Conditional κ = 1.013357935292321·10⁻³: exact value-level re-synthesis of two targets on #401's response-subspace R781 E8 complex program (+4.70·10⁻⁸ over #401)

[chafreaky]

Conditional κ = 1.013357935292321·10⁻³: exact target re-synthesis on #401's response-subspace R781 E8 complex program (+0.0046 % over #401)

κ = 1013357935292321/10¹⁸ ≈ 1.013357935292321e-3, a gain of +4.704783·10⁻⁸ over Dugongue's #401 (101331088745971/10¹⁷).

The complex side binds:

b = 1014385871264415626437/10²⁴ ≈ 1.014385871e-3      (#401: 1.014338728e-3, +4.714·10⁻⁸)

The bit side is #401's, unchanged: c ≈ 1.021539e-3 > b.

Commit: 7883c63ec724999b921c2304b66648cbb9e7a112. The package is research/e8-response-resynth, and the workflow is .github/workflows/e8-response-resynth.yml.

What is new

The base is #401's word. entrance.py is our own re-implementation of #401's 26 entrance subspaces (119 gauged pivots, 240 unit shears) on #398's word. It reproduces #401's frozen candidate hash 3b78069c….

We re-synthesise two targets, y₁₇₇ and y₂₂₀. Each target's B-phase feeds are replaced by another exact rational combination of register values that already exist at frames their registers already hold. These are helper values plus y₁₈₀'s restored prefix, read through a #398-style setup/restore. The new frames form a more unbalanced nested chain:

y177, y220:  climbs (3, 5) -> (6, 2)      local histogram {2:+2, 3:-2, 5:-2, 6:+2}

Original feed gates become #398's source-touching empty markers, so no helper walk, bank or residual changes.

The full raw rational operator is identical to #401's on every register and all 1,021 input columns, including every helper-dirt column. #401's K, its 119 zero gauged columns and the retained totals are therefore unchanged. gx.check1 (E0–E7, including the gauged-dirt rule), the gxcore mirror and the source-span rule pass.

Discovery is an exact value-level search (search.py):

• It runs over all 120 targets and enumerates nested frame chains whose climb partition beats the current one under Σ r ln r.
• It runs an exact breakpoint scan over register windows.
• It solves over ℚ with the x-part, the y→y identity and the 119 gauged columns fixed.

It finds 27 improvable targets:

• 25 are Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398 donors read by an existing restore; their improvement would need their own destination as donor, which is circular.
• 151/157 need gauged dirt on research: response-subspace gauges and two-sided normalization, kappa = 1.013310887459710e-3 #401's word.
• That leaves exactly 177 and 220. --regen reproduces them.

Bracket. We emit with #390's dirty-preconditioned bracket, as in our #399. On #401's unmodified word our emission reproduces #401's 24-digit coarse root and κ exactly, which is the control. Since our word has #401's exact raw operator, #401's own bracket (K, main, complete cleanup, two-sided R₀⁻¹/L⁻¹) also applies unchanged.

Negative result recorded in the package. #394/#399 padding fusion does not compose with #401. The only same-frame zero-row sinks (301/324/387) each need exactly a quarter of a #401-gauged register (1018/1019/1020), and the solution is unique. gxcore's sixths rule (E5) rejects the 1/4 gauged dirt that leaves in the sink. Fusion would otherwise have added about +1.7·10⁻⁷.

Verify

bash research/e8-response-resynth/verify.sh --quick /tmp/e8r-q
bash research/e8-response-resynth/verify.sh /tmp/e8r-full      # adds all regenerations

This needs Python 3.11+ and the standard library only. The script runs:

1. the manifest check and git-blob pins, including research: response-subspace gauges and two-sided normalization, kappa = 1.013310887459710e-3 #401's selection.json, BANK-PATTERNS.json, RESULT.json and LICENSE;
2. our Conditional κ = 1.002661106042177·10⁻³: our #394 padding fusion on #398's R781 signed-prefix E8 complex program (+0.0166 % over #398) #399 pipeline as a control (E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390 bracket regression, Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398 and Conditional κ = 1.002661106042177·10⁻³: our #394 padding fusion on #398's R781 signed-prefix E8 complex program (+0.0166 % over #398) #399 κ);
3. the research: response-subspace gauges and two-sided normalization, kappa = 1.013310887459710e-3 #401 word check, M₄₀₁ = M₃₉₈ · diag(I, I, S) on all columns;
4. the sink negative check;
5. the re-synthesis checks: operator identity, gx/gxcore, walks, histogram and mutation controls;
6. the local-program replay on all 1,021 live columns in both orientations, for research: response-subspace gauges and two-sided normalization, kappa = 1.013310887459710e-3 #401's word and for ours, with controls;
7. the re-implemented research: response-subspace gauges and two-sided normalization, kappa = 1.013310887459710e-3 #401 bank patterns (748 banks per stage, 482 rank-1 + 1 rank-10 paddings);
8. the price: research: response-subspace gauges and two-sided normalization, kappa = 1.013310887459710e-3 #401's b and κ reproduced, then ours with both PR Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315 engines, the 10⁻¹⁶ fallback and six bootstrap levels with 47 strict constraints;
9. cxcheck.

Both modes pass from a fresh clone.

Admitted conditions

• Re-opened:
  • Gaussian-phase compilation of the changed target edges (0→892, 892→179 and 892→222);
  • the scalar invoice delta of 10 helper→target deliveries, 2 restores and 2 setups per invocation, against research: response-subspace gauges and two-sided normalization, kappa = 1.013310887459710e-3 #401's reserve (row 15,445 < 20,161).
• Inherited and not re-run:
  • research: response-subspace gauges and two-sided normalization, kappa = 1.013310887459710e-3 #401's entrance phases, two-sided adapters, routes, owner injectivity, invoice and bit component (through its pinned RESULT.json, as research: response-subspace gauges and two-sided normalization, kappa = 1.013310887459710e-3 #401 does through receipts);
  • Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398/Conditional κ=0.0011208649522582786: split centres with complete signed admission #376's reconstruction, Conditional kappa 0.001000220224856110: six fixed levels on #390 #392's bootstrap, and κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377's and κ = 8.01083628465007e-4: #346's w3 bit word with Sussman's Lean-checked E8 unit as the complex supplier (+0.96 % over #350) #352's interfaces.

This is a conditional finite construction, not a formal verification.

Credits

• Dugongue: research: response-subspace gauges and two-sided normalization, kappa = 1.013310887459710e-3 #401 (response subspaces, two-sided normalisation, mixed-residual banks, the base here), Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398 (signed target prefixes; the marker and setup/restore form we reuse) and E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390 (target prefixes, the dirty-preconditioned bracket).
• rohanarun: Conditional kappa 0.001002683712533373: R781 heterogeneous banks and padding fusion #400 (R781 heterogeneous banks with fusion, re-implemented in our exploration), Conditional kappa 0.001001098432210898: signed-prefix donor gauges #396 (the entrance response-kernel route research: response-subspace gauges and two-sided normalization, kappa = 1.013310887459710e-3 #401 extends), Conditional kappa 0.001001256225581264: signed prefixes and padding fusion #395, and Conditional kappa 0.000998634658090835: finite bootstrap and exact coarse precision #387/Conditional kappa 0.001000220224856110: six fixed levels on #390 #392 (bootstrap pricing).
• applecrusher (Conditional κ=0.0011208649522582786: split centres with complete signed admission #376), jacklightChen (κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377), κ = 8.01083628465007e-4: #346's w3 bit word with Sussman's Lean-checked E8 unit as the complex supplier (+0.96 % over #350) #352, Jacob Sussman (E8 unit, gx/gxcore), DaysSky and utcorvusvolat-dotcom (bit lineage), DreamingOfClouds (Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315).
• Ours: Conditional κ = 9.5464544449649·10⁻⁴: Sussman's E8 complex unit re-timed by far piece relocation, in #375's unsplit B₃ (+0.042 % over #375) #378, Conditional κ = 1.000385913854264·10⁻³: bank padding fused into three new free sink helpers on #390's E8 complex program (+0.0166 % over #390) #394/Conditional κ = 1.002661106042177·10⁻³: our #394 padding fusion on #398's R781 signed-prefix E8 complex program (+0.0166 % over #398) #399, and here the target re-synthesis search, the re-implementation of research: response-subspace gauges and two-sided normalization, kappa = 1.013310887459710e-3 #401's word and the checks.

Prepared by Chafik Boukhalfa (chafreaky) with substantial Anthropic Claude assistance. Apache-2.0.

Repository CI. This branch includes PR #318's unchanged three-file joint-dual source-pin repair (certificates/joint-dual-kappa.json, certificates/joint-dual-validation.json, research/joint-dual/SOURCE.json), as #366 and #368 do, so the repository-wide verify (joint) job passes. No mathematics is changed by it.

[Mister-Meeseeks]

Thank you @chafreaky. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Value-level resynthesis of two target prefixes.

Review and limits: Exact local scalar/checker equality and price analysis; source-listed phase, owner and finite-invoice obligations remain explicit.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: ac91a9ce6780bdea6b1753cee90fcf437ca4c783. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.