Certify the complete staged544 joint width-price region

[ToikY]

Change and mathematical scope

PR3862 gives sufficient single-basis ranges for the staged544 width prices. This independently reconstructs the complete optimal face and determines exactly which simultaneous adjacent-width inventory changes attain those prices, at the same fixed exponent. Adds only research/staged544-joint-price-region/; no new kappa or frontier change.

The two additional tight columns enlarge the sharp ranges, in helper-calls per owner replica:

Move	Previous sufficient basis range	Complete sharp range
6 to 5	28	48
7 to 6	28	88/3
9 to 8	28	48

The other axial ranges stay 18 for 5 to 4 and 30 for 8 to 7. The complete joint region for nonnegative flows is

x6+x7+x8+x9 <= 48
x5 <= 18+x6
x7+x8 <= 30+x6+x9
3*x7 <= 88+x6+x8+x9

Each boundary has an exact rational packing, with integer aggregate helper/bin counts at T360. The coupled counterexample x6=x7=28 lies within both old individual ranges but outside the joint region. An explicit feasible packing and a separating-row certificate prove a strictly positive idle-cost gap above the common-dual prediction, greater than 0.0006757 per replica. PR3862's global lower bound remains valid; its claims are not described as erroneous.

Validation

CPython 3.12.14, standard library, one numeric thread. The registered replay passed in 69.31 seconds (maximum child RSS 75,710,464 bytes): two independent implementations each classify all 2,551 patterns as 12 symbolic tight and 2,539 strictly positive; exact affine elimination establishes the continuous region, with finite grids as controls; seven independent semantic controls and six actual optimized CLI mutations reject as expected. All three scripts also pass from a read-only, regular-only, inode-disjoint Git-free export with all 14 final package files unchanged. Committed-byte, scoped diff and compile checks pass. Separate adversarial AI review found no mathematical or prose defect.

python3.12 -B scripts/research_package.py --package research/staged544-joint-price-region

Task and overlap

Task independent-moment, refined from its historical #2979 starting point to a specific changed-inventory sharpness implication. Repository assignment a1fddd441a9e9145c53ad1534170023f; original card and scope rationale are retained in TASK.json. Entry overlap inspected all 592 open PR bodies over six API pages, recent #1166 handoffs and the staging #3606 thread; pre-submission refresh found only unrelated new #3887/#3888. The local overlap cache lacked remote coverage; live reads supplied it, without claiming exhaustive historical code-diff coverage. PR3815 establishes the incumbent ceiling; PR3862 supplies the price screen and sufficient basis ranges; PR3857/3875 endpoint refinements and PR3883/3886 admission/retiming audits have distinct scopes.

Source binding: canonical base 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision 6, refreshed before submission. PR3815 77ffbf287b6ca198b4a31e3835db3bae60718b93 and PR3862 b12d58d7b391b0d8490d6ebebe5cd0d4f57653b6 remain unchanged. Exact retained source bytes, Git blobs and SHA256 are checked without executing predecessor code; standalone replay requires no pending Git object. The optional strict source-history gate declined the shallow checkout and is not claimed passed. Full native/selected cold replay and Ruff were not run; no shared supplier code changes. Hosted CI run 39193422908 completed successfully at exact head 772953c9bda6d41db7b89e4cd6bae3dc05d7636e: plan and the registered replay both passed. The replay job 117871474350 log contains both mathematical PASS statuses, the actual CLI control marker and registered outcome completed (finished 2026-10-19T03:15:09Z).

Remaining obligations

Scope: a=0.001730578437266246818741, widths 4..9, fixed bins, fractional clock-free paid-remainder ideal. The new symbolic dual uses exact h; predecessor rational proxy prices are enclosure-consistent rather than claimed bit-identical. Net inventory flows may reuse newly produced widths; disjoint original-helper changes have additional availability limits. Chronology, per-call collisions, literal staging, dirty restoration, local-profile compensation, physical/recursive invoices, supplier admission and inherited all-size/precision/analytic/BIT hypotheses remain unresolved. No all-exponent or all-concave optimality is asserted.

Credit

Credit Erwan Secousses / ChaloupeMusic (PR3815 model and dual), Sagar Thakrar (PR3862 screen/ranges), dataisfire (PR3734), zxv (PR3682), Ross Richardson (PR3605), Jeremy Kothe and the retained lineage; hexapode's PR3795/3802 results retain separate priority. New work by ToikY with substantial OpenAI Codex assistance and a separately implemented AI checker; no independent human or proof-kernel review claimed. Apache-2.0. Open next questions and exact proof are in the package README/TASK/PROOF.

Open next questions

The package README/TASK specifies two follow-ups: construct one literal width-6..9 staging inside the enlarged region with a complete dirty/physical bill, or certify the next optimum outside the region using the coupled counterexample. Both remain unimplemented; the present result completes only the exact planning/audit question.