Helper minimality (no κ claim): the shipped helper layers of #656 / #799 are minimal for their op multiset

[DreamingOfClouds]

No κ claim and no word. Knowledge package research/helper-minimality/: frame-side lower bounds for the helper layers of the frontier words, each with its scope and hypotheses, plus a single verify.sh that rechecks them. It continues #717 (rows F3-F5 there are the value-side records) and the merged #595.

What is shown (22 rows: 13 CHECKED, 4 ARGUED, 5 RECORDED)

Layer	Result	Scope
S1 scratch	In each of the 11 eight-line blocks of #656, 15 registers are infeasible: whole-bound registers ≥ plane stoppers = 8, perfect matching forced, every whole-bound register (the "never read" ones too) is indispensable; the 38 "twin" couriers are whole-bound registers of this kind (38/38), none equal to its partner (value-forced, B176). CP-SAT (F₃ relaxation, dyadic coefficients, 4-level lemma argued in the card; any horizon) confirms for block (4,5,6,7): dropping 433 is INFEASIBLE.	given the shipped end-of-A frames as bounds (own 24 registers), FOM; true-cap case open
S3/S4 copies	No copy pair can be merged by any reordering: 268/268 FAIL on #656 and on #799 (Lawler 1973 backward rule, checked against exhaustive search); every copy already has a two-visit conflict.	FOM; shipped-order partner frames g (lower bound for arbitrary reorderings only where value-forced, B172 caveat); rerun skipping the 1921 / 2317 value-zero adds is still 268/268 FAIL on #656 / #799
Bridges, holders	The 12 bridge registers 716..727: 12/12 FAIL; holder/entering copies 450..457 and 484..569: 94/94 FAIL.	#799, FOM
Cut	Under H-CUT (justified by the spec lines 03-motifs.tex:78-83 and the emitter line hpe_emit_local.py:94, which are checked at the quoted lines), the cut needs ≥ μ ≥ ω = 8 centres; both shipped words attain 8 = rank(F−F₀).	#656, #799
Pre-read	Minimum touched pre-read targets on #799 within the 271-move family is 66 (CP-SAT OPTIMAL, bound 66) against threshold 62.	pairs / recorded triples only
Model	The oracle frames equal the AL frames on all 5726 A/B ops of the #656 word (B128).	#656 chain word

The S1 bound (given the shipped end-of-A frames) and the Lawler bounds (under the g hypothesis in the table) are support/frame bounds, so they hold for cancelling programs of the same op multiset too (contrast Boyar–Find 2013); undo/redo ADDs change the multiset and are open.

Open gaps, stated plainly

1. S1 blocks (0,1,2,3), (0,1,4,5), (0,1,6,7): the Hall bound is slack (21/21 matching with line registers alone); minimality of their 48 scratch registers is unproven, not refuted. B177 is running; no verdict yet.
2. Multiset-changing merges (undo/redo ADDs, holder merges) are outside every Lawler result.
3. S1 minimality is given the shipped end-of-A frames as bounds: the true-cap case (later readers' true caps replace them, as in Format limits part 2 (no κ claim): erratum to #595, exact invoice prices, frontier-word censuses, decoder floors, spec requirements, centre and fragmentation walls #717) is open, as is an interface change; H-CUT is not enforced by AL (a centre-free toy is accepted by AL and flagged by the H-CUT check).
4. Decoder n ≥ 9 (Format limits part 2 (no κ claim): erratum to #595, exact invoice prices, frontier-word censuses, decoder floors, spec requirements, centre and fragmentation walls #717 D1) is a separate question and not covered.
5. Pre-read is checked to singles, all pairs and recorded triple probes, not arbitrary k-sets.
6. The Lawler layers and the S1 lemma have no independent second-reader audit.

Verification

bash research/helper-minimality/verify.sh /fresh/dir from a fresh git archive: 53 PASS lines (2 in step 0), 0 FAIL, exit 0, about 52 s. Step 0 checks every file against MANIFEST.sha256 and every vendored input against SOURCES.json (git blob ids of the #799 / #635 / main files, hashes published by #656 / #799 for the rebuilt words) and stops on any mismatch (tamper test: a changed byte in a word or in a script exits 1 at step 0). HM_FULL=1 also regenerates the 36,501-pair pre-read table. No absolute paths, no personal names, no κ claim.

Credit

Builds on #717 and #595 (DreamingOfClouds); the words and verifier are #656, #659, #664, #799, #741, #685 (jeremykothe2, chafreaky), #645 (icekylinx), #622/#623/#635 (jeremykothe2); related: #698 (thakrarsagar; fixed gate order, exact end-of-B frames, against the order-free bounds here), #719 (rank-two obstruction for the 78 CK cleanup pairs), #705 (littleBro; different model and argument), #687 (cbethin), #683 (Mister-Meeseeks), #869/#870 (jeremykothe2). See NOTICE.md.

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code; Apache-2.0.

🤖 Generated with Claude Code

[Mister-Meeseeks]

100-PR batch review at 75e7699a691f321861ab8fe303f8bd8e6a6fc393.

This remains open after scoped triage. Next review step: Review solver-status and recorded-only minimality rows separately from exact certificates; execute verify.sh with its declared dependencies and preserve the proof scope.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

Batch review, attribution and evidence. Thank you for the contribution.