[petalas]

Summary

 fixed selected-T16 pooled integer inventory, p=1-a
- old profile certified only for 0 <= a <= 1/7
+ exact min(old,new) for every real 0 <= a <= 1/2

The sharp idle cost is min(2431+17^p,2429+5^p+14^p)+4887*48^p. This closes PR3364's first open question. The old profile remains optimal at a=3/20, despite its earlier fixed-potential failure, and stops being optimal at the unique root in (2/5,5/12) of 5^(1-a)+14^(1-a)-17^(1-a)-2=0. At a=1/2 the new exact optimum beats it by more than 7/50 idle units.

A continuum theorem for the pooled integer relaxation. The selected moment root, packing cap, physical supplier and kappa are unchanged.

Evidence

• Before: PR3364's failed fixed potential leaves an actual optimizer change unresolved. After: a minimum of 49 path candidates certifies all 56,987 path/pattern obligations uniformly. The 7,181 distinct comparisons comprise one identity, 6,850 integer concavity proofs and 330 rational Taylor/Bernstein proofs. Two explicit integer inventories satisfy all nine source-bound equations.
• The written derivative/Rolle argument proves the unique first transition. Exact integer-power and separate square-root controls establish its signs. Seven behavioral groups include independent pattern enumeration, wrong inventory/path/dual/sign controls and an actual optimized source-corruption rejection.
• Registered CPython 3.12.15 replay, optimized tests, Ruff E/F/B/SIM and whitespace checks pass. Committed-byte/source-history preflight passes for all 12 package/input files. The committed read-only Git-free export passes optimized Python 3.14 replay and all 13 exported files remain unchanged.

python3.12 -I -Werror -B scripts/research_package.py --package research/t16-integer-phase

Package research/t16-integer-phase/ includes the full proof, exact receipt, certificates, source hashes/blobs, reproduction instructions and Open next questions. Base/submission refresh 6f100d4bb5083f8c35d96fd95a539811c62f245c, priority revision 6. Canonical investigation: #1885 (comment).

Credit PR3352/PR3364, Ali Romman/PR1129, Hanson Sun/PR1914, Chafik Boukhalfa, Rohan Arun/PR1094, Jeremy Kothe, Hao Yan/PR858 and the retained lineage. Pending certificate bytes are retained so replay needs no unmerged Git object. Development used substantial AI assistance; no independent human or proof-kernel review is claimed.

Merge Danger

Door: two-way. Nine added files form one isolated theorem package.

Blast radius: package. Timing, owner/family eligibility, frames, physical chronology, paid local word, complete native invoice, bit/outer/all-size/compiler, precision, stock/tape/recovery and analytic obligations are not newly replayed. These pooled inventories do not supply physical supplier bills. Frozen inputs and the selected record are unchanged.