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…it hosting assumption
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…dle child, conditional kappa 1.330733966732373e-3 Builds on CrocSwap#442 (idle-axis hosting) without changing either local word. A hosted helper residual may be laid across the pending coordinates of two copies (two data families) of the same idle child, which share the child's window, so every child's capacity tiles exactly and the idle remainder children almost vanish. Bit: live 9,457 -> 8,592, c = 1.332507e-3. Complex: live 8,400 -> 7,680, b = 1.437532e-3. kappa = 1330733966732373/10^18, bit-bound, +17.44% over CrocSwap#434, +7.53% over CrocSwap#442. PROOF.md section 7 replies to the trace note of CrocSwap#451. Co-Authored-By: Claude Fable 5.1 <noreply@anthropic.com>
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…375517e-3 Both E8 suppliers of #434 keep their local words. Whole dirty-helper residuals move from bank streams onto pending idle-advance coordinates of data families that are untouched between the hosting invocation and the advance. Bit c 1.239085e-3, complex b 1.267453e-3, kappa 618775866623647/5e17 (+9.22% over #434). Fresh #434 replay, independent audit with nine mutation controls, two exact moments, fresh invoices and all 47 outer inequalities.
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Draft conditional κ = 0.001249035299298004, 0.23769014% above #444 at
8c4242ea61c67827fe62b07b2edc093d7ddfa631and our preceding verified result.Same additional idle-data hosting assumption as #442/#444. This is not an improvement under only #441's pre-hosting assumptions. The operator extension is inherited; finite interval audits do not prove it. All compiler, dirty-bank/compressed-window, restored-work, tape/common-grid, precision and analytic obligations remain. No unconditional theorem or new Lean proof is claimed.
The new search evaluates response-kernel relations together with idle-slot hosting costs. It admits nine additional rank-one relations that the unhosted model rejected. Starting from the pinned R740/PR443 lineage, the new word has 123 pivots, 118 donor slots and entrance rank 147. With 64 bit replicas, hosting leaves 74,528 live / 74,537 physical families, 321,536 hosted residuals and deficit 19,968. The complex supplier remains eight replicas, 8,400 live families and deficit 2,496. The bit supplier binds at exact lower endpoint
625298669760136826673 / 500000000000000000000000.Passed before draft publication: exact precursor and kernel reconstruction; all 980 local columns and integer source spans; the 7,620-column unbanked global identity; all 576 emitted invocations and 426,240 helper assignments; independent exact projector, reflected-program and residual-isometry audits; both hosted templates and nine hosting mutations; both rational moment engines; fresh 64-replica invoice retaining the extra route bound 144 plus two hosting factors; six bootstrap levels and all 47 strict outer inequalities. Next κ grid point rejected. Evidence is in
research/e8-hosting-selected-kernel/evidence/.Pending: full uninterrupted Python 3.11/3.14 wrappers and GitHub CI. Reproduce:
python -B research/e8-hosting-selected-kernel/verify.py /absolute/new/output(standard library only). It first replays the previous full rank138 composition and all 2,918 complex Gaussian phase edges, then reconstructs and checks the new kernel, global layout and assembly. The preceding result0.001246073505487942remains fully verified on Python 3.11/3.14, with receipts inresearch/e8-rank138-idle-hosting/local-evidence/.Rohan Arun with substantial OpenAI Codex assistance: hosting-aware kernel discovery and integration. Credit chafreaky #443/#444/#417/#439, eumemic #436/#440/#441, DaysSky #442 hosting with Claude assistance, Rohan Arun #434/#435/#438, icekylinx #433, the retained #426/#389/#376/#370, #425/#386 and Jacob Sussman contributions. Source pins, licenses and assistance disclosures are retained.