Complex supplier b ≈ 1.327·10⁻³ (+4.7%) on whole-residual hosting: shared-donor kernel and entrance gauges with paid residual wrappers (conditional on two lemmas)

[DreamingOfClouds]

A stronger complex supplier: b = 1326789535340266/10¹⁸ ≈ 1.326790·10⁻³ (+4.68% over #442), whole-residual hosting, one owner per family

This adds one self-contained package, research/e8-complex-gauge-kernel/. It is a complex-side supplier: #433's split-centre word (R = 560, m = 73), #442's idle-axis hosting with the one-helper-per-family rule of #458, and two tools of the bit side ported to it. No new κ is claimed in this package. With #442's bit word the bit side binds (κ = 1.237552·10⁻³; 1.279171·10⁻³ with our pinned R724 word). #493 reports κ ≈ 1.325·10⁻³ when its R489 bit word is paired with this supplier; there the complex side binds. #496 and #497 re-place this word's helpers and report 1.332077·10⁻³ and 1.325118·10⁻³. The current lemma-less κ frontier is #478 (Dugongue) at 1.295509·10⁻³, which sits at #458's complex cap; this supplier raises that cap; #458, #460 and #462 are bit-bound, not complex-bound. On the complex side b is +2.28% over #458 (1.297204·10⁻³) and +1.65% over #471 (1.305232·10⁻³; #471 admits our kernel79 through normalisation plus same-owner spectator borrowing, and is credited below). This raises the complex cap of this lineage to κ ≤ 1.325031·10⁻³ (the #448/#450 lineage used in #464 and #469 has complex roots of about 1.44·10⁻³). It is conditional on two lemmas, the review points: Lemma A (paid residual wrappers with explicit spectators; the entrances are isotropic, not coordinate entrances) and Lemma G (nonzero-frame compensation and the centre scatter). It supersedes the draft #449.

complex word	entering helpers	entrance rank	b (·10⁻³)	gain over #442
#442	0	0	1.267453
shared-donor kernel	79	79	1.305684	+3.02%
entrance gauges	80	106	1.306413	+3.07%
both	122	148	1.326790	+4.68%

Relation to #458 and #471

#458 (chafreaky) puts the 79-pivot kernel of #449 on an eleven-stage complex word and pays its entrance transforms as emitted, checked adapters, with at most one helper per host family per invocation. This package adopts that treatment on the nine-stage word and adds the entrance gauges; #449's hosting allowed several helpers on one family and left an entering helper's spectator coordinates unplaced, which #458 showed is not sufficient. #471 normalises the complete invocation and borrows spectators in the same owner array; this package applies the same borrowing to its banks (an entering helper is the only owner of its bank family in an invocation, its spectators are the first cells of the bank's pad) and extends the one-owner rule to every register class of an invocation, with a control for each.

Change

A helper's frame-zero compensation (the response of its arbitrary dirt) pins it to start frame {0}, so its residual has width 9. Two exact ways avoid it:

• Shared-donor kernel (E8 bit supplier c ≈ 1.044064·10⁻³ in the nine-stage word (+2.2 % bit cap; no new κ, complex-bound) #417/κ = 0.001128832830332521: #417's shared-donor kernel on #433's R758 bit word (+0.71%) #435/Conditional kappa 0.001249035299298004: hosting-selected rank147 kernel #434/Conditional κ = 0.001133928043872860: R740 shared-donor kernel with explicit residual bank routes #439/κ = 1.136403735815528e-3: R740 shared-donor kernel with 57 primitive frame changes #440): a pivot whose response column is a rational combination of donors' columns starts at an entrance E contained in every member's first frame; each donor receives d += −q·pivot at E before the main word (inverse after cleanup).
• Entrance gauge (research: response-subspace gauges and two-sided normalization, kappa = 1.013310887459710e-3 #401/κ = 1.017022657082799e-3: 205 shared-donor gauges with 64-replica banks #408–κ = 1.017798035290733e-3: propagated frames admit 214 entrance gauges #416): the compensation of a helper is executed at its entrance frame E instead of frame 0; targets climb through the nested entrance chain, and each centre copy descends from its frame through the levels it serves (mass-neutral, rank deficit 2,496).

A helper that starts at E has the residual S_E = T_FULL·T_E⁻¹ = L_E (C_[w] ⊗ I) R_E, w = 9 − dim E, exactly (global phase included). The paid wrappers R_E⁻¹ before the whole invocation and L_E⁻¹ after its cleanup turn it into the coordinate factor and restore the e = 9 − w spectator coordinates; they are emitted, replayed, decoded and charged. Widths 8/16/32/48 of #442's idle children are tiled by width-8 residuals; the hosting is whole-residual with distinct stages in every copy and at most one helper per physical family per invocation, with the host chart (residual vectors, first unused basis vectors as spectators) in the route and audited. Residuals that are not hosted go to banks of their own, one entering helper per replica per bank family, with the spectators in the pad and the route and chart audited in the same way.

Result

Both rational moment engines certify b = 1326789535340266/10¹⁸ (next grid point rejected). Under the one-owner rule the word hosts 39,792 of 40,320 residuals whole; the other 528 (all of width 8) sit in 66 banks (live stock 7,746, maximum child 9, rank deficit 2,496). The finite invoices with the hosted stock and the wrapper charge, and the strict outer assembly (six bootstrap levels, 47 inequalities, next-grid, zero-η, zero-β and unit-β controls rejected), pass for #442's bit word (run here, κ = 1.237552·10⁻³ unchanged) and, as a pinned input from our own bit-side package that is not re-run here, for the eleven-stage R724 mixed word under whole-residual hosting (c = 1.280810·10⁻³): κ = 1.279171·10⁻³, bit-bound (no new κ). That pairing is informational; our R715 word is #463.

Checks (python -B research/e8-complex-gauge-kernel/verify.py DIR [--all] [--discover-check])

• Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442 runs unchanged (with Conditional kappa 0.001249035299298004: hosting-selected rank147 kernel #434 and Split centres and source-basis reuse for conditional κ = 0.001120864952258278 #433 below it), then the regression anchor: with the empty selection the generalised scripts reproduce Split centres and source-basis reuse for conditional κ = 0.001120864952258278 #433's canonical word, local programs, local checks, parent global template and independent replay, and Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442's hosted template and HOSTED.json.
• Gauge + kernel word: exact arbitrary-dirt and inverse-transpose replay (800 columns); emitted program replay in both orientations with centre lifetimes and wrappers; independent read-back of the serialized stream; exact Gaussian recipes with global phase for every used frame transition (complex_phases_full.py, unmodified); exact decoding of the wrapper monomials (L⁻¹ S_E R⁻¹ = C_[w] ⊗ I on every entrance frame); independent hosting audit (windows, slots, owners, spectator chart on data and bank families, one owner per family on every register class, private namespace, remainders, wrong-width control); fresh finite invoices; certified b.
• Independent Lemma G stage (code/lemmaG/, imports no package script): the label / scalar / accounting checker on both orientations, the exact operator identities of every compensation, centre and helper instance on all 512 basis columns, the complete emitted program end to end against the transform computed from definitions, and 14 necessity mutants of the clauses.
• Mutation controls, all rejected, several through the certifying scripts themselves: kernel and gauge gates, entrances, target levels, centre chains, wrappers, centre descent steps, two helpers on one data or bank family (accepted by Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442's rule, rejected here), spectator records, owners in the private namespace or on another replica's families, wrappers on non-helper registers, and selection-level conditions; the generators also refuse the eight-entering-helpers-per-bank tiling.
• --discover-check recomputes the frozen hosting selections with HiGHS and requires equality. Status: the full --all runs pass from fresh copies on Python 3.14 (--all --discover-check), 3.12 (--all) and 3.11 (--all --discover-check), with identical summaries. The .gz inputs of the Lemma G stage are pinned by the sha256 of their decompressed JSON, since gzip container bytes depend on the zlib of the Python in use.

Principal review points (conditional)

• Lemma A (PROOF.md §3). Exact and checked: S_E = L_E (C_[w] ⊗ I) R_E and the wrapped product on every entrance frame. Assumed: Conditional κ = 0.001274032589159993: complex kernel with paid residual adapters #458's argument that wrappers around a whole invocation of a family on which nothing else acts leave the spectators identical, at nine stages, for hosted families and banks.
• The entrances are not coordinate entrances. Every entrance frame consists of even-weight vectors of F2⁹ (15 of 16 frames totally isotropic). The bit-style criterion of κ = 1.136403735815528e-3: R740 shared-donor kernel with 57 primitive frame changes #440 (validate_isometries) is not met and cannot be met by any orthogonal map of O(9,2); restricted to entrances that meet it, the gain is 0. Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442's complex audit asserted w == 9 and Split centres and source-basis reuse for conditional κ = 0.001120864952258278 #433's emit start frame 0 for compensated registers; the marked copies here generalise exactly these.
• Lemma G (PROOF.md §2): reduces to κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425's conjugation lemma and Conditional kappa = 0.001274548075945587: full-nine-axis kernel normalization and same-owner spectator borrowing #471's / Conditional κ = 0.001274032589159993: complex kernel with paid residual adapters #458's virtual entrance, plus a schedule lemma (hypotheses (a)-(d)) whose clauses are checked on the emitted streams by the independent stage above; new relative to Conditional κ = 0.001274032589159993: complex kernel with paid residual adapters #458 and Conditional kappa = 0.001274548075945587: full-nine-axis kernel normalization and same-owner spectator borrowing #471, which do not use the gauge.
• All interfaces inherited from κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370, 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, Split centres and source-basis reuse for conditional κ = 0.001120864952258278 #433 and Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442 remain assumptions. This is a conditional finite witness, not a theorem.

Credits

Dugongue (#471), chafreaky (#458, #417, #439), DaysSky (#442), icekylinx (#433), jacklightChen / Zhihao Chen (#386, #425), rohanarun (#434, #435, #438), eumemic (#436, #437, #440, #441), #471 (normalisation of the complete invocation, same-owner spectator borrowing), the authors of #401 and #408–#416 (entrance gauges and prefixes), Dylan Crow (#370), Jacob Sussman (E8 unit); #451 for the cross-family obstruction.

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code. Licensed Apache-2.0.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Integrated into published main at 681e8923a0f23a19f93c7105c3a9c96149aa8362, preserving the original commits, notices and contributor credit. Thank you for the contribution.

All four gauge/kernel words were rebuilt and checked, including labels, phases, wrappers, end-to-end identities and negative controls.

The scoped review and pinned execution receipts record the exact coverage and remaining assumptions. This strengthens specific interfaces and preserves useful research; it does not change the selected κ or establish the complete multiplication theorem. Local checks passed; fresh Linux CI was queued at publication.