Record that the selected k8 word emits no paid child above rank 8.

[unpaidmoon]

Change and mathematical scope

Scoped rank bound. No new κ. The selected conditional result stays 0.001672660197076918.

On the selected k8 word (c2b294300760e1fd6eeab8c22c4d13e4633dca40e52cac8da0d68163a76e3ca8), every legal monotone frame assignment emits paid children of rank at most 8. Graph climbs have rank at most 5. The eight centre descents are rank 8 and attain the maximum. A rank-44 paid child, the threshold named by the rank-cap handoff, is not a child of this word.

This is a finite certificate on one fixed scalar word. It is not a supplier, a moment improvement, or a block-decoder construction.

Task and overlap

Base: bb4bccbf42844d2801487450727c2f63e10d1d0f (canonical CrocSwap/integer-mult-bounds main). Priority revision 6. Freshness check at submission: that commit was still main.

Task: complex-rank-shape-decoder, scope refined. The card's second method asks for the best effective child rank on one fixed support alphabet, counting copies, frames and restoration. This package takes that bound for the selected word's native F2^9 frame alphabet and does not synthesize a block decoder.

Item taken: item 1 below.

Ordered eligible list. Concrete #1166 next tasks still open after dropping work performed by a later DONE, an open PR, or this account's round through #4157. Order is the comment order used by the 03:24–03:25 UTC selection notes that still named the child-rank bound first. Comment 19021170 question 1, the (7,18)/(7,20) functional census, is older and still unanswered; #4148 only proves codimension-one soundness for n ∈ {3,4} and L = 2n, and those notes had already left the census off this list.

1. Largest physical child rank on one support alphabet (comment 19021222). This package.
2. Downstream consumer of (w, q, v_N) (comment 19030045). Distinct from the Price one (N-P) tensor M consumer at 199680 unit additions. #4143 (N-P)⊗M consumer.
3. Adjacent six-tape event after the Research: first F0 caller six-tape admission audit #3932 F0 bind (comment 19030049). Distinct from Ledger intermediate F0 magnitudes on the L359 dirty ray. #4114.
4. Caller that fuses the P104 source-normalization factor with the full-output interface (comment 19035513). Certify the screened P104 source-chart shortcut misses the debt-free … #4153 certifies the screened shortcut and does not build this caller.

Nearest overlap:

• PR Research: exact complete frame ceiling for the selected k8 word #1671 (research/selected-k8-frame-ceiling) proves the moment stays under the packing ceiling for every legal frame assignment, with symbolic charges through rank 9. It does not state the edge-interval climb cap of 5.
• PR Exact rank-cap ceiling curve for the selected complex local word (no new kappa) #3701 asks for this rank bound and leaves it blocked on a missing inventory.
• PR Block-climb complete profile: exact obstruction for charged block-climb realization - no kappa #3039, PR Correct the mixed-profile inference in the block-climb screen #3066 and research/rank-shape-coframe-bill-seed850bc3 price a different child identity, bundling rank-1 payloads. That identity is not certified here.

Overlap refresh at 2026-10-20 03:52 UTC covered 850 open PR records and 2678 discussion records. No open PR states this climb cap.

Validation

Python 3.12, standard library only. CPython 3.12.3.

python3.12 -I -B research/selected-k8-child-rank-bound/verify.py
python3.12 -B scripts/research_package.py --package research/selected-k8-child-rank-bound

Expected: PASS_SELECTED_K8_CHILD_RANK_BOUND max_paid_rank=8 max_climb=5 centre=8x8 rank44=0 free_gates=1741 increasable_edges=3222 no_new_kappa and PASS_REGISTERED_RESEARCH_PACKAGE.

Local evidence: both commands passed. check_research_commit.py --check-source-history passed at 324d14bce8cfc7cda174e089029be06c2bcea9dd. The section walk and the edge-interval census agree on the published profile. Interval sizes 2/5/16 and the 1,741 free gates match the published census. Controls reject a frame vector outside F2^9 and a false cap of 4. Dropping cone ceilings leaves the climb cap at 5.

Not run: Lean, Docker, a selected cold replay, and the moment duals of PR #1671.

Credit

Word, frame contract and interval census: PR #1671 (Gareth Davidson, using navinor's frame method and the gauge44 auditor). The rank-44 comparison is the question in Discussion #1166 comment 19021222 / PR #3701. Block-reblocking tariff: PR #3039, PR #3066, and research/rank-shape-coframe-bill-seed850bc3.

Prepared by unpaidmoon with substantial Cursor Grok assistance. No independent human or formal-kernel review is claimed. Apache-2.0.

Open next questions

• Can one co-frame block on this same word have rank above 8? The frames are now available; owned temporary lifetimes are still absent from the archive.
• Does one legal scalar edit open a register edge whose interval jump exceeds 5?