[unpaidmoon]

Change and mathematical scope

Scoped negative result. No new κ. The selected conditional κ stays 0.001672660197076918.

On the selected complex local word, every paid child is one strict nested climb in F_2^9 or one fixed centre descent. In the unrestricted-terminal event graph:

• all 10,222 positive-multiplicity edges have climb rank at most 6;
• the eight centre descents are rank 8;
• the 106 cone-restoration edges have multiplicity 0 and create no child.

So the largest paid child rank is 8. Ideal rank-8 and rank-9 delivery of the 5,586-unit local mass still have packing ceilings below a = 1/99. The rank-44 greedy histogram {44:126, 42:1} clears that target and is not realizable in this alphabet.

This is not a decoder, a supplier, or a change to the selected record.

Task and overlap

Base commit: bb4bccbf42844d2801487450727c2f63e10d1d0f (canonical CrocSwap/integer-mult-bounds main). This fork's main has diverged, so the GitHub file list against fork main is not the scientific diff. The package commit adds only research/f2-nine-paid-child-rank/.

Task: complex-rank-shape-decoder, scope refined. Method taken: bound the best effective child rank for one fixed support alphabet, including copies, frames and restoration. This is item 1 of the chronological open CrocSwap#1166 list below: discussion comment 19021222. The physical rank question was marked blocked there; the retained frame model is the missing inventory.

Nearest overlap:

• Open PR Exact rank-cap ceiling curve for the selected complex local word (no new kappa) CrocSwap/integer-mult-bounds#3701 prices the rank-cap relaxation and shows rank ≥ 44 is necessary for a ≥ 1/99. It does not bound realizable rank. This package uses that threshold and supplies the cap.
• research/selected-k8-unrestricted-ends supplies the word, the climb rule and the unrestricted graph. Its LP and duals were not rerun.
• research/complex-local-word-ceiling supplies the ceiling arithmetic. The selected bracket is re-derived and matches.

Validation

python3.12 -B scripts/research_package.py --package research/f2-nine-paid-child-rank

CPython 3.12.3, standard library, about 8 seconds. Marker:

PASS_F2_NINE_PAID_CHILD_RANK H=9 graph_climb_cap=6 centre=8x8 paid_rank_cap=8 positive_edges=10222 cone_edges=106 domain_frames=39711 selected_hi=0.0016900541090890 cap8_hi=0.0027395043845670 cap9_hi=0.0028862793175050 cap43_hi=0.0099997320169050 cap44_lo=0.0103695753555750 target=1/99 cleared_by_physical_cap=no no_new_kappa

scripts/check_research_commit.py --package research/f2-nine-paid-child-rank --check-source-history passed at facf0a631f52e60d443d797119ab2d1b5be33fc0.

Failure controls: idle rank 52 is rejected by the halving guard; a rank-10 histogram is priced by the ceiling model and still misses 1/99, but rank 10 is outside F_2^9. Centre ranks are recomputed from the word without the event graph and agree with the model.

Not run: selected-k8 LP/dual replay, selected cold replay, Lean, Docker.

Credit

Gareth Davidson (frame contract and unrestricted-end model, PR CrocSwap#1995 / PR CrocSwap#2013 lineage). Douglas Colkitt with Claude (Anthropic) assistance (research/complex-local-word-ceiling). Nick Petalas, PR CrocSwap#3701 and discussion comment 19021222, for the rank-44 threshold and the open physical question. unpaidmoon, with substantial Cursor Grok assistance. Apache-2.0. No independent human or formal-kernel review.

Open next questions

1. Can a bill that charges one child for the sum of several F_2^9 climbs, or a larger alphabet, reach rank 44 with copies and restoration priced? Screen that histogram with ceiling.ceiling before any search. Blocked until the contract exists.
2. Does the rank-8 cap survive a scalar rewrite? Terminal changes inside the fixed word are already inside this model. Blocked on a rewritten scalar program.

Workflow feedback (optional)

CrocSwap#1200