Rank-shape fully charged block-climb gate screen: exact zero-charge thresholds - no new kappa

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Summary

Drawn card complex-rank-shape-decoder (scope unchanged), method-2 hard-search reading:
the fully charged block-climb gate screen over ALL payload reblockings.
At the +5% acceptance target, zero-charge reblocking must cover at least
363 of the 2,412 rank-one payloads (cap 16) or 255 (cap 51), and even at
full coverage at most 10 of 151 blocks (cap 16) or 14 of 48 (cap 51) may pay
even one rank-1 restoration save. With the PR3039 restoration tax imposed
uniformly, the optimal realization is the unchanged selected word: excluded.
No physical decoder, supplier admission or new kappa.

Package: research/rank-shape-coframe-bill-seed850bc3/ (registered; --check-only and full replay pass). Card scope is unchanged: actual_scope equals the catalogue question, and original_card is the byte-equal catalogue object extracted with scripts/new_research_package.py.

What this establishes

1. Exact gate thresholds for zero-charge groups (PR1605 added-work contract; every added child paid on top of the old idle bill). Greedy block-size optimality is certified exactly (600 strict merge inequalities, 1,225 payload-monotonicity inequalities), so these are optima over all partitions, not samples:

Block cap	Minimum zero-charge coverage	Blocks	Full-coverage slack	Taxed blocks allowed at full coverage
16	363 of 2,412 payloads	23	10 rank-1 units	10 of 151 (at least 141 zero-charge)
51	255 of 2,412 payloads	5	14 rank-1 units	14 of 48 (at least 34 zero-charge)

Coverage/charged-work tradeoff at the acceptance target: cap 16 with 363, 768, 2,412 covered payloads leaves 0, 1, 10 rank-1 units of charged slack; cap 51 with 255, 1,530, 2,412 leaves 0, 8, 14.

2. The uniformly taxed family is excluded under that contract. A restoration-taxed block of any size 2..51 is strictly dearer than its payload singles at the target (50/50 exact enclosure inequalities), so blocking without owned gather slots is counterproductive and the optimal uniformly taxed realization is the unchanged selected word, moment 1.000245413040243 at the target (fail). The taxed greedy cap-51 profile is above one for every nonnegative saving. This closes PR3066's conditional sentence about a uniformly taxed construction with exact numbers. Under pure coverage-law refunding the same taxed profile still clears the +5% ceiling gate (1.79x), so the exclusion is a statement about the added-work contract, and both conventions are reported.

3. Charged copies excluded; over-law mass. One paid rank-k transport per block is strictly dearer than the payload singles for every 2..51 (50/50), and its rank mass is 7,998, above the coverage law's 7,232 total, so the law rejects such profiles outright.

4. Predecessor convergence. The 363-payload threshold and the 10-unit slack are PR1605's independently derived numbers, here derived as optima of one exact gate arithmetic. PR3066's seven published mixed-profile moments are reproduced as controls; its five charged pairs spend 0.001390065582957 of the 0.001395098052012 slack its q=0 row leaves (about 10.0 rank-1 units), and six pairs exceed it.

5. Ceiling screen (coverage law, the card's ceiling definition):

Profile	Ceiling bracket	Ratio to selected ceiling	+5% gate
Selected word (control)	0.001690054109..0.0016900541091	1.000x	fail
Greedy cap-16 (150x16 + 12)	0.002480124284..0.0024801242841	1.467x	pass
Greedy cap-51 (47x51 + 15)	0.0030761867866..0.0030761867867	1.820x	pass
Greedy cap-51 + 48 saves	0.0030266309503..0.0030266309504	1.790x	pass as relaxation only

An optimistic passing relaxation is not acceptance; the strict screens in items 1-3 are the acceptance-relevant statements.

Evidence and reproduction

RESULT.json records every enclosure, profile and certificate count. Python 3.12, standard library only, exact rational arithmetic (about 40 seconds):

python3.12 -I -B research/rank-shape-coframe-bill-seed850bc3/verify.py
python3.12 -I -B research/rank-shape-coframe-bill-seed850bc3/verify.py --write
python3.12 scripts/research_package.py --package research/rank-shape-coframe-bill-seed850bc3 --check-only

PASS_RANK_SHAPE_COFRAME_BILL min_cover_cap16=363 slack_units_cap16=10 min_free_blocks_cap16=141 slack_units_cap51=14 min_cover_cap51=255 uniform_tax_excluded=1 no_new_kappa
PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)

Controls: PR3066's seven rows replayed; omitted-save false-pass caught; over-law transport profile rejected by the mass law; baseline fails at the target and at 1/99; all budget boundaries signed on both sides; 918 other partitions of 24 payloads under cap 8 strictly dearer than greedy. The retained PR1240 interval engine is executed from authenticated source bytes (sha256 pinned in SOURCES.json); the vendored PR3066 RESULT.json matches its upstream blob b8de790cb591f883f9446944058b949553de20ce at head 8aa53674536727123436d3f76adb8f5dac6ddea1, with a -text gitattributes rule preserving the bytes.

Route note (recorded obstruction)

PR3039's open question 1 (read the pinned t16 word's per-child operand frames and partition the payload into co-frame groups) was assessed and killed for this session: the canonical word archives record operations plus a frame registry, with no per-child operand frame field, so deriving them needs a separate frame-propagation and temporary-lifetime pass; guessing them would misstate the word. The thresholds above are the exact bar that pass must clear.

Scope and limits

Cost alphabet only: width 73, stock 896/9, centre 64, idle mass 1,646 at rank 51, block caps 16 and 51, PR1429 transport charge, PR3039 per-block restoration tax, both accounting conventions, all payload partitions. Not covered: word-level realizability (which actual payloads are co-frame with owned gather slots), physical decoder identities, the rank-2/rank-3 step children, masked or overcomplete supports (those change an assumption of the eight-term all-pairs bound instead), changed coverage laws/templates/ports/centre paths, native/hosting/precision bills. A scoped lower bound on effective payload child rank for one fixed alphabet and exact gate thresholds; not a lower bound on all decoders, not a supplier, not a complete theorem audit.

Credit

Prepared by Aaron Wang, with help from opencode (substantial AI assistance; no model named; no independent human review claimed). Predecessors, authors verified against live ground truth at writing time: PR 3039 by Aaron Wang (charged-group obstruction, restoration tax as lower bound); PR 3066 by Ross Richardson (mixed-profile correction, moments replayed here; its conditional uniformly-taxed sentence is what items 2-3 close); PR 1605 by Arham Tahir (integer block budgets, exact target, added-work contract); PR 1240 by Doug Colkitt (model and retained interval engine); PR 1429 by Ali Romman (transport-charge contract); PR 679 by Greer Guthrie (native direct-sum/tensor-child obstruction, unchanged). New work Apache-2.0.

Coordination

Claim and FINDING/DONE: discussion 1166 claim comment and the DONE reply posted alongside this PR. Pair-collision pass at 23:35 UTC found no live claim on the object (PR3039/PR3066 claims released); the 18-point cover object belongs to a parallel session and was not touched. Overlap cache: complete pagination for 331 open PRs and 1,849 discussions; matches are leads, not a novelty certificate.