[petalas]

Summary

Scoped proof with exact finite corroboration. With exactly two arbitrary dirty integer rows, one preserved source, integer scaled ADDs only and positive coefficient-independent prices, demand (5,12) has the exact tariff

OPT = min(2 mu, mu + 6 c).
mu=10, c=1: optimum 16; extending the finite-ring formula would predict 14.

One insertion requires a primitive integer column. Neither signed target column is reachable from either coordinate vector with two shears. The inverse prefix then forces six paid row additions. An explicit conjugation attains that bound; two direct insertions attain the other branch. Arbitrary integer coefficients and arbitrary finite input-independent interleavings are covered by the written proof.

Only research/integer-fanout-shortening-gap/ is added. This contribution was produced by Nick Petalas. Credit Ali Romman #5312/#5310, lee-phys #5139, Krzysztof Wojt #5045, Ross Richardson #3121, the older #4916 contract and their retained dirty-fanout lineage. Elementary-matrix and Bezout methods are classical. No predecessor scientific suite was audited or replayed as the deliverable.

Evidence

Before: #5312's finite-ring shortening formula explicitly excludes infinite rings. After: a sharp integer tariff shows that principality alone does not preserve its finite-length price. This does not contradict the finite-ring theorem.

Local CPython 3.12.15, standard library only:

• Both full 3-by-3 dirty/source operators and the attaining inverse pass.
• 1,458 literal dirty executions and 729 rational price fixtures pass.
• Complete modulo-60 two-shear searches enumerate 1,617 reachable columns per origin and 14,400 transitions in total. Independent normal-form sets agree; both signed target columns are absent.
• All 16 semantic rejection controls pass. Warning-strict normal/optimized receipts agree exactly.
• Both registered checks pass, timed locally at 0.024 and 0.026 seconds. Package metadata, committed bytes/modes, source ancestry and official export/input integrity pass.
• Read-only Git-free normal/optimized export replays agree with the checkout receipts, with empty PATH and all eight exported files unchanged. Actual source-byte and attaining-word corruptions both reject under optimized Python.
• Source compilation, function-size inspection and whitespace checks pass. No separate Python linter is installed.

From the contribution checkout root:

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

Another CPython 3.12 executable can replace the local interpreter path. Expected markers are PASS_INTEGER_FANOUT_SHORTENING_GAP, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The direct verifier generates a receipt bound to all six package files and writes nothing.

Limitations and unrun obligations

This is an abstract exact integer tariff, not a physical supplier or a complete multiplication theorem. Extra dirty rows, free scales, rational/dyadic coefficients, adaptive words and coefficient-dependent prices change the contract. A matching actual caller, coefficient preparation, routing, precision/tape, stock and full recursive invoice remain unprovided. All registered commands ran locally; no selected cold replay, native suite, Lean or Docker build was run. The selected frontier and all five framework assumptions are unchanged. No new kappa, independent human review or formal proof-kernel acceptance is claimed.

Base main remains 2485805a6d47f36ea83ddde6327d453ac82d44dc, priority 8. Source #5312 is pinned at 5cbdbc8e3858e67c8bfbd160f742ef1e9ced9d38. Live overlap was refreshed before committing, including current reservations and main/merged searches. Sparse notes and unpublished work remain coverage limits. Advance CLAIM.

Open next questions

• Can one-insertion restoration require arbitrarily many row additions for primitive integer demands? Start with signed Euclidean paths and retain inverse-prefix costs. A general shortest-column classification remains open here.
• Does an actual helper caller enforce integer coefficients and provide the complete coefficient-independent prices assumed here? Pin that caller and its physical arithmetic/access contract before applying this tariff.

The first exact result is shipped without extending either question.

Related open work

• Research: exact four-call minimum for an integer-shear target #3905 studies a four-call target over a formal group algebra of integer address shears. Here integer coefficients mix dirty output rows with a preserved common source; no common minimum is computed.

• Research: nonconsecutive commuting C3 fanout schedule #1865 studies commuting C3 fanout intervals in a selected scalar word. Its scheduling and rank profile differ from this unrestricted two-row integer tariff.

• Prove exact heterogeneous three-source clean-row tariffs (no κ) #6254 studies heterogeneous three-source clean-row connector tariffs. This result has one immutable source and two dirty output rows, with different paid operations.

• Copied-information floor of the selected centre path #3120 studies copied-information rank for the selected centre path; no source geometry or copied-information quantity from it is evaluated here.

• research: centre sink duality, dependent-triple braid sparsity (|B| <= 12), and circle-cancellation sharing ceilings #3147 studies centre-source incidence and sharing ceilings; this package supplies no native centre schedule or comparison of its reported quantities.

• Add division-free dual-pair exchange: 134 to 64 scalar units (no κ) #6280 supplies a division-free dual-pair exchange with a restored spare coordinate and coefficient-dependent dyadic pricing. This theorem uses exactly two dirty output rows, one preserved source and coefficient-independent integer gate prices; no common optimum is computed.

The completion refresh inspected all sixteen account open PRs created since the advance CLAIM. No new submission computes the same integer demand tariff, and no numerical disagreement was found in the inspected stated results. No additional scientific computations checked other contributions.

CI notes

Both hosted checks passed at exact contribution head cd2404bfcacaa98fe1281946d0a549c11f396614 in Registered research packages run 41067773387: plan in 4m35s and registered package replay in 1m16s. The completed checks were read once; run metadata confirms the matching head, completed status and successful conclusion. The hosted replay executes both registered warning-strict normal and optimized checks. CI certifies execution of those bounded checks; the universal tariff rests on the written proof. Independent human and formal proof-kernel acceptance remain unclaimed. All registered commands also passed locally.

Merge Danger

Door: two-way. Blast radius: isolated. Six new package files; shared code, predecessor packages, frozen evidence and selected records are unchanged.