Conditional κ = 0.001308501945841842: kernel79 complex supplier under #471's footprint rule at nine stages, with #493's bit word (+0.38%)

[rohanarun]

Conditional κ = 654250972920921/(5·10¹⁷) = 0.001308501945841842, +0.3814 % over #493 (0.001303530097714685). The result is complex-bound with #493's R489 bit word. It uses only interfaces that #493's own κ already relies on: no Lemma G, no cross-copy.

The complex endpoint is b = 0.001310216366506891. For comparison:

• Conditional kappa = 0.001274548075945587: full-nine-axis kernel normalization and same-owner spectator borrowing #471's eleven-stage supplier: 1.305231506·10⁻³;
• 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) #484's one-owner nine-stage kernel79: 1.305683990·10⁻³.

Conditional variant (#484's Lemma G):

• b = 0.001333854156208409, with all 40,320 residuals hosted whole and no banks;
• with Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493's bit word, κ = 0.001332077359286333;
• that is +0.532 % over the 0.001325031497415498 that Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493 lists with 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) #484's supplier.

Change

This takes #484's nine-stage pipeline: #449's kernel79 word, paid wrappers, explicit spectators and banks. It replaces #458's one-helper-per-family hosting rule with #471's footprint rule (Dugongue), the admission behind #471's complex supplier that #493 cites. Several helpers of one invocation may share a host family when their full nine-coordinate footprints (residual cells plus explicit spectators) are pairwise disjoint.

For a copy of an idle child of rank r, the capacity is:

• Σw ≤ r;
• 9k ≤ 73;
• Σ(9 − w) ≤ 73 − r.

This is #471's inequality set with 73 in place of 89.

Kernel79 under the footprint rule:

• 286 width-9 banks;
• live stock 7,966;
• max child 8;
• deficit 2,496.

Our model reproduces #484's one-owner kernel79 b exactly before the rule change.

Package

research/complex-footprint-kernel/ is a copy of #484's e8-complex-gauge-kernel with these additions:

• code/footprint_gk.py;
• --footprint paths in the hosting generator, the independent hosting audit and the variant runner;
• frozen footprint selections;
• Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493's bit supplier pinned from its EXPECTED.json, as Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493 pins complex suppliers.

Checks run here

• Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442 in full (Linux, Python 3.13).
• Per word, through 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) #484's unchanged scripts:
  • exact arbitrary-dirt replay;
  • emitted programs in both orientations;
  • Split centres and source-basis reuse for conditional κ = 0.001120864952258278 #433's exact phase recipes;
  • exact wrappers;
  • independent read-back.
• Hosting under the footprint rule, asserted by the generator and by the independent audit. The audit rejects 17 mutations, including:
  • overlapping footprints of two same-invocation helpers;
  • a spectator on a residual cell;
  • a helper on an active family;
  • cross-replica hosting.
• 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) #484's certification: both moment engines (next grid point rejected), finite invoices, and the 47-inequality outer assembly with controls.

The full verify.py (anchor, both words, Lemma G check stage, assembly) is running now. Its receipt and EXPECTED.json will follow in a commit and a comment. CI is not claimed.

python -B research/complex-footprint-kernel/verify.py /absolute/new/output

Scope

Review point: #471's footprint admission at nine stages, used here with #458/#484's paid wrappers. As in #471, every simultaneous footprint is disjoint, and spectators are explicit and restored by the whole-invocation identity.

All other inherited interfaces (all-size compiler, hosting, restored work, analytic) remain. No unconditional theorem is claimed.

Credits

• Dugongue: Conditional kappa = 0.001274548075945587: full-nine-axis kernel normalization and same-owner spectator borrowing #471, footprint rule and normalization.
• DreamingOfClouds: 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) #484 package, Lemma A/G and gauges; [Draft] Complex supplier b ≈ 1.334e-3 (+5.2%): shared-donor kernel and entrance gauges on the E8 complex word #449 kernel79.
• eumemic: Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493 bit word.
• chafreaky: Conditional κ = 0.001274032589159993: complex kernel with paid residual adapters #458.
• 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 = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425.
• All retained authors.

Footprint hosting, selection and packaging by Rohan Arun with Claude Opus 5.5 (Anthropic).

🤖 Generated with Claude Code

https://claude.ai/code/session_01TGoMCHAqctZe4wTb4Qr7VV