[petalas]

Summary

Scoped written proof with exact finite corroboration: the binary terminal return-stock bound min(b+s,b+2h) is sharp for every even b, with fixed maximal source libraries and at most h changed unmarked arrivals. The new attaining exchange is

unmarked u -> p+q
marked   p -> u       difference p+u
marked   q -> q+u     difference u

The three final arrivals form a basis. The two marked differences are independent modulo the new unmarked span. Direct sums attain the bound; a synthetic 523+22 instance reaches stock 22 with five changed arrivals and preserves terminal rank 545, including the base support partition 3+3+2+4.

ToikY #7258 supplies the credited dimension-only upper bound and motivating altered-arrival question. The new result is its sharpness construction, not a recheck of that package. A stronger obstruction must use actual libraries, donor projections, scheduling or costs. CLAIM.

Evidence

Before, the dimension argument supplied a five-arrival necessary condition without an attaining abstract construction. The new proof and verifier attain that condition in the declared maximal-library relaxation.

CPython 3.12.15 local checks passed on 825 small construction cases, six 545-dimensional synthetic witnesses, and all 26,286 assignments across ten complete small singleton models. Independent elimination and span-cardinality ranks agree. Four mathematical controls and three invalid-parameter controls pass. Normal and optimized warning-strict receipts agree, with result SHA-256 9d319389498ee9feea465fc7cf4d578b996320a41690919c0a0369dd6fb5fa38. Each registered check was timed locally at under one second.

Metadata/pins, exact committed bytes, modes and source ancestry passed. A six-file read-only Git-free export replay with an empty PATH produces the identical receipt. No predecessor scientific verifier ran.

From the checkout root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error scripts/research_package.py --package research/bit-terminal-two-for-one
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error research/bit-terminal-two-for-one/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/bit-terminal-two-for-one --check-modes --check-source-history

The registered runner executes both normal and optimized commands. The verifier prints a generated receipt bound to all six package files. Its optional --receipt path must be outside the checkout.

Limitations

The libraries are synthetic. This does not assert stock 22 on the actual BIT word or give a chronological implementation, restored named endpoint, physical tariff, complete supplier, improved kappa or selected-frontier change. Free final-basis reconstruction is part of the relaxation. The theorem is over F_2; it fails over F_3. All five selected framework assumptions remain. Native, compiler, physical caller, tape, selected cold replay, independent human review and formal proof-kernel acceptance are unclaimed and unrun.

Open next questions

Can funded changed paths or expanded marked libraries realize one such exchange with actual payloads and distinct eligible terminal ports? Use #7260's source-pinned donor map as a cited input, then enforce the shared U component, quotient projections, joint endpoint and complete timed/owned bill. Do not repeat its census. These obligations do not follow from this abstract attainment.

Related open work

• Bound endpoint-preserving BIT terminal stock (0% κ change) #7258 fixes the actual marked libraries and derives the motivating dimension bound. This contribution proves sharpness with maximal synthetic libraries; it does not compute its literal stock.
• Bound altered BIT terminal donor stock and split charges (0% κ change) #7260 bounds actual old-source unmarked donor stock and separately charged refinements. Its source restrictions are absent from the sharpness relaxation; no same-instance numerical comparison is claimed.
• Bound BIT main-terminal transfer stock (0% κ change) #7209 studies individually restored same-chart transfer at the marked BIT terminal banks. The new exchange grants joint terminal-basis reconstruction and arbitrary unmarked payload changes.
• Exclude cached strict-prefix BIT donors (0% κ change) #7200 excludes cached strict-prefix BIT donors. This construction supplies no such cached donor or physical path.
• Propose conditional κ 0.003645372367832 from two funded cuts #7244 proposes a complete complex-side improvement and motivates the BIT target in Bound endpoint-preserving BIT terminal stock (0% κ change) #7258. This package supplies no complete invoice or competing kappa.

The closeout own-open-PR check found eight other contributions created since the CLAIM. None studies this terminal return-stock capacity question; no additional same-object entry or numerical comparison was needed.

CI notes

CI did not certify this package at unchanged head 19f6c9e5962ae11519661d070c3bdcd3aa6537a3. In run 43029427643, plan job 130552352656 failed in Onboarding helper tests on exactly test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record. The log reports 760 tests, one failure and 24 skipped. Its AssertionError says that `make frontier7000-verify` was not found in the CI verification doc. The package replay job was skipped because plan failed.

The cause is inherited from main e54523517d0f03e58352e647c403a5af3e525a57: certificates/selected-result.json names make frontier7000-verify, while docs/ci-verification.md still describes #6030 and make frontier6030-verify. Both main files were inspected. The record, doc and failing test are unchanged at this PR head.

I read the failed-job logs for each comparison below and confirmed the same sole failing test, missing-command assertion, 760 tests, one failure and 24 skips:

• 42905094314
• 42903313790
• 43024693032
• 43029125610
• 43029249083

All scientific checks listed above passed locally. No fix, retrigger commit, push or job rerun was made. Hosted package replay remains unrun.

Credit

The exchange, sharpness proof and standalone verifier were produced by Nick Petalas. Credit ToikY #7258, exact head 90a63fbefcf79c3e0c0453ef0741dff8517b6167, for the upper bound and open altered-arrival route; Jerry Chen / CubeJerry #7001/#6726 and their retained contributors for the BIT word; Clear1nsight #7244 for the complete comparator. Binary elimination, rank-nullity and direct sums are classical. Independent human and formal proof-kernel review are unclaimed.

Merge Danger

Door: two-way

Blast Radius: package

Six new scoped package files; shared execution and selected-result files are unchanged.