Post-center rank-four kernels for conditional κ = 0.001584931111785787

[icekylinx]

This extends the merged #531 supplier with post-center compensation and rank-four response kernels. It is one new commit based on main at f19cb621be2caf3a7291a37c343c1c55afe935e0.

Under the retained all-size compiler, common-label/mutable-basis realization, normalized Lemma A, whole-call hosting, restored-work, precision/tape and analytic-transfer interfaces,

$$ T(n)=O!\left(n(\log n)^{1-\kappa}\right),\qquad \kappa=\frac{1584931111785787}{10^{18}} =0.001584931111785787. $$

The saving is 4.42561% above #531 and 3.98392% above #533. These percentages compare exponent savings, not practical runtime.

Construction

The 112 ordinary-M helpers remain untouched through the source prefix and center cut. Their initial dirt can therefore be compensated at the beginning of delivery. In each of fourteen disjoint groups, four response columns span the other four, and every helper's first native frame contains a common rank-four frame.

Perform the gauge compensations first, then inject each pivot into its donors at that rank-four frame. A donor can now serve both its own gauge and the larger kernel: its compensation has already read its initial dirt before the new pivot contribution arrives. Exact response relations, target chains, nested frames and the full inverse cleanup are checked on the emitted word.

The entrance inventory is 40 rank-one kernel pivots, 112 rank-one gauges and 56 rank-four pivots. The 504 residual widths are consequently 56 of width five, 152 of width eight and 296 of width nine. A fresh whole-owner assignment and finite invoice produce the global profile.

Quantity	#531	This increment
Ports / helpers	112 / 504	112 / 504
Total entrance rank	152	376
Local rank mass	6,240	6,016
Live / physical families	7,168 / 7,682	7,168 / 7,682
Global rank deficit	2,560	2,560
Certified complex saving	0.001520068077832496458155	0.001589991533606044174031

The complex supplier now exceeds the previous bit endpoint. The final composition therefore adopts the published #510 R480 nine-stage bit supplier at 0.001587447106092548137108; the bit side binds. This bit improvement is adopted and credited, not claimed as new work here. The unchanged #493 bit supplier gives the separate control κ 0.001562995248531632.

Both component moments are repriced by two rational engines. Six bootstrap levels and all 47 strict outer inequalities certify the displayed global κ; adjacent-grid and invalid-parameter controls are rejected.

Contribution sources and prior PRs

Source	Contributor	Retained contribution
#531, e07a8a3f9f79c22113398a0c5dbbc7601f489b3e	icekylinx	SAME504 scalar architecture, geometry, frame coalescing, phase compiler and owner/controller baseline.
#510, 64edee18af02aab0c1696b677a0287dd0146381b	DreamingOfClouds, with disclosed Claude assistance	Adopted R480 nine-stage bit supplier and its conditional interfaces.
#484 and its credited predecessors	DreamingOfClouds and upstream contributors	Response-kernel, gauge and normalized-wrapper primitives.
#493, #503, #443/#444/#472, #442, #386 and #425	Credited upstream contributors	Mutable-source bit word, R480 change, replication and whole-call hosting, geometry and common-label realization.
#514/#507, #361/#315 and retained framework	Credited upstream contributors	Scalar construction ancestry, rational moments and outer/analytic interfaces.

My #523 supplied the 112-port restriction; #433 supplied split centers and source-basis reuse. Earlier contributions in #10, #18, #24, #32, #36, #104, #115, #130 and #144 concern batched boundaries, partial swaps, gauges, certified assembly and frame/cover constructions. These are provenance, not additive numerical gains. The earlier packages retain Dugongue's asynchronous scheduling and other source-specific credits.

The new ordered gauge/kernel composition, rank-four construction, physical integration and carryover exposition are contributed by icekylinx, with GPT-6 Astra assistance. Local review and packaging used OpenAI Codex. NOTICE.md, SOURCES.json and the separate original #531/#510 notices preserve the contribution boundaries.

Proof and validation

The incremental package is research/post-center-rank4-global/; it reuses the baseline already present in main.

• proof/POST-GAUGE-RANK4-PROOF.md: ordered compensation, exact relations, frames and cleanup.
• proof/GLOBAL-HOSTING-PROOF.md: actual owners, paid profile and finite accounting.
• proof/PR510-MUTABLE-SOURCE-CARRYOVER.md: role-independent common-label argument and explicit bit dependency scope.
• evidence/audit/EXACT-ASSEMBLY.json: both component moments and all strict assembly margins.
• BASELINE-PINS.json, MANIFEST.json, LOCAL-VALIDATION.json: inherited inputs, submitted-file bindings and incremental validation.

python -B research/post-center-rank4-global/replay_delta.py \
  --baseline research/direct-delivery504-global \
  --work /tmp/rank4-check

Local validation also used --rebuild-local --delta-phases with Python 3.14.8 and NumPy 2.3.5. The full changed-word construction, scalar/frame replay, physical owner audit, fresh invoice and exact assembly passed. The newly introduced intermediate phase cache was checked against #531 (194 changed edges), followed by the final 126-edge delta. Independent local replay covered both orientations on all 728 columns; all 21 residual wrappers passed, including global phases, with 11 local and five wrapper mutation controls rejected.

The manifest was generated from staged Git files, excluding all logs. A clean archive of those staged files passed the same incremental reconstruction in 14.85 seconds. Every baseline pin also resolved and matched inside that archive. The PR510 data copy differs from its pinned source only by an appended terminal LF; both byte hashes and the unchanged JSON data are documented in SOURCES.json.

Package manifest SHA256: 23143f650947a2b95b443a7ea5ded79d1d5644783b7f8a54967ba137b8b801f8.

Global template content SHA256: b9f4b60e56e5153c1fc89370e9ceb09f24b23e1c8ff56f2b47ef9d867aa04f2c.

Dedicated Python 3.11/3.14 CI covers reconstruction, changed phases and local wrappers. Remote results are pending. No predecessor cold replay, full bit-operator rebuild or PDF generation was performed.

This remains a finite conditional witness. The cited #510 physical bit operator and the inherited all-size/analytic contracts remain dependencies; the package does not claim an unconditional multiplication theorem, new Lean formalization or unrestricted optimality.

[Mister-Meeseeks]

Integrated on main in 9449300, with original commit history and individual credit retained.

Your post-center gauge/kernel ordering, rank-four construction and physical packing are selected on main with DreamingOfClouds’ freshly replayed #510 supplier: κ = 0.001584931111785787. The review additionally lowered all 2,927 local edges and 224 endpoint adapters, checked all 21 wrappers, local and owner mutations, and bound the fresh bit invoice/profile to the 47-constraint combined assembly.

Maintainer review and exact scope · Contributor record. The multiplication claim remains conditional on the inherited all-size/realization/analytic interfaces; these checks do not establish a complete unconditional theorem.

Thank you for contributing this work.