[ToikY]

This quantifies the shared-delivery opening already stated by zxv's #6083, on the complete selected #6030 recursive invoice. It is a hypothetical histogram relaxation, with no native delivery or new kappa claim.

For any power tariff r^p, 0<p<1, the sharp maximum credit from four binary helper splits is 4(4^p+5^p-9^p) per LOCAL, even allowing descendant splits. Four distinct rank-nine roots attain the relaxed histogram bound; the pinned source declares 385 per LOCAL.

At selected k=781451/250000000, retaining all eighteen LOCALs, original targets and original paid jobs:

• Two splits per LOCAL cannot pay one additional global rank-one job.
• Three per LOCAL can; with uneven allocation, the exact minimum is 52 total splits.
• Four per LOCAL leave an additional recursive raw-moment allowance strictly between 0.385280036 and 0.385280037 after that one job.
• Even this optimum cannot pay a rank-two job or a second rank-one job.

The relaxation grants two eligible coordinates at each cut with a single global rank-one delivery charge. This sharing, actual split eligibility/target independence, native chart, timing, lifetime and arbitrary-input/owned-field restoration are unknown. The allowance uses recursive moment units, not scalar, physical, precision or compiler fees. A common coordinate budget d must be at least two; no per-cut minimum is asserted for heterogeneous allocations. This does not refute #6083, whose sharing caveat and coarse control are explicitly credited.

Written proof, complete edited-profile receipts, task/scope/exposure and exact source pins are in the package. The original helper-inventory card is explicitly changed/partial; its full productive/native acceptance gate remains unmet.

Validation: first new verifier 0.119 s / 21.125 MiB, 36 split cases, five uniform cases, total-ticket threshold and ten controls. Normal/optimized and both registered checks have identical 21,164-byte scientific receipts. Committed-byte preflight, 11-member official export/input coverage, read-only Git-free empty-PATH normal/optimized replays and eight exact-reason source/typed-result corruption rejections pass. No predecessor checker, word, census, constructor or native program ran. Related source-aware AI proof/code/metadata reviews found no blocker; they are not independent human or kernel acceptance. Original hosted run 42308214257, attempt 1 at this exact head, passed 711 helper tests (24 skips) and both registered checks under CPython 3.12.16. Both complete hosted scientific packets equal the frozen local packets byte for byte. The synthetic test-ref tree and canonical/public parents were verified; this is not a repository merge or native admission.

python3.12 -B scripts/research_package.py --package research/coupled-cut-delivery-fee-budget
python3.12 -B scripts/check_research_commit.py --package research/coupled-cut-delivery-fee-budget

Public head bca4d4a90675b88b45d154935716c8eaecf15956, tested tree fd88ce33f4905e6a415c31be28e09688855980c5, canonical parent 7c49fe32c55de02c2c873655c804b16e4c3fef0a / priority revision 8. Refresh before submission: 767 open bodies/heads, only unrelated #6819 new; narrow all-state searches and live Hub checked, with no claim of exhaustive novelty. Advance investigation/claim.

Credit zxv #6064/#6083, DreamingOfClouds #6030, Ali Romman, Matarener #5994, Nick Petalas #6033, Pierre-Loic Doulcet, Cinna Davis, Jacob Alan Sussman and retained predecessors; classical concavity and rational series methods. Prepared for ToikY with substantial OpenAI Codex proposal, implementation, source/overlap and review assistance. Pre-first receipt-binding correction and separate decimal-display correction are preserved in local receipts; scientific code was unchanged after its first run.

Next: one actual native two-coordinate delivery contract and its complete restoration/physical invoice. All five selected assumptions remain. No merge, selected/frontier change or supplier adoption. Maintainer edits disabled.