Conditional κ = 1.016526770666368·10⁻³: response-gauge re-pivoting (checker-clean) and padding fusion on #404's shared-donor R781 E8 complex program (+5.71·10⁻⁸ over #406)

[chafreaky]

Change and mathematical scope

Adds research/e8-repivot-fusion, a conditional finite construction with

κ = 7941615395831/7812500000000000 = 1016526770666368/10¹⁸ ≈ 1.016526770666368 × 10⁻³ (complex-bound).

That is +1.044502·10⁻⁷ over Dugongue's #404 (1.016422320428301e-3) and +5.71·10⁻⁸ over rohanarun's #406 (1.016469657190768e-3).

	b (complex coarse)	κ
#404 (control, reproduced from our emission of its word)	254364121477661453089/2.5·10²³	1016422320428301/10¹⁸
ours	508780574407464422817/5·10²³ ≈ 1.017561149e-3	1016526770666368/10¹⁸

The bit side is #404's pinned coarse bound, c = 255384852701747/2.5·10¹⁷ ≈ 1.021539e-3 > b, so the complex side binds.

Profile delta vs #404 is {1:−216, 3:+72, 4:+72, 5:−144, 6:+72}. Live is 14,322, m = 73, the deficit is 2,496, and there are 738 banks per stage.

Base. We re-implement #404's word from its data: its 194 shared-donor entrance gauges on #398's signed781 word, its 124/125 recovery and #376's frame edit. Both of #404's frozen hashes are reproduced (65b0f9dd…, d18ccdc5…). Our emission of that word under #390's dirty-preconditioned bracket reproduces #404's 24-digit b and κ exactly. None of #404's scripts were run.

New: gauge re-pivoting. A response-gauge plan is a span of relations e_p + Σ c_{p,d} e_d. Choosing a different pivot column for the same span zeroes the same response space, so M = M₃₉₈ · diag(I, I, S′) still holds with all 194 pivot columns zero. The rank is checked exactly, and the operator is checked on all 1,021 columns. Pivot dimensions, the residual inventory and #404's bank patterns are unchanged.

The pivot choice is what fixes denominators and which register carries which gauged dirt. We swap four pivots:

• 555 → 748 (plan 28). research: shared-donor chains and overlapping target recovery, conditional kappa = 1.016422320428301e-3 #404's word fails gxcore's sixths rule (E5) at register 947, because 555 has five half-coefficient donors. After the swap every coefficient is an integer.
• 1019 → 958 and 1020 → 1001 (plans 0 and 5). 1019 and 1020 become donors with coefficients −2, −2.
• 585 → 636 (plan 27). The inverse column of sink 387 stops meeting a gauged coordinate.

We do not use #404's 124/125 recovery, because it fails gx E4 at target 125. That costs −5.69·10⁻⁸, and it is included in the totals above.

Padding fusion (our #394/#399), now composable on the response-gauge line.

• The only same-frame zero-row sinks are 301, 324 and 387. Each needs exactly −1/4 of one helper (1019, 1018 and 1020), and the coefficient is unique: there is no solution without that helper.
• On research: response-subspace gauges and two-sided normalization, kappa = 1.013310887459710e-3 #401's and research: shared-donor chains and overlapping target recovery, conditional kappa = 1.016422320428301e-3 #404's gauges, 1019 and 1020 are pivots, so the quarter breaks sixths. That was the blocker we recorded in 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) #402.
• After re-pivoting, a quarter of their even-coefficient dirt stays in sixths, so 301 and 387 are fused.
• 324 is not fused. Its 1018 is the only donor of pivot 941, which leaves nothing to re-pivot. Ungauging 941 needs a 739th bank, and an advanced-donor uncompute prices at −3.1·10⁻⁷.
• We also check the premise of research: shared-donor chains and overlapping target recovery, conditional kappa = 1.016422320428301e-3 #404's research note on sink isolation: for both fused sinks, a_G = b_G = 0 on all 194 gauges.

Target re-synthesis (our #402). y₁₇₇ and y₂₂₀ climb (6, 2) instead of (3, 5). The full raw operator is identical.

Banks. The 16 fused occurrences own slot 0 of 16 distinct rank-1-padded banks, with the padding on coordinate 72.

Retained hypotheses.

• Inherited from research: shared-donor chains and overlapping target recovery, conditional kappa = 1.016422320428301e-3 #404/research: response-subspace gauges and two-sided normalization, kappa = 1.013310887459710e-3 #401, not re-run: two-sided normalisation and entrance-chart phases (same 37 entrance spaces), bank routes and owner injections, the finite invoice, and the bit component via receipts.
• Re-opened:
  • phases of the four re-pivoted entrance tokens;
  • owner placement of the 16 fused slots and the fused final climbs;
  • the 177/220 target edges;
  • the scalar invoice.
• Inherited from Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398/Conditional κ=0.0011208649522582786: split centres with complete signed admission #376/Conditional kappa 0.001000220224856110: six fixed levels on #390 #392/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 as in our earlier packages.

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

Attribution.

• Dugongue: research: shared-donor chains and overlapping target recovery, conditional kappa = 1.016422320428301e-3 #404 (shared-donor gauges, the base), research: response-subspace gauges and two-sided normalization, kappa = 1.013310887459710e-3 #401, Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398, E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390.
• rohanarun: Conditional kappa 0.001016469657190768: shared-donor chains with target re-synthesis #406, Conditional kappa 0.001013417593791468: target re-synthesis with 64-replica banks #405, Conditional kappa 0.001002683712533373: R781 heterogeneous banks and padding fusion #400, Conditional kappa 0.001001098432210898: signed-prefix donor gauges #396, Conditional kappa 0.001001256225581264: signed prefixes and padding fusion #395, Conditional kappa 0.000998634658090835: finite bootstrap and exact coarse precision #387/Conditional kappa 0.001000220224856110: six fixed levels on #390 #392.
• applecrusher: Conditional κ=0.0011208649522582786: split centres with complete signed admission #376 (helper substitutions, frame edit 203 → 942).
• 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: the local invocation and emitter.
• Jacob Sussman: E8 unit, gx/gxcore.
• DaysSky (Complex supplier b = 9.32783e-4: E8 in a seven-stage bridged word (+6.45 % over #352; no new κ, raises the complex cap) #363, κ = 9.3191395651209e-4: #370's E8 bit word and Sussman's E8 complex unit in the seven-stage bridged word (+6.45 % over #370) #373, κ = 9.5424356568133e-4: seven-stage E8 suppliers with the complex idle climbs at halving degree two (+2.40 % over #373) #375, κ = 9.95656966441037e-4: #370's E8 bit word in the nine-stage bridged word with #377's nine-stage complex E8 (above 2^-10; +2.94 % over #380) #381) and utcorvusvolat-dotcom (κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370): the 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 engines.
• 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, 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) #402, and here the re-pivoting.

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

The branch also carries 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) for repository CI. No mathematics changes.

Validation

Commit 2eca2715826bb899a17d013552d5cf35d42567d1 is on top of our #402 branch (ac91a9c).

From a fresh clone:

bash research/e8-repivot-fusion/verify.sh --quick /tmp/e8p-q   # EXIT 0
bash research/e8-repivot-fusion/verify.sh /tmp/e8p-full        # EXIT 0 (all regenerations)

This needs Python 3.11+ and the standard library only. The verify covers:

• the manifest, which matches git ls-files, and the git-blob pins, including research: shared-donor chains and overlapping target recovery, conditional kappa = 1.016422320428301e-3 #404's selection194.json, bank-patterns.json and RESULT.json;
• 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 and 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) #402 pipelines as controls;
• research: shared-donor chains and overlapping target recovery, conditional kappa = 1.016422320428301e-3 #404's hashes, b and κ as controls;
• the recorded gx/gxcore rejections of research: shared-donor chains and overlapping target recovery, conditional kappa = 1.016422320428301e-3 #404's word;
• re-pivot span and operator identities;
• gx.check1 + gxcore on the base, the sinked word and ours;
• the 324 negative control;
• regeneration: the free-sink scan finds exactly 301/324/387, the isolation screen holds, and re-synthesis discovery finds exactly 177/220;
• both-orientation replay on all 1,021 live columns with mutation controls;
• bank placement and price;
• Sussman's checkers with the flipped-sign and source-span controls on ours.

The CI workflow is .github/workflows/e8-repivot-fusion.yml (Python 3.12 and 3.14).

[Mister-Meeseeks]

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

Contribution: Re-pivoting and padding fusion compatible with gx checks.

Review and limits: Provides a checker-compatible alternative with exact controls. Local acceptance does not discharge newly changed phase, owner, invoice or sink-isolation obligations.

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: 2eca2715826bb899a17d013552d5cf35d42567d1. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.