[petalas]

Summary

Scoped metric proof with exact finite corroboration. A carrier starting at zero, visiting m distinct prescribed entrance lines and ending at a c-dimensional destination has exact minimum tariff

H(1) + (m-1) H(2) + H(c-1), if a required line lies in the destination;
H(1) + (m-1) H(2) + H(c+1), otherwise.

Here m>0 and H is nonnegative, nondecreasing and subadditive, with H(0)=0. The complete subspace-distance graph allows arbitrary intermediate frames and repetitions. With no requirements the minimum is H(c). The result holds over every field and includes H(r)=r^s for 0<s<=1. Two distinct contained entrance lines require c+2 rank units, so sharing their carrier trip cannot retain the original linear bill c in this model.

Only research/multiple-entrance-carrier-tariff/ is added. The proof and standalone verifier were produced by Nick Petalas. Credit DreamingOfClouds for PR6375's shared-carrier construction and format-limits-corrected C.2's distance/single-detour bound; Ali Romman for PR6349 native input lifting and the retained PR6372/6353/6348, P104 and query-caller lineage. The metric and shortcut principle are classical. The new step is the exact multiple-required-line tariff. No predecessor executable is imported or replayed.

Evidence

Before, the cited single-detour result did not price one carrier's complete itinerary through several distinct required entrance lines. After, a written uniform shortcut proof gives matching lower and upper prices, with a separate exact shortest-path corroboration.

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

• Every entrance-line subset and endpoint in all 16 subspaces of F2^3 and all six subspaces of F3^2, under six exact tariffs: 12,864 formula comparisons pass. Full frame/visited-line Dijkstra states allow intermediates, extra visits and repetitions.
• Separate modular row reduction checks geometry ranks and join/intersection distances. Controls distinguish false free sharing, false containment and destination omission; concrete routes show why monotonicity and subadditivity are needed.
• Warning-strict direct normal/optimized replays pass in 0.043/0.047 s. Both registered commands pass, with 0.364 s total package-runner time. These are completion times, not speedup claims.
• Metadata/pins, AST compilation, exact committed bytes and declared source history pass at a083e6e421bffdf2e8ebeddacc32b8ea0c3694df.
• An optimized read-only package copy passes with empty PATH and Git unavailable; all seven files retain their bytes and modes. --emit generates a receipt bound to package hashes.

Reproduce from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/multiple-entrance-carrier-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/multiple-entrance-carrier-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/multiple-entrance-carrier-tariff
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/multiple-entrance-carrier-tariff --check-source-history

Any CPython 3.12 installation can replace the absolute interpreter path. Expected direct output is PASS_MULTIPLE_ENTRANCE_CARRIER_TARIFF instances=12864 no_new_kappa.

Limitations

This is a scoped proof in the declared transport metric, not a native supplier admission or complete multiplication theorem. A restricted native route table inherits the lower bound only after proving that each transition costs at least this metric. Its attainment, scalar response delivery, native absolute phases/all-width implementation, dirty restoration, productive target debits, complete supplier bill, compiler, stock/precision/tape and analytic recovery remain unrun obligations. No selected cold replay or proof-assistant build was run. Independent human and formal proof-kernel review are unclaimed. No kappa or selected-frontier change.

The CLAIM also proposed an arbitrary-subspace zero-excess characterization. This contribution stops at the first verified line-tariff result and defers that secondary extension.

Open next questions

1. Can different response groups use one common entrance line after a fully charged change of scalar basis? Start with two exact response maps, retaining all target debits and correction costs.
2. Can the actual native chart grammar attain the metric tariff for two specified entrance lines? Pin one carrier and compile every column, phase and restoration operation. Physical applicability remains open.

Related open work

• #6375 supplies the shared-carrier input lifts. This result prices a new multiple-distinct-entrance itinerary; it does not check that word or its kappa.

• #6378 studies the frozen6375 complete relative power-tariff threshold. This result studies a new carrier route, with no comparison of the predecessor's numerical threshold.

• #6350 composes native input lifting with the tree caller. The route theorem is a separate transport constraint for later extensions.

• #6353 contracts helper outputs; its unchanged source underlies later input lifts, while this result changes no boundary basis or word.

• #6355 checks6353's rational boundary. No scalar endpoint is checked here.

• #6359 recounts6353's finite invoice. This result gives a route tariff and does not recount that invoice.

• #6360 checks four native frame1094 matrices. Native realization of the present route remains a separate obligation.

• #6361 checks the fixed tree's Laurent endpoint. The present result assumes no caller endpoint.

• #6372 changes two helper output boundaries and supplies6375's source. This result does not change or recheck those boundaries.

• #6376 checks6372's scalar endpoint. This result supplies no native or scalar audit.

• #6381 excludes fixed-schedule stop deletions after helper respanning. Its fixed-contact word model differs from this complete metric route model.

• #6386 bounds three-carrier merges with source-normal retention and a rank-two tail. This result concerns one carrier visiting prescribed entrance lines; no common numerical value is compared.

• #6394 optimizes the fixed PR1940 multi-carrier prefix-tail assignment for concave tariffs. This result fixes one carrier and prescribed entrance lines, permits arbitrary intermediate subspaces, and computes no shared numerical quantity.

CI notes

Both hosted checks, plan and replay (research/multiple-entrance-carrier-tariff), passed in run 41511715994 at exact head a083e6e421bffdf2e8ebeddacc32b8ea0c3694df. The completed checks were read once: plan took 5m2s and replay took 1m35s. Separate run metadata confirms the matching head and successful completion. The replay log contains both scientific PASS markers and the registered-package PASS. Local scientific, metadata, committed-byte, source-history and read-only export checks also passed. Hosted finite replay does not establish independent scientific acceptance, physical metric applicability or a proof-kernel verification of the written theorem.

The completion refresh inspected all fourteen other open account PRs created since the CLAIM, #6380 through #6394 excluding #6388. It added #6394 below. No additional computation audited another contribution, and no conflicting value for this same required-line itinerary was identified.

Merge Danger

Door: two-way. One isolated research package can be reverted.

Blast Radius: package. Shared tooling, predecessor bytes and the selected frontier are unchanged.