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Conditional κ = 0.001729736166387916: six restore carrier/pivot skips (R1_E42, R2_E7) on PR2979's words and literal allocation - #3042

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Conditional κ = 0.001729736166387916: six restore carrier/pivot skips (R1_E42, R2_E7) on PR2979's words and literal allocation#3042
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dcharb78:rank7-restore-carrier-skips

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@dcharb78 dcharb78 commented Oct 16, 2026 •
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Summary

Conditional κ = 0.001729736166387915986812 − 10⁻⁴⁶ − 10⁻⁷⁰ (17297361663879159868119999999999999999999998999999999999999999999999 / 10^70 in research/rank7-restore-carrier-skips/CLAIM.json), +7.0218e-7 above PR #2979's archived author claim (0.0017290339826465, partially_reviewed / merged_for_archive_only) and +5.8519e-7 (6×) above open PR #3015 (0.001729150979425695).

Selected frontier & priority-directive status (docs/research/current-priorities.md, Priority revision 6, Oct 16, 2026):

Credit first:

Overlap, Supersession, and Follow-Up Composition Record

Related PR Author Scope / Range Relationship to PR #3042 (84cc406ad91ec53fff329a11a445d7effe564a77)
#2979 @mdumitrean research/rank7-joint-frames Base package (partially_reviewed, merged_for_archive_only at 30f016dd). Supplies the base words and T74 fill.
#3015 @jeremykothe2 restore[178:180] (R1, skip 63) Overlapped & superseded: restore[178:180] lies inside R1_E42 (restore[162:188]), which retains #3015's carrier-61 skip on 63 as Sub-block C while adding skips on 22, 40, 54 (R1_E42) and 5, 73 (R2_E7).
#3024 / #3030 @petalas T740 cap-attaining allocation check_cap.py in #3042 (using #2979's T74 fill) leaves a 1.7836763625644302e-9 gap below the fractional cap (0.001729737950064279). #3024/#3030 provide the T740 allocation that closes this packing gap.
#3050 @petalas research/rank7-restore-substitution-audit Code-disjoint independent substitution & frame audit of 84cc406ad91ec53fff329a11a445d7effe564a77 + prose description correction.
#3051 / #3054 @jeremydiviney / @petalas T740 + #3042 six-skip composition Compose pinned 84cc406ad91ec53fff329a11a445d7effe564a77 with #3024/#3030's T740 allocation, reaching conditional κ = 0.001729737950064278551242 − 10⁻⁴⁶ − 10⁻⁷⁰.

Literal Mechanism from edits/EDITS.json (R1_E42 and R2_E7)

Correction note vs. initial PR text: The initial PR prose and README.md at 84cc406ad91ec53fff329a11a445d7effe564a77 contained stale gate counts (26 -> 29 and 13 -> 14), misassigned register 22 to R2_E7, and cited draft register numbers (369, 450, 19, which do not appear in edits/EDITS.json). The table and algebraic derivation below are generated directly from the literal research/rank7-restore-carrier-skips/edits/EDITS.json and frame solver output.

Literal Edit Summary (edits/EDITS.json, identical on base and extra)

Edit ID Section & Slice (original WORD-base indices) Gate Count (old -> new) Registers Touched in new (edits/EDITS.json) Registers Skipping Rank-8 Hyperplane Per-Call Profile Delta
R1_E42 restore[162:188] 26 -> 33 [18, 22, 34, 40, 45, 54, 61, 63, 92, 103, 108, 226, 228, 230] 22, 40, 54, 63 skip $E_{42}$ ($7 \to 9$ instead of $7 \to 8 \to 9$) −8 H1 + 4 H2 (−19,984 frame units)
R2_E7 restore[190:203] (restore[197:212] post-R1_E42) 13 -> 15 [5, 59, 73, 96, 363, 395] 5, 73 skip $E_7$ ($7 \to 9$ instead of $7 \to 8 \to 9$) −4 H1 + 2 H2 (−9,992 frame units)
Total 39 -> 48 20 distinct registers 6 registers (22, 40, 54, 63, 5, 73) −12 H1 + 6 H2 per call (−29,976 frame units/call; −7,992 H1 + 3,996 H2 over 666 paid calls)

Exact Algebraic Structure of Each Sub-Block in edits/EDITS.json

Every sub-block uses the same three-stage algebraic pattern—(i) Rank-7 Subspace Fold before the pivot climbs to rank 8, (ii) Rank-8 Pivot-Carrier Swap with a live non-restored target already passing through $E_{42}$ or $E_7$, and (iii) Free FULL(9) Uncomputation among non-restored targets after they reach $\mathbb{F}_2^9$:

  1. R1_E42 Sub-block A — $W_B = V_{25,42,51}(7)$ Twin Coalescence & Self-Pivot Fold (40, 54 via 226 and 18):

    • Old gates (restore[162:170]): 226 updated 103, 72, 70 at $W_B(7)$, swapped with 18 at $E_{42}(8)$, and then 18 updated 40 (40 += 2*18) and 54 (54 -= 18) at $E_{42}(8)$, forcing both 40 and 54 to visit $E_{42}(8)$ before FULL(9).
    • Literal EDITS.json gates:
      • At $W_B(7)$: coalesce twin targets ['add', 40, 54, -2, 1], fold ['add', 226, 40, 1, 4] into pivot 226, and run 226's $W_B(7)$ updates to 103, 72, 70.
      • At $E_{42}(8)$: pivot 226 swaps with rank-8/9 carrier 18 (['add', 18, 226, 1, 1], ['scale', 18, -4, 1], ['add', 18, 226, -1, 1], ['add', 226, 18, 1, 1], ['add', 18, 226, -1, 1]), so 226 is restored to its exact output at $E_{42}(8)$ while 18 carries the $-4 \cdot (\tfrac{1}{4}(v_{40} - 2v_{54}))$ perturbation.
      • At FULL(9) (end of R1_E42): once 18, 40, 54 reach FULL(9), uncompute via ['add', 40, 18, 1, 1], ['scale', 18, 1, 2], ['add', 18, 40, 1, 1], ['add', 40, 18, -1, 1], ['add', 54, 18, -1, 1].
    • Result: 40 ($7 \to 9$) and 54 ($5 \to 7 \to 9$) skip $E_{42}(8)$ (-4 H1 + 2 H2).

  2. R1_E42 Sub-block B — $W_A = V_{30,42,52}(7)$ Twin Fold & Dual-Pivot Handoff (22 via 34, 228, 230 and 92, 108):

    • Old gates (restore[170:178]): 92 updated 22 (22 += 92) at $E_{42}(8)$ (34 was already updated at $W_A(7)$), and 228 swapped with 108 at $E_{42}(8)$ and updated 22 (22 -= 4*228), forcing 22 to visit $E_{42}(8)$.
    • Literal EDITS.json gates:
      • At $W_A(7)$: coalesce ['add', 22, 34, 1, 1], fold ['add', 228, 22, 1, 4] into 228 across 228's $W_A(7)$ updates (45, 40, 54), and fold ['add', 230, 22, -1, 4] into 230 across 230's $V_{45}(7)$ updates (63, 61).
      • At $E_{42}(8)$: 228 swaps with 108 (['add', 228, 108, 1, 1], ['scale', 228, 1, 4], ['add', 108, 228, -4, 1]), and 230 absorbs 18, 45, 61, 92, 103, 108 (['add', 230, 92, 1, 1], etc.) so 228 and 230 end cleanly at $E_{42}(8)$.
      • At FULL(9) (end of R1_E42): uncompute on 22 and 34 via ['add', 34, 92, -1, 1], ['add', 22, 108, -1, 1], ['add', 34, 108, -1, 1].
    • Result: 22 ($7 \to 9$) skips $E_{42}(8)$ (-2 H1 + 1 H2), while 34 preserves its existing $5 \to 7 \to 9$ path.

  3. R1_E42 Sub-block C — $V_{45} = V_{7,42,45}(7)$ Carrier Fold (63 via 61, from PR Conditional κ = 0.001729150973534224: one declared restore edit (R1) on PR2979's words and literal allocation #3015):

    • Literal EDITS.json gates: fold ['add', 61, 63, 2, 1] at $V_{45}(7)$ before ['add', 230, 61, -1, 1] at $E_{42}(8)$, then uncompute at FULL(9) via ['add', 61, 63, 2, 1], ['add', 63, 61, -1, 1], ['add', 61, 63, 2, 1].
    • Result: 63 ($7 \to 9$) skips $E_{42}(8)$ (-2 H1 + 1 H2).

  4. R2_E7 Sub-blocks D1 & D2 — $W_C = V_{7,25,30}(7)$ and $W_D = V_{7,51,52}(7)$ Self-Pivot Folds (73 via 363, 59 and 5 via 395, 96):

    • Old gates (restore[190:203]): 363 swapped with 59 at $E_7(8)$ and updated 73 (73 += 4*363), while 395 swapped with 96 at $E_7(8)$ and updated 5 (5 -= 4*395), forcing 73 and 5 to stop at $E_7(8)$.
    • Literal EDITS.json gates:
      • D1 ($W_C$): fold ['add', 363, 73, 1, 4] at $W_C(7)$, swap 363 with 59 at $E_7(8)$ (['add', 59, 363, -1, 1], ['scale', 59, 4, 1], ['add', 59, 363, -1, 1], ['add', 363, 59, 1, 1], ['add', 59, 363, -1, 1]), and uncompute at FULL(9) via ['add', 73, 59, 1, 1].
      • D2 ($W_D$): fold ['add', 395, 5, -1, 4] at $W_D(7)$, swap 395 with 96 at $E_7(8)$ (['add', 96, 395, -1, 1], ['scale', 96, 4, 1], ['add', 96, 395, -1, 1], ['add', 395, 96, 1, 1], ['add', 96, 395, -1, 1], ['scale', 96, -1, 1]), and uncompute at FULL(9) via ['add', 5, 96, 1, 1].
    • Result: 73 ($5 \to 7 \to 9$) and 5 ($5 \to 7 \to 9$) both skip $E_7(8)$ (-4 H1 + 2 H2).

Structural Completeness & Boundary Analysis (Why 6/6 Restore Skips Are Exhausted)

To answer whether any further carrier skips or restore-enabled rank-7 helper lowerings exist on PR #2979's words, we performed a complete orbit and frame-cone census across all 95 registers touched in restore[0..211] and all 91 restored helpers (word['helper_final']):

  1. Exhaustion of Fixed-Frame Carrier Skips in restore (57 intermediate stops across restore[0..211]):

    • After R1_E42 and R2_E7 eliminate 6 stops (22, 40, 54, 63, 5, 73), the remaining 51 intermediate stops in restore partition into three locked classes:
      • Class I (37 stops, Subspace-Span Lock $\dim(\text{span}(F_{\text{prev}}(r), F_{\text{prev}}(u))) = \dim(F_{\text{mid}})$): Every candidate carrier $u$ present at $F_{\text{mid}}$ has a prior frame $F_{\text{prev}}(u)$ whose span with $F_{\text{prev}}(r)$ already has dimension $\dim(F_{\text{mid}})$ (including the two remaining $7 \to 8 \to 9$ stops outside $E_{42}/E_7$: register 23 at $E_{85}$, where $\text{endB}[23] = V_{42,85,127}(7)$ and $\text{endB}[511] = F(6, [85,145,196,290])$ span $E_{85}(8)$; and register 35 at $E_{120}$, where $\text{endB}[35] = V_{42,82,120}(7)$ and $\text{endB}[499] = F(6, [41,81,120,414])$ span $E_{120}(8)$).
      • Class II (6 stops on 306, 308, 310, 322, 324, 326, $6 \to 8 \to 9$): Routing through their rank-7 partner shifts $6 \to 8 \to 9$ ($c_2 + c_1$) to $6 \to 7 \to 9$ ($c_1 + c_2$), saving 0 on the target while costing the partner -4,996.
      • Class III (8 rank-5 twin pairs (105,408), (414,412), (44,424), (430,428), (34,331), (54,299), (73,363), (5,395), $5 \to 7 \to 9$): Coalescing at $\dim 6$ reshapes the target from 2 H2 to H1 + H3 (+1,881), but forces the restored twin from H2 ($5 \to 7$) to 2 H1 ($5 \to 6 \to 7$, -4,996), a net loss of -3,115.
      • Within restore[136:212], register 61 ([1, 1, 1, 2, 1, 1, 1]) is the unique mandatory carrier delivering the $E_{42}$ center 92 (and 18, 108) into 230 at $E_{42}$; by causal-cone lower-bound propagation, at least one of {61, 63} must touch 230 at $\dim 8$, so 63 skipping $E_{42}$ forces 61 to pass through $E_{42}$.

  2. Why Restored Helpers 586 and 588 Sit at -3,345 and Cannot Be Rescued in restore:

    • Across all 91 restored helpers in FINAL-WORD-base.json.gz, 89 have tight lower bounds $\dim(L_h) = \dim(U_h)$; only 586 ($U_{586} = E_{30}(8), L_{586} = V_{30,76,82}(7)$) and 588 ($U_{588} = E_{30}(8), L_{588} = V_{30,97,127}(7)$) have $\dim(L_h) = 7 < \dim(U_h) = 8$.
    • Lowering 586 (or 588) saves -2,087,955 frame units against the packing break-even threshold BE = 2,091,300 (a net margin of -3,345):
      • Direct chain (-2,099,815, beating BE by +8,515): 586 ([2,3,1,1] -> [2,1,2,1], -2,109,794), 136 at B[784] (+3,102), 187 at B[1230] (+6,877).
      • Indirect chain (+11,860, triggered by restore[85]: 586 -= 438): non-restored helper 438 (+1,881), 140 at B[794] (+3,102), and 144 at B[1214] (+6,877).
    • Commuting restore[85] (586 -= 438) before B[1214] (144 -= 1/2*438) pushes $\text{lo}(438)$ to $L_{586} = V_{30,76,82}(7)$ at B[1214], which contains vector 18 ($18 \cdot 30 = 18 \cdot 76 = 18 \cdot 82 \equiv 0 \pmod 2$) outside target_final[144] = E_{74} ($18 \cdot 74 \equiv 1 \pmod 2$), making 144 frame-infeasible.

  3. Causal Lower-Bound Lock on the 5 Stage-2 Pivots (226, 228, 230, 363, 395):

    • Symbolic replay of A + B shows that entering restore[0], each pivot's initial variable $(h, ())$ resides inside its rank-8 swap partner (18 for 226, 108 for 228, {45, 103} for 230, 59 for 363, 96 for 395), so no restore-only rewrite can lower $\text{lo}(\text{end}[h])$ below $\dim 8$.
    • Permuting the preload center-cancellation hubs A[4031:4043] (Hub 92 for $E_{42}$ or Hub 57 for $E_7) to place $W_B, W_A, W_C$, or $W_D$ ahead of $V_{45}$ lowers one Stage-2 pivot to $\dim 7$, but pushes 3 or 4 of the $V_{45}$ helpers (249, 251, 265, 267) up to $\dim 8$ (a net loss of +4,182,600to+8,417,326` units).

Comparison Against Selected Bound and Archived / Open Predecessors

Package / PR Status in Repo Per-Call Profile Delta vs #2979 Paid Profile Delta (666 calls) Conditional κ Delta vs PR #3042
PR #1094 (certificates/selected-result.json) Selected conditional result — — 0.001672660197076918 +5.7076e-5
PR #2969 (research/rank7-normal45-extractions) merged_for_archive_only — — 0.001724035275795395 +5.7009e-6
PR #2979 (research/rank7-joint-frames) partially_reviewed, merged_for_archive_only (archived author claim) baseline baseline 0.001729033982646500 +7.0218e-7
PR #3015 (research/rank7-restore-skip) Open (superseded on restore[178:180]) −2 H1 + 1 H2 (63) −1,332 H1 + 666 H2 0.001729150979425695 +5.8519e-7 (6× #3015's gain)
PR #3042 (research/rank7-restore-carrier-skips, T74 fill) This PR (84cc406ad91ec53fff329a11a445d7effe564a77) −12 H1 + 6 H2 (22, 40, 54, 63, 5, 73) −7,992 H1 + 3,996 H2 0.001729736166387916 baseline
PR #3051 / #3054 (T740 + #3042 composition) Open follow-ups (#3024/#3030 T740 + #3042) −12 H1 + 6 H2 + T740 cap −7,992 H1 + 3,996 H2 + T740 0.001729737950064279 −1.7837e-9 (closes T74 cap gap)

Verification & Reproduction

  1. Fast registered replay (~5s): 
    python3 -B scripts/research_package.py --package research/rank7-restore-carrier-skips
    Runs verify_manifest.py and run_registered.py (check_cap.py, check_allocation.py, verify.py --fast, and edits/edits_review.py with 11 tamper controls per word).
  2. Full end-to-end native reproduction (~15–20 min): 
    python3 -B research/rank7-restore-carrier-skips/reproduce.py --from-scratch
    Rebuilds Stage 1+2 (WORD-base.json.gz, WORD-extra.json.gz), applies the declared edits R1_E42 and R2_E7 via edits/declared_edits.py, verifies exact matrix equivalence and frame steps (FINAL-WORD-base.json.gz cost 10,707,715,131 vs 10,707,745,107 in 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), compiles the native circuit (663,549 physical instructions, 1,479-bit prefix), runs audit_local.py (all_passed: true) and outer_check.py, and verifies all MANIFEST.json SHA-256 digests.

Merge Danger

  • Door: Two-way (additive research package under research/rank7-restore-carrier-skips/ plus registration in scripts/research_package.py).
  • Blast Radius: Isolated. Does not modify certificates/selected-result.json or any upstream package.

@dcharb78
research/rank7-restore-carrier-skips: six restore carrier/pivot skips…
84cc406 
… on PR2979 words (κ = 0.001729736166387915986812 − 10⁻⁴⁶ − 10⁻⁷⁰)
@jeremydiviney jeremydiviney mentioned this pull request Oct 16, 2026
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Mister-Meeseeks merged commit 5d93a57 into CrocSwap:main Oct 16, 2026
@Gonzalo-Mier Gonzalo-Mier mentioned this pull request Oct 17, 2026
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Mister-Meeseeks pushed a commit that referenced this pull request Oct 17, 2026
@mdumitrean
research: prove the restore-only optimum for PR #3042's six-skip words
c88769c 
Every exact restore-only rewrite of dcharb78's six-skip words (both
providers; A/B, warps, cut, helper ends, start frames and buffers
fixed; any restore gate sequence; all frames re-solved; exactness =
PR2833's auditor) costs at least LB_forced + forced restore-phase
visits = 10,867,765,164 + 337,923 = 10,868,103,087, the certified cost
of the words, which attain it. No new kappa.

The stdlib verifier recomputes the exact frame optimum (numpy-free port
of the certified solver), the free-algebra restore-entry rows and the
forced coordinates, the 88 entry caps and LB_forced, and the sound
classification (dual-vector departure lemma; 42 blocked registers, 9
payer groups). It also checks the PR2979 calibration (29,976 = the six
PR3042 skips), the per-block erratum of PR3042's README, the
auditor-semantics and attainment controls, twelve rejection controls,
and the deletion and tail-exchange censuses.
@rohanarun rohanarun mentioned this pull request Oct 17, 2026
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@dcharb78 @Mister-Meeseeks