[petalas]

Summary

For dirty-fanout demand (2,2,2) over Z/4, prove the exact seven-vector Pareto frontier for every positive source price and every positive destination-only ADD price. This settles the first all-nonunit example explicitly left open by lee-phys #5163. The same written proof gives the common nonzero-demand theorem over every commutative unital ring and every row count:

a_i = min(mu_i, 2c_i)
OPT = sum_i a_i + min_i(mu_i-a_i)

Claim type: scoped written algebraic proof with exact finite corroboration. At source prices (1,9,9) and unit ADD prices, the exact optimum is five. Every nonempty insertion support attains its row-count lower bound by an explicit restoring echo word. There are exactly 2^q-1 Pareto-minimal aggregate count vectors at q rows.

Only five files in research/equal-nonunit-fanout/ are added. The result is produced by Nick Petalas. Credit lee-phys #5163 for the first-example handoff, Krzysztof Wojt #5045 for source-count classification and the two-row nonunit theorem, Ross Richardson #3063/#3121 for the tariff/restoration framework, Nick Petalas #4916 for the ring-valued contract, and Cinna Davis, Anthony, Oscar Fennell, Tekalign and their retained fanout lineage. The return lemma and echo construction are credited premises, not claimed as new techniques. No predecessor verifier is replayed as this contribution.

Evidence

Before: #5163's theorem requires at least one unit demand and explicitly leaves (2,2,2) over Z/4 open. After: an arbitrary-word row-count lower bound and matching words prove its full destination-price envelope, including the arbitrary-row common-demand statement.

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

• 23,433 attaining words and inverses, 298,716 formal-column executions, and 1,792 exhaustive scalar input assignments for the seven three-row witnesses.
• Twelve ring fixtures with 6,595 element triples; all nonzero common demands and all source supports/chosen roots at one through six rows.
• Seven separately implemented complete lower-cost searches on flattened three-row Z/4 affine matrices. Forward/reverse balls certify optima five, four or three at their declared prices; this is not enumeration of the entire affine graph.
• 480 heterogeneous-price comparisons, 120 uniquely exposed frontier vectors, and seven rejecting controls. Warning-strict normal and optimized direct/registered replay pass, with identical evidence output. Direct checks took 0.42 and 0.45 seconds locally, below their 60-second registered limits.
• Ruff 0.17.2 diagnostics E,F,B,I,UP,C4,SIM, formatting and staged whitespace pass. Publication-byte/source-history ancestry checks pass for all five files at c5c71fd61d06e9580673fe4d2475d5f8b4c95e01. Official export integrity and read-only Git-free normal/optimized replay pass; all five exported package files remain unchanged.

From the contribution checkout root:

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

The direct verifier prints PASS_EQUAL_NONUNIT_FANOUT and a generated JSON receipt binding counts to SHA256 hashes of all five package files. The registered runner also prints PASS_REGISTERED_RESEARCH_PACKAGE.

Limitations and unrun obligations

The universal conclusion rests on the written proof; finite checks cover exactly the documented fixtures. Independent human and formal proof-kernel review are unclaimed. Unequal demands, arbitrary directed-edge tariffs, extra dirty workspace, physical source/coefficient/routing/frame/precision costs and complete supplier admission remain outside scope. No new kappa, selected-frontier change or complete multiplication theorem is claimed. Lean, Docker and selected supplier cold replay were not run.

Scientific base is freshly fetched canonical main ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c, priority revision 7, unchanged at submission refresh. TASK.json preserves the historical bit-helper-contract card and records the change to #5163's explicit handoff. Advance CLAIM. Entry and submission searches cover live PRs, current handoffs, active Discussions, account submissions, distinctive main README and merged searches. Cached entry coverage was 314 open PRs and 4,389 Discussion records; sparse local notes, arbitrary diffs and unpublished work remain coverage gaps.

Open next questions

• Determine the destination-priced optimum for (e,h,e+h) over the eight-element square-zero ring, beginning with source-support/ideal bounds. A multi-insertion sharing argument is missing.
• Determine the independently directed-edge-priced optimum for (2,2,2) over Z/4, beginning with a cheap directed cycle and a matching cut bound.
• Bind one physical caller to complete stationary tariffs before inferring a supplier gain.

Related open work

• Research: nonconsecutive commuting C3 fanout schedule #1865 studies commuting C3 fanout intervals in a fixed selected scalar word. This result prices unrestricted abstract equal-demand ring fanout; it does not compute that word's native profile or ceiling.

• Research: price the first profitable helper-star interface word (no κ) #3891 prices a three-target helper-star response with a restored mutable helper. This theorem preserves one disjoint source and restores all dirty output columns; it does not supply that helper contract.

• Copied-information floor of the selected centre path #3120 studies copied-information rank for the selected centre consumer. No native centre geometry or shared rank quantity is computed here.

• 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. Its geometry and complete native bill differ from this abstract destination-priced common-demand contract.

• Prove the exact 60-call full signed ternary connector optimum (no κ) #5201 proves a signed 26-output, three-immutable-source connector optimum. This theorem has one common ring-valued source and mutable dirty output rows, with different paid gates and no shared optimum comparison.

• Prove a principal-ideal dirty-fanout obstruction over K[x,y] (no new κ) #5211 proves a one-source-insertion obstruction for unequal polynomial demands (1+xy,x²) over K[x,y]. This theorem treats common nonzero demands on arbitrary row counts; neither contribution computes a competing optimum for the other's demand.

The completion refresh inspected this account's open PRs created since the CLAIM at 2026-10-23T12:24:55Z. The newly published fanout work is distinguished above; the other submissions study different geometric or operator quantities. No extra computations were run to audit or compare their results.

CI notes

Both hosted checks passed at exact contribution head c5c71fd61d06e9580673fe4d2475d5f8b4c95e01 in Registered research packages run 39954237453: planning in 4m36s and registered package replay in 1m11s. Run metadata confirms the matching head, completed status and conclusion success. The checks were read once on completion.

The hosted replay executes both registered warning-strict normal and optimized checks. Local direct/registered replay, publication-byte/source-history checks, export integrity and read-only Git-free normal/optimized replay also pass. This certifies execution of the bounded checks; independent human and formal proof-kernel acceptance of the universal written proof remain unclaimed. No selected supplier cold replay, Lean or physical caller admission was run.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Five new package files. Shared code, frozen predecessor packages, pinned evidence and selected records are unchanged.