Selected bit profile alone exceeds the κ=0.1 moment budget

[Gonzalo-Mier]

Result

This follow-up to PR #3107 certifies an exact local-only lower bound for the selected bit supplier at the requested target κ = 0.1 (a = 1/9 under the retained conversion).

The pinned local histogram {1:1323, 2:1342, 3:79, 4:44, 5:102, 6:11, 7:19}, normalized by w = 296/3, has outward-rational moment 1.195679601525 by itself. This deletes the centre charge and every idle child. Therefore, any change that keeps this histogram and call width cannot pass by changing unpaid-coordinate coverage alone: the local terms already exceed one.

For comparison, under the hypothetical u=2, R=53 law, the full relaxed profile costs 1.355508 centre-free and 1.365696 with the centre. The ideal aggregate floor of 0.999452 does not describe the selected word. Uniform fractional repacking of the same rank mass first passes at rank 48 centre-free or rank 53 with the centre; this is a profile sensitivity only, not a realizable supplier.

This result is scoped to the selected word and call width. A different local word or changed w remains open. It constructs no endpoint and claims no κ improvement; the selected κ remains 0.001672660197076918.

Validation

• python3.12 scripts/research_package.py --package research/kappa010-rank-cap-floor passed.
• python3.12 scripts/check_research_commit.py --package research/kappa010-rank-cap-floor and the corresponding complex-package byte check passed at head fcba8eab38c51f4cdb685559be505222d44f8ddc.
• git diff --check origin/main...HEAD passed; the PR contains only the five-file package delta.

Exact intervals, source pins, limitations and next questions are in the rank-cap package. Canonical main was refreshed to 997e6a9ca7b4f95d50e366f61412fd640d72a88a (priority revision 6); the selected result and pinned quantitative inputs are unchanged.

Prepared with substantial OpenAI Codex assistance; predecessor attribution is recorded in the package.