P112: sharp cost-45 floor for all low-tariff frames on five triads

[flyingviper]

Change and mathematical scope

On five explicit P112 triads, every finite rational decomposition of the strict I15 endpoint whose individual whole-group tariffs are at most 3 costs at least 45. Direct copies attain 45. This rules out all attempts to save a local unit using only such terms, including the previously unresolved 16-term and 17-term profiles; it does not exclude terms costing 4 or more or improve the global P112 target of 296.

The proof uses a rational auxiliary lift and a complementary-projector rank argument. The exact census has just four disjoint exceptional rectangles, each with lifted rank 4 versus tariff 3; repeated exceptions force enough surplus rank to pay back their apparent savings. The older 21-branch certificate is retained as secondary evidence, not a premise of the stronger theorem.

Validation

Two standalone Python 3.12 standard-library replays use independently implemented Fraction and primitive-integer elimination, checking all 961 rectangles. Both pass with -I -O -B, including exact endpoint, repeated-type projector and mutation controls. Metadata/source pins pass, a clean Git-free export replays without writes, and all nine published files match the validated local package byte for byte. The selected cold replay and Lean suites were not run for this isolated scoped proof.

python3 -I -O -B research/p112-five-group-projector-lift/verify.py
python3 -I -O -B research/p112-five-group-projector-lift/independent.py
python3 scripts/research_package.py --package research/p112-five-group-projector-lift

Task and overlap

Based on canonical main d8e0f83f00191df62174b641e9fff055c357f641, refreshed before submission; priority revision 6 and the selected frontier are unchanged. The only external mathematical input is the authenticated selected replay archive/member. The custom triads are explicit local data. TASK.json records the original coverage-law-change card and the explicit user-directed scope change to whole-group rational decoder cost: this local obstruction eliminates one possible ingredient of a cheaper downstream consumer.

This follows the whole-group decoder investigation in #2757, but does not require its candidate partition or unpublished search artifacts. Related #2985 and #2832 concern different one-sided or singleton decoder contracts; their lower bounds are not imported here. Latest overlap check covers the 100 most recently updated open PR bodies; Discussions were unavailable after HTTP 406, so no completeness or novelty guarantee is claimed.

Credit

Prepared by flyingviper with substantial OpenAI Codex assistance, including separate AI proof and implementation audits. Apache-2.0; predecessor attribution is retained. No human scientific admission, native supplier, or improved multiplication bound is claimed.

Open next questions

Can factors costing 4 or more yield a verified saving on these triads? The smallest next test is one explicitly budgeted higher-tariff profile and an exact endpoint certificate; the present lift has additional exceptions there. TASK.json records this question and the missing full decoder/supplier obligations.