Conditional κ = 0.001266309796182835: rank151 bit kernel on #451's dual eleven-stage hosting (+0.0299%)

[rohanarun]

Conditional κ = 253261959236567 / (2·10¹⁷) = 0.001266309796182835, 0.0299% above #451 (0.001265931727546219 at f3a38153) and 0.0364% above #447. Bit binds: c = 0.001267915369836389691636 (#451: 0.001267536342017397); the unchanged complex endpoint is b = 0.001268142296690675.

One change: a larger bit response kernel on #451's construction. With every helper residual hosted on idle coordinates (no banks), one unit of entrance rank is worth ≈4.4·10⁻⁷ of bit saving, and an extra donor slot costs one local child per invocation (≈0.9·10⁻⁷). A hosting-aware search over donor sets, with closed entrances drawn from intersections of first frames, finds two more pivots on #451's own 57-retimed primitive R740 word:

• 123 pivots, entrance rank 151 (Conditional κ = 0.001265931727546219: dual eleven-stage whole-residual hosting #451: 121 / 149), 120 donor slots;
• κ = 1.138798055343319e-3: R740 block retiming after shared-donor kernel #441's 112 pair and 42 window B retimings apply unchanged afterwards. The full e8core replay (legality, F₂ on all columns, source spans) passes;
• 18 regenerated explicit residual isometries. A re-solved Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442 selection still hosts all 65,120 helper occurrences, with no banks and no padding.

Everything else is #451's code, byte for byte: the eleven-stage geometry (m = 78), three-level halving (max child 56), row bound 19,431 < 20,161, the complex supplier, both moment engines, the invoices and the 47-inequality outer assembly. The new package research/r740-hosted-rank151/ is a copy of #451's r740-hosted-rank149 with only the kernel, word, hosting selection, isometries, expected values and kernel constants changed.

Bit profile: {1:224264, 2:93744, 3:23712, 4:33912, 5:9512, 6:1936, 7:5104, 8:792, 28:8, 56:456}.

Status: the complete #445 parent replay, rebuild, official C++ legality/columns/k4, record mutations, positive Gram, geometry, emit, independent audit (22 mutations), complex phases/emit/audit (19 mutations) and exact assembly all pass locally (Linux, Python 3.13). A clean full uninterrupted verify.py run is in progress and will be posted as a comment. CI is not claimed.

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

All inherited conditional interfaces stay explicit: #442 idle-data hosting, the all-size compiler, mutable basis, correlated dirty helpers, compressed windows, restored work, tape/common grid, precision, row divisor and analytic transfer. No unconditional theorem is claimed.

Credits: chafreaky (#451 package, #443–#445, #417 kernel lineage), eumemic (#436/#440/#441), DaysSky (#442), icekylinx (#433), Zhihao Chen (#377/#386), Henry Grant (#234) and all retained parent authors. Kernel search and repackaging 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/verify.py /tmp/v151full on Linux, Python 3.13.16.

• status PASS_DUAL_ELEVEN_RANK151_HOSTING, mode full
• κ = 253261959236567/200000000000000000 = 0.001266309796182835
• manifest sha256 038d4f4c5b8f3123c2df4a510351aff85b98c3aa343a9f278642f5a673478740
• emitted program sha256 61813e20b1100d1c1b8257cf3a30c844766ebc49ddf051e9ca006e5107886649
• steps (16, total 1773 s): parent445 1155s, rebuild 294s, export-records 2s, stage-cohort 0s, official-legality 0s, official-columns 0s, official-k4-columns 0s, record-mutations 0s, positive-gram 0s, geometry 11s, emit 18s, independent-audit 44s, complex-phases 71s, complex-emit 27s, complex-independent-audit 151s, 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: Larger rank151 kernel on dual eleven-stage hosting.

Review and limits: All 65120 helper occurrences require a new legal allocation. Inherited rank149 CI does not execute this changed rank151 package.

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: 3e89db7725785f8e8e172d04d1d4804f57db72f3. Publication: 459f9dbdf41ea86f17fa21a6c71df00010a62c9b.