Certify the coupled P104 helper-history cost ceiling

[Arham-Tahir64]

PR2640 leaves the P104 crossing inequalities without a coupled price test. This contribution couples all 256 complete monotone helper histories with the credited source-departure and target-arrival floors, the entire original channel profile, and the declared fixed-template ideal-idle budget. A rational primal/dual certificate brackets the optimal relaxed primitive-saving threshold strictly between 0.0023596069 and 0.0023596070.

The result is a budget decision: these necessary constraints still do not reject a correction competitive with the selected complex budget. An explicit integer profile with 137 (1,7,1) histories and 71 all-rank-one histories also passes that relaxation. Neither profile supplies compatible actual subspaces, chronology, coefficients or dirty restoration. No supplier or new kappa is claimed.

Only research/p104-coupled-profile-budget/ is added. It contains the written implication, complete certificates/receipt, source pins, an exact rational proposer, a verifier that trusts no optimizer, seven behavioral groups and an opt-in replay hook. All seven cut censuses and both endpoint-floor premises are separately reconstructed from immutable main inputs. Every dual price comparison uses inherited outward rational intervals. Main, the selected frontier and predecessor evidence are unchanged.

Validation on Windows:

• CPython 3.12.10 registered replay passes, including all seven behavioral groups and exact proposer regeneration.
• Optimized CPython 3.14.6 direct replay and behavioral controls pass; Git-free exports preserve every input byte, and source/certificate/receipt/output faults reject.
• All 65 research workflow tests pass, with 3 skips. Selected baseline bindings, metadata/source pins and staged whitespace pass.

Reproduce:

python3.12 -B scripts/research_package.py --package research/p104-coupled-profile-budget

Scientific base: b4560d9e7216f72b73c5aa940ac099c05e33ccf9, priority revision 5. Publication compatibility was also checked against newly advanced main 1884c50; relevant guidance, selected record, live inputs and replay tools are unchanged. The entry task draw, refined scope, complete cost/unknown-obligation ledger and final overlap coverage are in TASK.json. Canonical advance claim. Final snapshot covered 841 open PRs including drafts and 1,305 Discussion records, with complete pagination and explicit metadata-search limits; exact dependency heads were checked separately.

Credit Nick Petalas/PR2640 for the seven-cut theorem and handoff; ywz65/PR2614 for the departure implication; Hiroki Kumazaki/PR2353 for the arrival implication and explicit model; Nick Petalas/PR2604 for correcting cheap-shape exclusion; Philippe Van Velzen for the original floor, relay design and template screen; retain Rohan Arun/1972, Chafik Boukhalfa/1991, Adam Gall/1940, Josie Elliston/1071, Ali Romman/1125 and the geometry/compiler lineage. Exact heads and written proof links are pinned in the package.

Geometric/chronological realization, native correction/dual, actual hosting/idle/residual charges, physical precision, all-size admission and the selected cold replay remain unrun. The family decision is to finish this audit and stop before construction expansion; a passing relaxation does not price unknown obligations at zero.

Prepared for Arham Tahir with substantial OpenAI Codex assistance in mathematics, implementation, checking and writing. Apache-2.0. No subagents, independent human, formal-kernel or hosted-CI review is claimed.