,3) over Z/6), three-row nonunit sharing, and one actual physical caller. Their blockers are the missing one-insertion orbit criterion, multi-row return argument and complete physical tariff matrix respectively. This contribution stops at the proved two-row result.