[lee-phys]

Change and mathematical scope

PR #5045 leaves incomparable demand ideals open, starting with (2,3) over Z/6. This proves the exact optimum for that instance, gives all five attaining words and proves that their count vectors are the complete Pareto frontier. For arbitrary positive stationary coefficient-independent prices, with c_i charging ADD into row i:

OPT = min(mu0+mu1, 2mu0+2c1, 2mu1+2c0,
          mu0+2c0+2c1, mu1+2c0+2c1).

The additional algebra also extends #5045's piecewise five-term formula to every demand pair over every Z/n, n >= 2, including incomparable ideals with a nonunit common generator. Given demands x,y, the new construction chooses beta=x-a*y with (beta)=(x,y), using an elementary CRT argument. Writing y=r*beta then gives a single-insertion, four-ADD word with exact dirt restoration. The arbitrary-word source-count lower bound and directional return lemma are credited to #5045 and restated in the self-contained proof.

Only research/z6-incomparable-fanout/ is added (five files). This is a scoped written algebraic proof with exact finite corroboration, no new κ. Exactly two arbitrary dirty rows, one preserved disjoint source, and full dirt restoration are required. Extra rows, observers, coefficient-dependent physical prices, arbitrary noncommutative incomparable ideals, a physical caller and compiler/outer admission remain outside scope. No independent human or proof-kernel acceptance is claimed.

Task and overlap

• Scientific base: freshly fetched canonical main 81e9bd69e4a5b00ba9c713a1620e36d576d5522e; unchanged at the submission refresh. Priority revision 7; selected κ remains 60403/20000000.
• Explicit donor assignment takes Prove exact two-row nonunit dirty-fanout tariffs (no new κ) #5045 Open next question 1. bit-helper-contract is marked changed, preserving the full original card and explaining the algebraic side-result boundary.
• Canonical investigation comment in #5030.
• Prove exact two-row nonunit dirty-fanout tariffs (no new κ) #5045, Krzysztof Wójt, retained head 8639d7b6348d9b09683e71d74afcceedf2d4bdc0: proof and README are pinned by Git blob and SHA256 at the new base, as provenance. No predecessor executable is imported or rerun.
• Initial and submission overlap checks include targeted all-state incomparable/fanout/5045 searches and the live DONE: exact two-row nonunit dirty-fanout tariff (#5045) #5030 thread. The submission snapshot covers 307 open PRs (08:39 UTC) and 4,212 Discussion records (08:41 UTC), with complete reported remote pagination. No resolving successor or conflicting active claim was found in that coverage. Local note coverage is sparse; arbitrary diffs, omitted notes, unpublished work and later activity are not exhaustively covered.

Validation

At contribution head ba325c0b94bc69d2a0d264bec581899f220d4613, standard-library Python 3.12 on macOS:

• 119 complete affine-state Dijkstra runs, comparing 3,654 exact optima. All demand pairs over Z/2 through Z/6 are covered; the 5,184-state Z/6 graph is searched at 91 price vectors. The oracle uses only primitive transitions and does not call the formula or word constructors.
• 9,830 ordered CRT checks, 24,575 attaining-candidate words, 147,450 forward/inverse basis checks and 2,160 literal Z/6 scalar-state checks. The word checks cover every demand pair over Z/2 through Z/24 plus larger composite fixtures.
• Sixteen built-in controls cover invalid domains, the false unit-demand shortcut, missing dirt restoration, wrong coefficients and necessity of every envelope term. Three additional disposable semantic mutations reject: missing restore gate, omitted mixing charge, and wrong oracle prices.
• Registered normal and optimized replay, exact source pins, committed-byte preflight and official export integrity pass. Normal and optimized direct replay also pass from a read-only Git-free export with PATH=/nonexistent; all six exported files remain unchanged.

python3.12 -B scripts/research_package.py --package research/z6-incomparable-fanout --check-only
python3.12 -B scripts/research_package.py --package research/z6-incomparable-fanout
python3.12 -B scripts/check_research_commit.py --package research/z6-incomparable-fanout

The universal conclusion rests on the written CRT and arbitrary-word proofs, not extrapolation from the finite price/modulus checks. No selected-construction cold replay, Lean or Docker was needed for this isolated result. Optional full source-history ancestry checking was not run in the shallow checkout. Hosted CI passed both planning and the registered package replay at this exact head in run 39926839937. This is replay evidence, not mathematical acceptance.

Credit

Prepared by lee-phys with substantial OpenAI Codex assistance in derivation, implementation, checking and writing; Apache-2.0. Credit Krzysztof Wójt #5045 for the source-count lower bound, directional return lemma and comparable-demand result; Nick Petalas #4916 for the ring-valued contract; Ross Richardson #3121/#3063; Cinna Davis #1944/#1968; Anthony #774; Oscar Fennell #759; Tekalign #3130; and the retained lineage. No novelty is claimed for CRT or the inherited lower-bound argument.

Open next questions

README and TASK give first steps and blockers for the one-insertion orbit over other rings, three-row nonunit demand (2,1,1) over Z/4, and one actual caller with a complete physical tariff matrix. The residue-ring two-row question is resolved within the stated contract; these extensions remain separate.