Conditional κ = 0.001274415506145115: complex response kernel + rank151/mixed bit word on #451 (+0.670%)

[rohanarun]

Conditional κ = 254883101229023 / (2·10¹⁷) = 0.001274415506145115: +0.670% over #451 (0.001265931727546219) and +0.622% over our #459. Bit binds at c ≈ 1.276041713·10⁻³ (#459's word, unchanged). The complex endpoint rises to b ≈ 1.293526619·10⁻³ (#451: 1.268142297·10⁻³).

New: a shared-donor response kernel on the complex supplier

#433's split-centre complex word cancels each helper's dirt with frame-zero target additions (compensation, 12,760 rational adds), just as the bit word does with its D gates. The #417 kernel therefore carries over with rational coefficients:

• an entry is (e, p, d, λ) with response(p) = λ·response(d) exactly over ℚ, and e in the first frames of p and d;
• p starts at ⟨e⟩ and leaves the compensation;
• d += −λp at ⟨e⟩ before A, and d += λp at FULL after cleanup.

The word has exactly 56 proportional helper pairs, and every pair shares an entrance vector. That gives 56 pivots, 56 donors and entrance rank 56, so 56 complex residuals shrink from width 9 to width 8. Width-8 residuals pack exactly into the rank-8k idle children, so the remainder children drop:

• complex profile {1:174864, 2:78848, 3:8680, 4:19712, 5:38120, 7:2944, 8:8000, 30:8, 64:2160};
• deficit 3264, all 49,280 occurrences still hosted, no banks.

Checks, all regenerated by verify.py

• Kernel program: exact rational replay of the complete kernel program on all 800 columns, plus 4 mutation controls (code/ckernel.py).
• Split centres and source-basis reuse for conditional κ = 0.001120864952258278 #433 emitter, unchanged: run on the new canonical word, with its literal column and paid-profile checks.
• Clifford phase recipes: exact recipes for all 2,925 nested edges, regenerated with Split centres and source-basis reuse for conditional κ = 0.001120864952258278 #433's unchanged complex_phases_full.py (≈3.5 min).
• Independent local replay of both orientations (code/complex_kreplay.py):
  • common-frame, nesting and centre-lifetime checks, and the 192-bit prefix;
  • 6 mutation controls, including pivot entry at frame 0 and an omitted kernel exit correction.
• Residual routes: one exact route per helper start frame (8 frames). Each lowers start → FULL by lower_child as monomial·F_r·monomial with r = 9 − e, re-verified on both tableau and phase.
• Hosting and audit:
  • re-solved Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442 hosting;
  • the hosted eleven-stage template;
  • the independent complex audit: 8,520 phase equalities and 20 mutations, including a wrong width-8 residual width;
  • both moment engines, invoices and the 47-inequality outer assembly.
• Bit side: identical to Conditional κ = 0.001266536148630309: rank151 kernel + target-prefix mixing on #451, complex-bound (+0.0477%) #459, whose full verify passed.

Status:

• Passed locally: every new and changed step, reusing the identical Conditional κ = 0.001263365464781156: eleven-stage R740 hosting with complete row bound #445 parent output.
• Running: a clean full verify.py. The result will be posted.
• CI: not claimed.

Review point: as in the bit kernel lineage, a helper may start at a nonzero frame. Its hosted residual is the rank-(9−e) child from its start frame to FULL, carried by the stored exact Clifford route. PROOF.md ("Complex response kernel") gives the scalar identity and the route argument.

Bit side: #457's rank151 kernel plus #450's target-prefix mixing (DreamingOfClouds), as in #459. Inherited interfaces are unchanged, including #442 hosting. Cross-copy hosting is not used. No unconditional theorem is claimed.

python -m pip install -r research/e8-r740-replica32/parent436/requirements.txt
python -B research/r740-dual-kernel/verify.py /absolute/new/output

Credits:

• icekylinx: Split centres and source-basis reuse for conditional κ = 0.001120864952258278 #433 word, emitter, phases and replay.
• chafreaky: E8 bit supplier c ≈ 1.044064·10⁻³ in the nine-stage word (+2.2 % bit cap; no new κ, complex-bound) #417 kernel lineage, Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443–Conditional κ = 0.001263365464781156: eleven-stage R740 hosting with complete row bound #445 and the Conditional κ = 0.001265931727546219: dual eleven-stage whole-residual hosting #451 package.
• DreamingOfClouds: Conditional κ = 0.001347744547714835: eleven-stage cross-copy hosting with target mixing (+1.278% over #448) #450 mixing.
• DaysSky: Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442.
• eumemic: Conditional κ = 1.128983115440183e-3: an E8 bit word with 740 helpers under #433's complex supplier #436/κ = 1.136403735815528e-3: R740 shared-donor kernel with 57 primitive frame changes #440/κ = 1.138798055343319e-3: R740 block retiming after shared-donor kernel #441.
• Zhihao Chen: κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377/Compressed nine-stage E8 windows and whole-block banking for a conditional κ = 9.98243154231047e-4 > 2^-10 bound #386/κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425.
• Henry Grant: Conditional κ ≈ 7.037017435 × 10⁻⁴ > 2⁻¹¹: five-stage bit and complex suppliers #234.
• All retained authors.

Complex kernel, composition and packaging by Rohan Arun with Claude Opus 5.5 (Anthropic).

🤖 Generated with Claude Code

https://claude.ai/code/session_01TGoMCHAqctZe4wTb4Qr7VV

[rohanarun]

Full verification PASS (uninterrupted, not --quick): python -B research/r740-dual-kernel/verify.py /tmp/vdkfull on Linux, Python 3.13.16.

• status PASS_DUAL_KERNEL_ELEVEN_HOSTING, mode full
• κ = 254883101229023/200000000000000000 = 0.001274415506145115 (bit-bound; complex b = 0.00129352661943759)
• manifest sha256 b4c8b2f34b950d1f3a09aacaac119b5745796fb89ea08d24455abc24a3d1c159
• emitted program sha256 be8f43c36e8b961d72a05e23456b49f6479c980d19b66436d7212787d4997d66
• steps (20, total 1533 s): parent445 956s, rebuild 208s, export-records 1s, stage-cohort 0s, official-legality 0s, official-columns 0s, official-k4-columns 0s, record-mutations 0s, positive-gram 0s, geometry 6s, emit 9s, independent-audit 23s, complex-kernel 1s, complex-kernel-local-programs 1s, complex-kernel-clifford-phases 201s, complex-kernel-replay 4s, complex-phases 31s, complex-emit 13s, complex-independent-audit 77s, exact-assembly 1s

[rohanarun]

Correction: please do not admit this κ as it stands.

The complex kernel here hosts each entering helper's residual as a width-8 interval. Its route is the exact lowering of T_FULL·T_E⁻¹ (monomial·C₈·monomial), but nothing places the helper's ninth, spectator coordinate. The same gap was identified for #449's hosting by #458, and #484 notes that complex entrances are isotropic rather than coordinate entrances. #471 closes the gap with full-invocation normalisation and same-owner spectator borrowing (credit to Dugongue). This package does neither.

What remains valid here:

• the exact 800-column scalar replay;
• the regenerated phase recipes;
• the local replays.

The hosted complex accounting, and therefore κ = 0.001274415506145115, depends on the unplaced spectator, so I am withdrawing that claim and returning the PR to draft. The bit-side packages #457 and #459 are unaffected: bit entrances are coordinate entrances with explicit residual isometries.

The 56 proportional-pair entries found here are of the same type as #449's kernel79 entries; I have not checked whether they are a subset.