Conditional κ = 0.001274032589159993: complex kernel with paid residual adapters

[chafreaky]

Conditional κ = 0.001274032589159993, 0.591885% above #454 (d14eb0dd425bd681e3b43f208d089e679d600176). The complex endpoint rises to b = 0.001297203880559577927527; the unchanged #454 bit endpoint c = 0.001275657818793753037875 now binds.

The 79-pivot complex kernel from #449 has isotropic entrances, so an orthogonal coordinate relabeling does not justify hosting its residuals. This package reconstructs the kernel, proves each actual residual has an exact two-sided Gaussian monomial factorization, and pays for the inverse monomials before and after the entire local invocation. A new assignment permits at most one helper per physical host family in each invocation. All 49,280 residuals remain whole, with every spectator restored and no uncharged helper entrance transform. The written proof covers arbitrary borrowed values and both scalar orientations.

The reproducible package is research/e8-complex-adapted-kernel/. It includes frozen inputs, independent local/global audits, complete instruction hashing, source inventories and a fresh #454 parent replay. It retains m=89, 9,600 live families, 10,178 physical families, maximum child 64 and the 20,161 row contract. The finite invoice adds 788,480 rank-zero compiler groups conservatively for 13,904 nontrivial wrapper events.

Validation:

• Linux full replay and macOS quick replay with E8_REPLAY_PHASES=1 both pass; quick mode skips only inherited Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443 exhaustive controls. Every new check runs in both modes, and the complete exact result matches across platforms.
• Both 800-column orientations, 2,932 local phase edges, 14 adapter frames, 8,520 global phase equalities, 1,320 data routes and 49,280 residual routes pass.
• Seven local and 25 global mutations, 24 omitted-bridge controls, both rational moment engines, both finite invoices, six bootstrap levels and all 47 strict outer constraints pass.
• Repository make verify-joint passes. Receipts and exact results are in research/e8-complex-adapted-kernel-validation/. GitHub CI is pending; this PR remains draft until its checks finish.

python -m pip install -r research/e8-r740-replica32/parent436/requirements.txt
E8_REPLAY_PHASES=1 python -B research/e8-complex-adapted-kernel/verify.py /absolute/new/output
python -B research/whole-hosted-format-ceiling/verify.py

A separate scoped proof in research/whole-hosted-format-ceiling/ shows that the fixed 120-port, copied-rank72, bank-free compressed-window format has κ < 2/127 throughout its declared stage family, and κ < 1/87 for the present eleven-stage geometry. These are limitations of the stated format and outer assembly, not lower bounds for integer multiplication. Exact controls supplement the uniform written inequality. Reaching κ=1 requires different hypotheses and a different outer reduction.

The all-size Gaussian/bit compilers, restored work, mutable-basis realization, compressed geometry, common grid, precision, row-divisor and analytic transfer interfaces remain conditional. This does not adopt #449's entrance-gauge argument or any cross-family residual split. It is not an unconditional theorem or a κ=1 result.

Credits: DreamingOfClouds/Rohan Arun and Claude Opus 5.5 for #449's kernel and generalized local tools; DaysSky for #442's hosting construction; icekylinx for #433's complex word; eumemic for the retained bit retimings; all source-pinned parent authors. New adapters, hosting proof, audits, composition and scoped ceiling by chafreaky with OpenAI Codex assistance. See NOTICE.md for detailed provenance.

Verification update: fixed the Python 3.11 discovery-metadata portability failure reported by PR460. Reproduced the old tolerance assertion on a freshly generated Python 3.11 parent (gain estimate delta -2.0889956431346945e-12). Only the two finite discovery floats are normalized; every exact field still compares identically. Eight regression controls pass on Python 3.11 and 3.14. A fresh full Python 3.11.15 replay, including regenerated phases, passed with RESULT.json byte-identical to the original release. New source-bound receipts are in research/e8-complex-adapted-kernel-portability; κ and both suppliers are unchanged.

[Mister-Meeseeks]

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

Contribution: Complex response kernel with paid two-sided Gaussian adapters.

Review and limits: Nonorthogonal entrances require charged monomial wrappers and spectator restoration. The portability repair changes discovery floats only, preserving exact fields.

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: 44e9eae8287951c1e6dbbd3c553f5743fe2240b0. Publication: 64039f1bd3b6a523b96083fd858ad040a75e3030.