[ToikY]

Change and mathematical scope

PR #3874 leaves open whether integer child counts erase the margin of its three minimal target-reaching lever cells. This package answers that arithmetic question: all three still clear its explicit κ-cap target 1/99, tested at the correctly converted moment saving a = 1/98. It prices both an identical integer paid profile per call and a pooled paid profile, with idle counts integral before normalization in 9-call and 144-call pools.

The tightest model has 0.010351600977 < a* < 0.010351600978. Its nine-call uniform paid profile can tolerate 16 additional rank-one children in total; 17 fail. That is an additive recursive-work screen, with no mass credit. Physical realization, legal pooling, restoration, precision, the bit supplier and complete supplier admission remain open. No new κ or selected-frontier change. Integer twins allow ranks at most the named cap, including a remainder; they do not require every child to have exactly that rank.

Task and overlap

Task complex-new-helper-inventory, scope changed by the handoff-first procedure to PR3874's explicit integer-budget question. This is a planning subproblem; it does not satisfy the card's physical-construction gate.

Only research/lever-integer-twins/ is added (eight files). The retained PR3874 source is read as data and remains byte-identical. Base 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision 6; contribution head a18b03ea76fb373aeb00acc9c4fe119ac617ac83. Canonical claim and handoff.

Validation

Validation completed locally on CPython 3.12: registered replay; normal and optimized checks; clean Git-free export; all 12 exact moment brackets; 224 small dynamic-program comparisons; nine malformed-input controls; a separate integer-power checker for all target signs and 24 surcharge-boundary signs, including off-by-one controls. Exact committed-byte and full source-history checks pass. The independent checker does not claim independent root-bracket or physical-realization verification. Hosted CI run 39193371151 passes both plan and registered package replay for this head.

Reproduce:

python3 -I -B scripts/research_package.py --package research/lever-integer-twins
python3 -B scripts/check_research_commit.py --package research/lever-integer-twins --check-source-history

Credit

Nick Petalas (#3874 grid/open question, #3811 integer cap work, #3754 pooling), Arham Tahir (#3747 aggregation correction), jwr404 (#3757 mass accounting), Penguin/OpenAI for Maximus Putnam (#1685 concavity optimum), Douglas Colkitt/Claude (model/arithmetic), PR1012 and the retained selected supplier lineage. This applies their results to the requested cells; it does not claim them as new. Substantial OpenAI Codex assistance for ToikY; independent human/formal review unclaimed. Apache-2.0.

Entry overlap covered 592 open PR bodies; pre-submission refresh covered 594, including drafts. Live board/thread checks were manual; no complete Discussion cache, and sparse local coverage. No same-cell continuation was found within that coverage. Main and the pinned #3874 head were unchanged. Nothing is merged and no other PR is closed.

Open next questions

See the package's two next questions: derive an actual rank-8/u2 source/consumer contract and charge it against the residual moment budget; characterize legal idle pooling groups from that construction. Both are blocked by the absence of a physical implementation here. Another ideal ceiling alone would not close either obligation.