Block-climb complete profile: exact obstruction for charged block-climb realization - no kappa

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Drawn card complex-rank-shape-decoder (scope unchanged). Package research/block-climb-complete-profile-seed1ed712/. Scoped exact obstruction; no kappa claim; the selected record is unchanged.

Agent: Aaron Wang, with help from opencode (substantial AI assistance; no model named; no independent human review claimed). Branch research/agent-82-rank-shape-blockclimb-1ed71298, head f0d6a48245e44fc29cfe077d21ca2521c2a8d941, base d8e0f83f00191df62174b641e9fff055c357f641 (canonical main, re-fetched and still identical at the pre-submission refresh). Chooser OS seed 1ed71298f6f32dd18e15e5ed217adb95, scope key 02b31ebd160542c86299, priority revision 6, selected-record SHA-256 bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129.

Question and result

The card asks for one exact small decoding identity whose frame path climbs in blocks, priced with its complete per-call profile: can a decoder deliver the local paid mass in fewer, larger-rank children instead of rank-1 to rank-3 steps? This session runs the card's own construction method at the smallest falsifying instance (k = 2) and one bounded pivot (the PR 1605 threshold partitions and full reblock), with every frame transport and restoration charged.

Result: an exact obstruction for every charged realization of the block-climb lever. A climb group of k payload units delivered by one rank-k child with alpha charged transports pays only when k^a \gt \alpha + 1, i.e. a \gt \ln(1+\alpha)/\ln k. Since $\ln 2 / \ln 51 = 0.17629143\ldots$, every group with at least one charged transport is strictly dearer than the k rank-1 children it replaces for every a at or below 1/99, every k = 2..51 (100 grid points, two independent exact enclosures). The lever survives only for per-group zero-charge realizations (both operand vectors already at the block frame, owned gather slots), whose complete charged invoice must then fit the call-level budget this package reproduces.

What is established

1. Exact identity classification at k = 2. A block child's outputs are rank-1 tensors. For a matching payload pattern the rank-1 tensors inside the required span are exactly the coordinate ones (determinant conic ab), so only coordinate-aligned child outputs deliver the two products; for shared-row/shared-column patterns mixed outputs exist and decode back with exact integer additions, but mixing never reduces the charged transport count. The k = 3 diagonal fixture witnesses the alignment; the general argument is written, not machine-checked.
2. Dirty gather-slot restoration forces paid work. Writing a gathered operand over a dirty slot is non-injective in the prior content (exact collision witness), so restoration needs a paid rank-1 save per overwritten slot. This turns the "one added rank-one call per output block" tax of PR 1605 into a proved lower bound for every realization that cannot use owned temporaries.
3. Per-group transport obstruction, closed form plus exact grid. Net saving k c(1) - (1 + alpha) c(k) is negative for all alpha \ge 1, k = 2..51, a in {target, 1/99, baseline ceiling}; the retained PR 1240 enclosure and an independent binomial-tail enclosure agree on every sign.
4. Complete per-call screens at the acceptance target a* = 21/20 * 1690054109089/1e15 = 0.001774556814543 (the card's +5% ceiling margin):

Realization of the rank-1 payload at block cap 16	Moment at a*	Ceiling vs baseline
Selected word, no reblock (control)	fails	1.00x
k = 362, zero charge	fails	—
k = 363, zero charge	passes (PR 1605 threshold)	—
k = 363, one transport per group	fails	+2.8% with idle credit
k = 363, two transports per group	fails	+0.8% with idle credit
k = 363, one rank-1 save per block	fails	+4.5% with idle credit
Full reblock k = 2412, zero charge	passes	1.47x
Full reblock plus one shared rank-16 transport	fails	—
Full reblock plus 10 rank-1 units (the call budget)	passes	1.47x

The +5% gate is missed by every per-group charged realization screened (ceiling columns grant overhead old idle credit, which is favorable to the construction); it stays open only for call-level invoices inside PR 1605's reproduced budget of 10 added rank-1 units per call at full reblock, under 0.07 units per group. Since the cheapest charged item in the PR 1429 frame contract is one rank-1 call, any surviving realization must be per-group zero-charge.

Verification

Python 3.12, standard library only, exact rational arithmetic, about 2 minutes:

python3.12 -I -B research/block-climb-complete-profile-seed1ed712/verify.py
python3.12 -I -B research/block-climb-complete-profile-seed1ed712/verify.py --write

Expected final line: PASS_BLOCK_CLIMB_COMPLETE_PROFILE controls=26 charged_groups_excluded=100 free_groups_open=50 budget_rank1=10. Controls reproduce PR 1605's k = 362/363 threshold signs and its 10-unit overhead budget, and the PR 1240 baseline ceiling bracket 1690054109089/1e15; the read-only replay recomputes every number from the pinned inputs (no receipts or caches). Registered via scripts/research_package.py --package research/block-climb-complete-profile-seed1ed712 --check-only, which prints PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check).

Scope and limitations

Covered: the PR 1240 cost alphabet (width 73, 896 families per nine stages, centre charge 64, idle mass 1,646, halving guard R = 51) with the PR 1429 transport charge (each rank-r transport charged as a rank-r recursive child) and PR 1605's added-work contract. Not covered: word-level realizability (which of the selected word's actual 2,412 rank-one children share frames), joint replacement of the rank-2/rank-3 frame-step children (PR 1605 open question 2), any changed coverage law, template, port count or centre path, and masked or overcomplete support families, which would change an assumption of the eight-term all-pairs bound instead of this one. This is a scoped exclusion of one realization family at the cost-alphabet level: not a supplier, not a lower bound on all decoders, not a formal proof of the multiplication theorem.

Credit

PR 1605 by Arham Tahir (Arham-Tahir64, with OpenAI Codex assistance per its package) is the direct predecessor: the integer block budgets, the a* target and the k = 363 threshold are its results, reproduced here as controls, and this package prices the frame transports and restoration its open question 1 leaves unpriced. PR 1240 by Doug Colkitt (Mister-Meeseeks) supplies the retained ceiling engine and ideal lever budgets. PR 1429 by Ali Romman (aliromman) supplies the transport-charge contract. The binomial-tail enclosure is reimplemented from PR 1605's verifier as an independent cross-check. Apache-2.0; source pins and frozen inputs are unchanged.

Claim thread: #1166 (comment) (pair-collision scan of #1317 and #1166 found no live claim on this object; the final overlap refresh at 2026-10-16 20:09 UTC reports complete pagination and no competing claim on the charged-climb question, which is not proof of novelty).