Conditional κ = 0.001266536148630309: rank151 kernel + target-prefix mixing on #451, complex-bound (+0.0477%)

[rohanarun]

Conditional κ = 1266536148630309 / 10¹⁸ = 0.001266536148630309, 0.0477% above #451 (0.001265931727546219) and 0.0179% above our kernel-only #457 (0.001266309796182835). The complex side now binds: the bit endpoint rises to c ≈ 1.276041713·10⁻³, against the unchanged complex b = 0.001268142296690675.

There are two bit-side changes on #451's dual eleven-stage, fully hosted construction. Everything else is #451's code.

1. The rank151 response kernel of Conditional κ = 0.001266309796182835: rank151 bit kernel on #451's dual eleven-stage hosting (+0.0299%) #457: 123 pivots, entrance rank 151, 120 donor slots, followed by κ = 1.138798055343319e-3: R740 block retiming after shared-donor kernel #441's 112 pair and 42 window B retimings.
2. Conditional κ = 0.001347744547714835: eleven-stage cross-copy hosting with target mixing (+1.278% over #448) #450's target-prefix mixing (DreamingOfClouds), applied verbatim (code/mix_target.py) to that word:
   • 46 receivers: 42 at dimension 6, 4 at dimension 5;
   • frame-0 preconditioners and scatter corrections;
   • helpers, sources, starts, finals and the A/C phases are unchanged. The kernel, the 18 residual isometries and the full Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442 hosting of all 65,120 helper occurrences (no banks) therefore carry over.

Under full hosting the mixing's local-child savings lift the bit supplier about 8.1·10⁻⁶ above #457, so the cap is now #451's complex supplier. Further gains need the complex side.

Bit profile: {1:218104, 2:94096, 3:23360, 4:31096, 5:8632, 6:5632, 7:5104, 8:792, 28:8, 56:456}.

Review point: #450's bit-label conjugation lemma, which allows a target as an addition operand at a common frame. Every inherited checker accepts and replays the mixed word:

• e8core F₂ replay of all 980 columns;
• official C++ legality, columns_h and columns_k4;
• record mutations and positive Gram;
• the emitted local-stream replay and all 9,340 global scalar columns;
• the independent audit (22 mutations).

Complex phases, emit and audit (19 mutations) and the exact assembly (47 strict inequalities and controls) all pass. assemble.py drops only its bit-binding assertion and reports the binding side.

Status:

• Passed locally (Linux, Python 3.13): every step after the Conditional κ = 0.001263365464781156: eleven-stage R740 hosting with complete row bound #445 parent replay, which was reused from the identical-parent run of Conditional κ = 0.001266309796182835: rank151 bit kernel on #451's dual eleven-stage hosting (+0.0299%) #457.
• Running: a clean uninterrupted full verify.py. The result will be posted as a comment.
• CI: not claimed.

python -m pip install -r research/e8-r740-replica32/parent436/requirements.txt
python -B research/r740-hosted-rank151-mixed/verify.py /absolute/new/output

All inherited conditional interfaces remain:

• Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442 idle-data hosting;
• the all-size compiler;
• mutable basis;
• correlated dirty helpers;
• compressed windows;
• restored work;
• tape/common grid;
• precision;
• row divisor;
• analytic transfer.

No unconditional theorem is claimed. Cross-copy hosting (#448/#450) is not used.

Credits:

• DreamingOfClouds (Conditional κ = 0.001347744547714835: eleven-stage cross-copy hosting with target mixing (+1.278% over #448) #450): target-prefix mixing and its lemma.
• chafreaky: Conditional κ = 0.001265931727546219: dual eleven-stage whole-residual hosting #451 package, 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, 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.
• 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.
• DaysSky: Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442.
• icekylinx: Split centres and source-basis reuse for conditional κ = 0.001120864952258278 #433.
• 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 parent authors.

Kernel search, 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-hosted-rank151-mixed/verify.py /tmp/vmixfull on Linux, Python 3.13.16.

• status PASS_DUAL_ELEVEN_RANK151_HOSTING, mode full
• κ = 1266536148630309/1000000000000000000 = 0.001266536148630309 (complex-bound)
• manifest sha256 44f86cfc10650c2108b566088dc22e50aedbf7b61096d74c24474e5b03fd6a42
• emitted program sha256 be8f43c36e8b961d72a05e23456b49f6479c980d19b66436d7212787d4997d66
• steps (16, total 1601 s): parent445 1222s, rebuild 215s, 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 26s, complex-phases 32s, complex-emit 13s, complex-independent-audit 76s, exact-assembly 1s

[Mister-Meeseeks]

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

Contribution: Rank151 kernel combined with target-prefix mixing.

Review and limits: The complete new word is checked under the common-frame bit-label interface. The unchanged complex supplier binds; inherited rank149 CI alone is insufficient.

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: e037bc50c3d5efc1bec0d258a9f274f79ba45ec7. Publication: b3c9e5c269fcfb25dfb4eda4ebc419c052d4c93d.