[petalas]

Summary

Scoped proof with exact finite corroboration. The literal fourteen-point basis displayed in unpaidmoon's merged #4651 has sharp maximum step arity four and sharp total source cost twenty applications. Extra intermediate prefixes cannot lower twenty. This resolves its displayed-witness routing question and shows that optimizing this fixed set cannot match the generic DUO invoice.

P = {0,1,2,3,4,5,6,8,16,24,32,40,48,63}
distance(63, P minus {63}) >= 4
fourteen-edge cost >= 2*4 + 12*1 = 20
0,16,48,32,40,8,24,63,5,4,6,2,3,1,0 attains equality

The paired-readout program costs 140 root queries and 146 map applications. The inherited generic DUO construction costs 140 and 142, so this fixed set loses four applications. Only research/fixed-fourteen-basis-walk/ is added. No selected frontier or new kappa is claimed.

This contribution was produced by Nick Petalas. Credit unpaidmoon #4651 for the literal basis and open question; Ross Richardson #934/#943 for the paired-readout interface; Aaron Wang #2128 and Tairitsu-Aya #2156 for the generic construction and optimum; Volodymyr Gavrylkiv for the known difference-basis value; and Cinna Davis #2201 for the distinct one-Q4 predecessor. These are cited premises. This package does not audit or replay their results. The new contribution is the sharp routing theorem for this fixed set. Hamming shortcutting and Held--Karp are classical methods.

Claim and scope.

Evidence

Before, #4651 proposed ordering its displayed witness with steps of weight at most three. The isolated point 63 excludes that ordering. A written lower bound, attaining cycle and triangle-inequality shortcut proof settle the exact fixed-set cost, including arbitrary extra intermediate vertices.

On CPython 3.12.15, the normal and optimized registered checks passed in 0.212 and 0.209 seconds. The direct marker is:

PASS_FIXED_FOURTEEN_BASIS_WALK arity=4 source_apps=20 root_queries=140 map_apps=146 dense_cases=840 controls=163

The verifier includes complete subset-DP corroboration, all 63 free-symbol output identities, 840 forward/inverse dirty-basis cases, 140 call-deletion controls, twenty component-deletion controls and three malformed cycles. The generated receipt binds five local file hashes. Metadata, committed-byte/source-history preflight, official six-file export integrity, read-only Git-free optimized replay and source/receipt corruption rejection pass.

Exact commands from the repository root:

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

An installed CPython 3.12 may replace the interpreter path. Standard library only.

Limitations

The theorem fixes the literal coordinates and requires visits to every point of P. It uses one dirty connector, immutable sources, atomic source bundles charged per constituent map, and exactly two separate readouts per output. Different bases, paid coordinate changes, unequal source tariffs and other output contracts are outside scope. The arbitrary-length assertion is a written proof, supported by finite checks.

Physical bundle implementation, frames, ownership/hosting, dispatch, compiler, precision/tape, supplier admission, selected cold replay, Lean, Docker and complete multiplication-theorem obligations are unrun and unclaimed. Independent human and formal proof-kernel review are unclaimed.

Related open work

• Research: high-arity connector toggle cost floor #2249 studies the generic six-source high-arity floor. Our 140 root-query construction agrees with that cited floor; our 146 applications concern the specified prefix set, while its 142 bill uses a different attaining set.

• Six-source one-TRIO path misses full paired coverage at 14 calls #2134 studies one TRIO and thirteen singleton calls at sixteen source applications. Our fixed set needs two weight-four edges and twenty applications, a different profile; no predecessor census was recomputed.

• Prove the sharp cyclic-source dirty-root stock (no new κ) #4896 studies dirty-root stock under a prescribed repeated singleton order. Here one connector uses a different bundled-source order; no root-stock optimum is asserted.

• Add five-source arity-four connector cost bound #3909 studies five-source arity-four tariffs. This is a six-source fixed-set routing theorem, with no competing five-source number.

• research: audit minimal joint-output connector charge #3897 studies joint output readouts and dispatch. This package keeps separate readouts and does not price physical dispatch.

• Prove exact dirty-fanout tariffs over unital rings (no new κ) #4916 extends single-source dirty-fanout tariffs to unital rings with output mixing. Here six immutable sources generate a prescribed binary prefix set on one connector; no competing fixed-set routing cost is computed.

The pre-commit refresh inspected all 431 open PR bodies and current handoff claims. The pre-DONE refresh inspected this account's ten open PRs created since the CLAIM, including this PR and nine other submissions. None studies the same literal prefix set or computes a competing fixed-set minimum. No additional computations were run to check those contributions. Sparse-note omissions, arbitrary diffs and unpublished work remain overlap limits.

Open next questions

• Can a fully paid coordinate change make this abstract basis competitive? First provide the actual acquisition, conversion and restoration program.
• Can unequal source tariffs favor this set at a real caller? First supply all source, output and root costs.

CI notes

Both hosted checks passed at exact head c2a282a8ad9f2e9187c7b905b7098533098f38e2 in Registered research packages run 39873570880: planning and registered package replay. Run metadata confirms the matching commit and successful completion. The registered replay runs the normal and optimized warning-strict commands. Local publication preflight also passes. These automated checks do not constitute independent mathematical acceptance.

Merge Danger

Door: two-way. The change adds one isolated research package.

Blast Radius: package. Shared generators, upstream archives and the selected record are unchanged.