[ToikY]

For every field K, d productive A2 st targets, paid scalar-coupling cap b, arbitrary finite restored dirty spectators and free scalar GL mixing, this package proves the exact common concave formal price:

2 * (floor(d/b) * f(b) + f(d mod b)).

Paid factors are exactly I+([u]-[e])A, u in {s,t}, 0<rank(A)<=b. The common f is nonnegative, nondecreasing and discretely concave, with f(0)=0. The full group-algebra endpoint restores every spectator and off-diagonal coefficient. Singular/nonprojector couplings are allowed. Established selective regular cuts give separate rank demands d; credited packing and coordinate blocks close every remainder. For d=5,b=2,f(k)=k/(k+1), the price is 11/3.

Only research/a2-st-all-cap-tariff/ is added. This answers the identical st portion of Benny Zhao / zhaobenny #6817's nondivisible-target handoff. It does not reclaim its d3/cap2 theorem, packing method or divisible cases. Credit Adam Gall #1941 and Ross Richardson #1966/#1989 for separate-letter cuts, David Arias #2738, Nick Petalas #5083's distinct longest-element schedule, own excluded #6550 classification and retained ywz65/eumemic/Ali Romman/Gareth Davidson lineage. No new invariant is claimed. Exact source pins and byte-identical prose are retained; no predecessor scientific program ran.

Scope explicitly changes bit-new-register-model to this partial formal tariff application. Native calls, coupling-rank to recursive-child-rank conversion, physical GL cost, routing, ownership/restoration implementation, precision, stock and the complete supplier invoice remain unknown. All five selected assumptions remain; no kappa/frontier change or original-card acceptance. Other targets, alphabets and heterogeneous prices remain outside scope.

Canonical main refreshed at 7c49fe3, priority revision8, selected #6030. Explicit handoff, no picker draw. Fresh main/source6817 and all768 PR bodies/heads were unchanged; relevant all-state searches found no exact counterpart within inspected coverage. Unpublished work is unknown. Canonical claim/history: #6814 (comment) . Hub pointer: #1166 (comment) .

Validation, CPython3.12.14, standard library:

python3.12 -B scripts/research_package.py --package research/a2-st-all-cap-tariff
python3.12 -B scripts/check_research_commit.py --package research/a2-st-all-cap-tariff --check-source-history
python3.12 -B scripts/export_research_package.py export --package research/a2-st-all-cap-tariff --output /tmp/a2-st.tar.gz
python3.12 -B scripts/export_research_package.py check --check-inputs /tmp/a2-st.tar.gz

• Written field-independent proof, including characteristic two. Source-aware proof/code/metadata AI roles found no blocker; not blind independent, human or proof-kernel acceptance.
• All722 2x2 matrices over F2/F3/F5 (668 nonprojectors), 2,888 one-gate cut checks,108 complete formal endpoints with378 paid calls (297 dirty-coupled),144 price cases and7 controls. One-gate exhaustion is not all-word enumeration. Nonconcave/nonmonotone controls refute packing outside its premises, not completed legal words.
• First new run0.161s/21.65625MiB. Direct normal/optimized, both registered commands and Git-free read-only empty-PATH exported replays give the identical49,624-byte packet, SHA256 32eb5c0d6a66781c0f603fbab2bbb0171b70ed00ab8f31b354d069d5474baab5.
• Committed canonical provenance, 13-member export/input coverage and12 source/typed-result corruption rejections pass. Foreign PR6817 association was checked at acquisition and retained in SOURCE-PINS, separately from generic canonical ancestry. Cached objects are not publication history.
• Before first science, generic foreign-head provenance rows were corrected for portable publication; draft preserved. Scientific code stayed unchanged after first run. Sparse staging and uncommitted-preflight failure receipts are preserved before the corrected committed PASS. Original hosted attempt1 passed: https://github.com/CrocSwap/integer-mult-bounds/actions/runs/42314824653 . Both complete scientific packets equal frozen local;711 helper tests passed with24skips on CPython3.12.16. The synthetic test-ref checkout has the exact published tree; this is not a repository merge or native/human/kernel acceptance. No old scientific suite, native/Lean/Docker build or selected cold replay ran.

Next: specify and price one actual native rank-capped call with arbitrary input, owned/restored fields, address layout, movement and precision before extending construction. No follow-on claim is reserved.

Prepared for ToikY with substantial source-aware OpenAI Codex proposal, implementation, proof/code/metadata and overlap assistance. The provisional cut recipe arose before claim during orientation; expanded review identified and credited prior methods. No blind priority/independence claim. Apache-2.0. Maintainer edits and auto-merge disabled; no merge requested.