research: restore-only optimum for PR #3042's six-skip words (scoped exact negative, no new κ)

[mdumitrean]

Summary

A scoped exact negative on dcharb78's PR #3042 six-skip words (both providers). These are also the words of Nick Petalas's PR #3054. Keep A (with its prelude), B, warps, cut, initial corrections, compensation, start frames, helper ends, removed helpers and buffers fixed. Replace the restore by any scalar gate sequence that PR2833's auditor accepts, and re-solve all frames. Then

cert(W') >= LB_forced + forced restore-phase visits = 10,867,765,164 + 337,923 = 10,868,103,087 = cert(PR3042 words)

PR #3042's restore is optimal in this class. No new κ; no selected-record change.

• LB_forced is the exact frame optimum of the empty-restore word under 88 forced entry caps. Every restored helper must end at its own input, a fixed combination M[c] of the independent restore-entry rows. M[c][r] ≠ 0 therefore forces r into E_c at restore entry.
• Forced visits (337,923):
  • 42 BLOCKED registers contribute Σβ = 292,959. They include the four normal-45 visits 2/70/71/72, whose eight 6-dim entry frames (with the pivots') are pairwise distinct inside V45.
  • 9 disjoint payer groups contribute 9 × 4,996: six zero-sum pairs, 69 and 444 (forced by the departure lemma), and 61's forced V45 → E42 lift for 230.
• Calibration: on PR Conditional κ ≈ 0.0017290339826465: joint rank-7 end frames, five paired extractions and #2957's rewrites on #2942's words (+5.00e-6 over #2969) #2979's words (same A/B) the same bound leaves exactly 29,976. That is dcharb78's six skips (63, 54, 40, 22, 73, 5), and PR Conditional κ = 0.001729736166387916: six restore carrier/pivot skips (R1_E42, R2_E7) on PR2979's words and literal allocation #3042 attains it.
• Erratum to PR Conditional κ = 0.001729736166387916: six restore carrier/pivot skips (R1_E42, R2_E7) on PR2979's words and literal allocation #3042's README (documentation only):
  • R1_E42 makes four skips (63/54/40/22) at −19,984.
  • R2_E7 makes two skips (73/5) at −9,992.
  • The total is −29,976.
  • The README says −14,888 / −14,938 / −29,826 and places 22 in R2_E7. The declared blocks are 26→33 and 13→15 gates, not 26→29 and 13→14.
  • The final words, profile and κ are unaffected.

Proof outline

1. Auditor semantics. We use only the auditor's forward end-of-restore requirements: restored helpers hold their own input rows, no register keeps a restored input, and targets are untouched. Other final rows are free; appending 0 += 1 or 62 += 70 at FULL passes forward and dual at delta 0.
2. Forced coordinates come from independence (rank 632).
3. Forced caps follow from the gate-path argument.
4. Merge: restore-phase climbs collapse into one end climb.
5. Delivery and cleaning lemmas.
6. Departure lemma (dual vectors): if M[O][{r} ∪ {y : L_y ⊆ G}] has full column rank, r cannot leave G.
7. Carriers and lifts.

Lane K's original screen used a partner-only rank test. The package replaces it with the sound departure lemma and enumerates every entry frame and every first frame. On these words the bound is unchanged. The ablation shows that the full-row lemma is load-bearing: restricting its rows lets eight (1,3) splits through and drops the bound by 15,048.

Verification

python3 -B research/rank7-skip-restore-optimum/run_registered.py      # ~70 s, Python standard library
python3 -B scripts/research_package.py --package research/rank7-skip-restore-optimum

verify.py uses only the standard library:

• frames_exact.py, a numpy-free port of the certified frame solver. It reproduces both certified optima, the 743/210/94 frame graph that PR Prove the complete joint A/B/restore frame optimum #3065 reports, and LB_forced.
• Free-algebra restore-entry rows from the vendored PR2833 auditor, the caps, and the classification.

Controls:

• auditor-semantics controls;
• PR Contract PR3042's four-ADD restore block to three #3056's and PR Prove an optimal native tariff for the rank-seven restore block #3060's restore[175:179] blocks, which tie the bound;
• 12 rejection controls: wrong cap, dropped visit, perturbed entry row, mislabeled provider, three cheaper restores that violate exactness, and others.

census.py adds two empirical controls:

• a complete single-gate deletion census: 10 exact deletions, all at delta 0;
• lane K's tail-exchange census: 8,098 candidates per provider, minimum delta 0. The candidate list is re-derived and the records are re-checked on a sample.

Lane K's current bound.py/lbforced.py (the version that takes the supplied M) are pinned in evidence/laneK/ and agree on all eight cross-check cases. Lane K's A/B censuses are cited as local evidence only.

Scope

• Covered: every restore gate sequence at fixed A/B scalar programs, with all frames re-solved, at the certified solver's weights.
• PR Prove the complete restore-frame optimum of the six-skip words #3057 changes only restore frames, and PR Prove the complete joint A/B/restore frame optimum #3065 changes A/B/restore frames jointly. Both keep the scalar order fixed.
• Not covered: A/B changes, helper ends, entrances, warps, buffers, the width inventory, packing or allocation, and native or physical costs.

Open next questions:

• A/B-phase re-synthesis: 161,558 forced-cap units and 192,384 kernel/preload visit units are forced by A/B.
• The fixed target port hyperplanes look like the common wall.

Credit

• dcharb78 (Conditional κ = 0.001729736166387916: six restore carrier/pivot skips (R1_E42, R2_E7) on PR2979's words and literal allocation #3042): the six-skip words and skips.
• Jeremy Kothe (Conditional κ = 0.001729150973534224: one declared restore edit (R1) on PR2979's words and literal allocation #3015): R1 and the declared restore-edit stage.
• Nick Petalas: feat(research): release the T740 six-skip composition #3054, Prove the complete restore-frame optimum of the six-skip words #3057, Prove the complete joint A/B/restore frame optimum #3065, Contract PR3042's four-ADD restore block to three #3056, Prove an optimal native tariff for the rank-seven restore block #3060 and T740.
• Marius-Paul Dumitrean: the Conditional κ ≈ 0.0017290339826465: joint rank-7 end frames, five paired extractions and #2957's rewrites on #2942's words (+5.00e-6 over #2969) #2979 lineage.
• icekylinx: PR2833's auditor.
• Mason Hale (Conditional κ ≈ 0.001721588647: nine additional paired helper extractions #2917) and Jeremy Diviney (Conditional κ ≈ 0.001720398337: two-sided helper extraction and charged T74 packing #2908).

Marius-Paul Dumitrean, with substantial AI assistance (Claude Code); author-checked, not independently reviewed.