[petalas]

Summary

Scoped algebraic proof with exact finite corroboration. For every field K, dirty-fanout demand (1+xy,x²) over K[x,y] generates the whole ring but cannot be realized with exactly one nonzero source insertion. This answers Open next question 1 of lee-phys #5139, which separates principal demand ideals from elementary-column reachability beyond finite commutative rings.

demand ideal = K[x,y]
one source insertion = impossible
OPT = min(mu0+mu1, 2mu0+2c1, 2mu1+2c0)

The contract has exactly two arbitrary dirty output rows and one preserved disjoint source, polynomial source insertions and elementary row additions, full dirt restoration, and positive stationary coefficient-independent destination prices. The three exact Pareto-minimal count vectors are (1,1,0,0), (2,0,0,2) and (0,2,2,0) in order (P0,P1,E0,E1). The theorem includes characteristic two and every positive data width.

The new step reduces a hypothetical one-insertion word to the elementary orbit of Cohn's classical non-elementary matrix, then combines the obstruction with Krzysztof Wójt's #5045 source-count/return argument and restoring templates. Cohn's theorem is explicitly cited, not rechecked or presented as new. Only seven files in research/cohn-fanout-obstruction/ are added. This contribution was produced by Nick Petalas. No new kappa or selected-frontier change.

Evidence

Before: #5139 proves a principal-ideal criterion over finite commutative rings and explicitly leaves nonsemilocal elementary-orbit obstructions open. After: the unit-ideal polynomial example has a complete all-word lower bound and a matching exact price envelope.

Local CPython 3.12.15, macOS 27.0.1 arm64, standard library only:

• Exact integer polynomial Bezout, inverse, full affine endpoint and completion-coset identities; three attaining words and ten single-call deletion controls.
• 60,621 enumerated scalar forward/inverse pairs over F2,F3,F5,F7; 625 positive price vectors and 17 controls in total.
• All 87 field-evaluated demand pairs have explicit checked one-insertion words. These finite evaluations cannot prove the polynomial obstruction; the generated receipt records that Cohn non-elementarity is not machine-verified.
• Warning-strict normal and optimized direct checks each completed in 0.19 seconds, below their separate 120-second registered limits. Both registered checks pass, as do package metadata/provenance, committed-byte/source-history preflight, static syntax/function-size inspection and whitespace.
• Official seven-file export integrity passes. Read-only Git-free normal and optimized replays produce identical stdout; all eight exported files, including the export receipt, remain unchanged. The verifier emits a generated source-hash receipt and writes nothing.

From the contribution checkout root:

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

Another CPython 3.12 installation can replace that local interpreter. Expected markers are PASS_COHN_FANOUT scalar_pairs=60621 specializations=87 prices=625 controls=17 no_new_kappa, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

The universal obstruction is the written reduction using Cohn, On the structure of the GL2 of a ring, IHÉS 30 (1966), §7, printed p. 26, the example after Proposition 7.3. Its classical non-elementarity theorem is an inherited cited dependency, not a finite certificate or a formal proof delivered here. Independent human and formal proof-kernel review are unclaimed.

Extra workspace, clearing, observations, free general linear maps, and input-dependent coefficients are outside the contract. Native coefficient preparation, routing, precision/tape, stock, supplier admission and a complete multiplication theorem remain unpriced or unproved. Lean, Docker and selected scientific cold replay were not run. Every registered package command ran locally; no predecessor verifier was imported or replayed.

Scientific base is freshly fetched canonical main ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c, priority revision 7, unchanged at submission refresh. TASK.json explicitly changes the historical bit-helper-contract card to #5139's open algebraic side question. SOURCES.json pins predecessor proof text as provenance. The advance CLAIM and entry/submission overlap checks cover live PRs, recent handoffs, active Discussions, account submissions, all-main README terms and distinctive merged searches. Cached entry pagination covered 315 open PRs and 4,395 Discussion records; sparse local notes and unpublished work remain coverage gaps.

Credit lee-phys #5139/#5089 for the criterion and open question, Krzysztof Wójt #5045 for the source-count classification, return lemma and repeated-insertion words, Nick Petalas #4916 for the ring-valued contract, and the Ross Richardson, Cinna Davis, Anthony, Oscar Fennell, Tekalign and retained fanout lineage. Predecessor files and credit are unchanged.

Open next questions

Can one extra restored dirty row remove this polynomial one-insertion obstruction, and at what complete price? Start with a stabilized Cohn elementary word, retaining all three independent dirt columns and every paid row direction. A complete three-row restoring construction and matching tariff are not supplied here.

Related open work

• Prove the exact equal-nonunit dirty-fanout tariff (no new κ) #5208 proves an arbitrary-row equal-nonunit-demand tariff over commutative rings. This PR has unequal polynomial demands and excludes one insertion through a nonsemilocal elementary-column obstruction; it computes no equal-demand optimum.
• Prove the exact 60-call full signed ternary connector optimum (no κ) #5201 proves a signed 26-output connector optimum with three immutable sources and a mutable connector. This PR has one source and two mutable dirty output rows, with different paid gates and no shared optimum computed.
• Research: nonconsecutive commuting C3 fanout schedule #1865 studies commuting C3 fanout intervals in a fixed scalar word. This PR studies an unrestricted polynomial-ring gate contract and does not compute that word's native profile or ceiling.
• Research: price the first profitable helper-star interface word (no κ) #3891 prices a helper-star response with a restored mutable helper. This PR has no helper row and retains a disjoint source; it supplies no H2-sem helper-star price or native supplier comparison.
• Copied-information floor of the selected centre path #3120 studies the selected centre's copied-information rank floor. This PR computes no native centre rank or geometry quantity.
• research: centre sink duality, dependent-triple braid sparsity (|B| <= 12), and circle-cancellation sharing ceilings #3147 studies centre-source incidence, braid sparsity and sharing ceilings. Its native object and bill differ from this polynomial two-row tariff; no shared numerical quantity is compared.

The completion refresh inspected all nine other open PRs from this account created since the CLAIM at 2026-10-23T12:27:09Z. The adjacent ring-valued fanout #5208 and signed connector #5201 are listed above. The others study A4/S4/order-21/dicyclic targets, a spinor relaxation, Gaussian Schur fill and integer-transvection length. No other submission in that snapshot computes this polynomial demand tariff. No additional computations were run to check other contributions.

CI notes

Both hosted checks passed at exact contribution head 514f3dfda983b7ed9ae0966e2e5d58860d92c25e in run 39954477496: planning in 5m56s and the registered package replay in 1m22s. Run metadata confirms the matching head, completed status and conclusion success. The hosted replay executes both warning-strict normal and optimized commands. Automated execution corroborates the bounded package checks; the universal result remains the written proof using the cited Cohn theorem, with independent human and formal proof-kernel review unclaimed. All registered local checks and publication preflights also passed.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Seven new files in one opt-in research package. Shared implementations, frozen sources and selected records are unchanged.