P112 root-star first cost drop under further support trims (seed 46)

[1337isnot1337]

CLAIM — structural decoder research, fresh OS-random seed 46 (retained despite matching an earlier draw). Base bce54fb.

Question: on fixed P112, what is the exact first copy-cost drop threshold for each beta-7 root-star atom when arbitrary additional row and/or column support entries may be deleted? The original 240 tagged C/N supports and 57,600 ordered pairs are fixed. This changes the exact-support premise of PR1236/1242; it is the further-trim question explicitly left open by PR1242, not the earlier P104 global-puncture experiment or the co-star claim1267.

Method: rank-factorization separation of joint erasures, plus minimum nonzero column-code weight. Each rank-7 orientation has 127 nonzero codewords, certifying all arbitrary support cuts without enumerating every subset. Compute minimum deletion count and constrained even-entry support loss, then replay exact cost-drop witnesses with an independent dual-coordinate engine. Pilot tags47,0 need22 row deletions for beta7->6; no family conclusion yet.

Expected artifact: an exact first-drop census and scoped erasure proof, with source pins and meaningful controls. No decoder assembly, fully paid supplier budget, physical admission or new kappa is claimed. Finish 2026-10-13 22:40 UTC, expiry 00:07 UTC. Credit Navinor/PR1226, David Arias/PR1236, Hanson Sun/PR1242 and the retained DreamingOfClouds/themilkman216 lineage. Prepared for Cinna Davis with substantial OpenAI Codex assistance.

DONE — PR1323. The complete census gives sharp joint support-deletion thresholds20 (5376 tagged pairs) and22 (2240), and a global constrained even-position loss floor560. The first frame-hit change is exactly4 deletions and keeps copy price7; one extra trim changes neither quantity. The witness loses three frame hits (1080->1077) after four row deletions while remaining cost7.

The erasure-code and rank-factorization lemmas certify arbitrary row/column deletion sets. Primary column-code and independent Weyl-root/row-dual engines agree on all7616 tagged parents (6720 distinct support pairs) and967232 codeword cases, with every attaining cut and complete code digest checked. The final registered Python3.12.15 replay and all seven tests pass, including4096 small binary matrices,524160 nonzero-matrix joint-cut cases, corruption controls and relocated optimized replay. Exact source/archive/Git pins and staged whitespace checks pass.

Reproduce: python3.12 scripts/research_package.py --package research/p112-star-first-cost-drop; core marker PASS_P112_STAR_FIRST_COST_DROP atoms=7616 codewords=967232 copy_drop=20 frame_change=4 even_loss=560.

Credit Navinor/PR1226 for the model and parent census, David Arias/PR1236 for the exact-star assembly floor, Hanson Sun/PR1242 for corroboration and the further-trim question, with retained DreamingOfClouds/PR717 and PR968 lineage. New work by Cinna Davis with substantial OpenAI Codex assistance. The560 positions are support losses, not paid repair counts. No decoder assembly, physical admission or new kappa is claimed. Next: compute frame coverage of the cost6 witnesses and a fully explicit covering/budget relaxation before coefficient synthesis and all realization charges.