Kappa 0.1 budget screens and conditional paired-plane sensitivity

[Gonzalo-Mier]

Result and scope

This PR adds exact budget screens for the requested target κ = 0.1, which requires primitive saving a = 1/9 under the retained κ = a/(1+a) conversion. The selected result remains κ = 0.001672660197076918; this PR does not claim a κ improvement.

The current-coverage/rank-cap screen still finds moments above one even after all mass is placed at the rank cap and the centre charge is removed: 1.005219793354 on bit (R=53) and 1.026397274382 on complex (R=51). These bounds apply to those selected coverage and rank-cap models, not to every architecture.

Fixed selected bit profile under two unpaid coordinates

The ideal bit aggregate floor reaches 0.999452 only after hypothetically leaving two coordinates unpaid at R=53. Applying that same law to the pinned selected local histogram {1:1323, 2:1342, 3:79, 4:44, 5:102, 6:11, 7:19} (rank mass 5,129) still gives moment 1.365696 with its centre charge, or 1.355508 with the centre removed. Thus the aggregate threshold does not rescue the actual selected bit word.

As a fractional uniform-rank profile screen, the first passing rank is 53 with the centre retained (rank 52 gives 1.001209) and 48 centre-free (rank 47 gives 1.000411). This repacking is a relaxation, not a constructed supplier or a universal obstruction. A bit endpoint would need both a second unpaid coordinate and a major shift toward high-rank children, with all source, frame, consumer and restoration charges accounted for.

Corrections and conditional paired-plane screen

The earlier version charged the mandatory q−2 = 110 dirty-target ADDs per block as rank-one-equivalent terms in the exponent moment. That was only a hypothetical tariff. The pinned arbitrary-dirty anchor accounts its scalar ADDs separately in the compiler envelope, so the revised verifier separates scalar additions from recursive rank costs.

A second distinction matters: the 49-support result is a scalar fitting-rank theorem, while the {2:2, 4:31, 6:16} values are full port-space Gram charges for the supports. The sources do not establish that these charges equal recursive child ranks or native COPY costs. The following moment table is therefore conditional on inserting those values as child ranks, and on the proposed u=2d coverage rule; neither mapping is source-bound to a hosted call.

Copies d	Proposed unpaid u=2d	Centre-free modeled moment	Centre-retained modeled moment
1	2	1.021884	1.033143
2	4	1.003115	1.014374
3	6	0.984346	0.995605
4	8	0.965578	0.976836
5	10	0.946809	0.958068
6	12	0.928040	0.939299
7	14	0.909271	0.920530

Under those assumptions, d=3 is the first pass in the relaxed complex moment model. The known arbitrary-dirty schedule for seven anchors uses 1,568 unit-addition macros; its conservative scalar bound is 10,688,681,187,742, below 2^48 = 281,474,976,710,656. That bound covers the anchor schedule only. The paired-plane decoder's native payloads, query preparation, child wrappers, frames, precision/tape, source placement, restoration and whole-supplier transfer remain unresolved.

Even a complete complex-side realization would leave the unchanged bit-side primitive endpoint at a = 0.001733427039589844234749, corresponding to a full κ cap near 0.001730427469823603. Reaching κ > 0.1 therefore requires a separate, major bit-side improvement as well. No supplier, new κ, or selected-frontier change is claimed.

Exact scopes, rational intervals, source pins and remaining obligations are in the complex budget package and rank-cap package. This corrects the prior fee-based rejection; that rejection is retained only as an artificial rank-one-equivalent tariff sensitivity.

Validation

On CPython 3.12:

• python3.12 scripts/research_package.py --package research/kappa010-complex-lever-floor passed, including the conditional d=1..7 moments and d=8 mass check.
• python3.12 scripts/research_package.py --package research/kappa010-rank-cap-floor passed, including the fixed-profile u=2 screen and adjacent uniform-rank boundaries.
• Both scripts/check_research_commit.py --package ... checks passed at head 7d40236bf22109ad64ae95a1d25eff7fa40478b5.
• scripts/research_baseline.py and git diff --check passed. The selected-record κ and source binding are unchanged.

Next work

The next complex-side test is one complete d=3 block bound to a selected-provider trace, with its query-factor implementation, recursive-rank mapping, child payloads, dirty-state inverse, frames and scalar operations charged. On the bit side, a separate endpoint must leave two coordinates unpaid and move the selected local profile toward high-rank children while pricing its complete call. Neither supplier is available yet; the family expansion gate still requires one complete endpoint or row/consumer invoice before another construction round. Canonical main was refreshed immediately before this update and remains 18aacb215fc3717b06be8ceb9fca62aeca0d3e99 (priority revision 6).

Prepared with substantial OpenAI Codex assistance. Predecessor attributions remain in each package.