Conditional kappa 0.001538356846647141: higher response spaces meet exact rank-32 saturation

[Dugongue]

Result

Conditional kappa = 0.001538356846647141, exactly 1538356846647141/1000000000000000000: +1.356990% over pinned #531's 0.001517760977820331.

The complex supplier binds at b = 385181758657419704781/250000000000000000000000. The unchanged cited #493 bit supplier remains c = 391360506745414227149/250000000000000000000000. The result is +4.794222% over our previous deliverable, but most of that comparison comes from incorporating #531; the new contribution beyond #531 is the stated +1.356990%. No separate gains are added together.

Mechanism: optimize response dimension together with actual residual capacity

The public direct504 word warms helpers using line relations. We instead compute exact response kernels over the common first-frame intersection of each active support. Eight rank-four components and sixteen rank-one components give72 pivots and32 residual helpers of width5. The resulting full local profile is

{1:2048,2:732,3:176,4:360,5:56,6:56,7:8}

with504 helper roles and widths 32x5,112x8,360x9. Local rank mass falls from6240 to6152, while total helper width also falls by88. That conservation law preserves the global deficit; it does not by itself establish an improvement.

The decisive new physical construction uses 2,304 exact fills 32 = 5+9+9+9. A timed integer pattern witness and explicit whole-owner matching realize every helper occurrence. The final histogram, including every positive idle remainder, is

{1:147456,2:52704,3:15258,4:25920,5:5311,6:4032,7:2816,23:1,48:998}

Its rank mass is520704 =73*7168-2560. Live stock7168, physical stock7682, ambient rank73 and deficit2560 stay at #531's values. The new map contains36,288 helper occurrences,16,128 pending bins and3,248 actual73-column charts;7,104 positive remainder children are paid. No helper alias, free storage, ignored spectator, recursive splitter or newly assumed oracle is used.

Four proved construction theorems, applied at different levels

1. Support-saturated response calculus. Exact pivot relations lift to any admissible common entrance of their entire active support; disjoint closed supports compose. The exact signed resource signature is derived from affected frame-chain edges. In a fixed affine resource regime, concavity gives an endpoint comparison principle. This finds higher-dimensional entrances that a pivot-count-only objective misses.
2. Integral residual-pattern calculus. A finite time/pattern certificate plus matching decomposition realizes whole physical owners. The proof includes the actual timeline, injectivity, source-specific charts, spectator restoration and all remainders. For this certificate, an exact rational logarithm enclosure proves strict complete-moment dominance for every 0 < b <= 0.002, not only one floating-point probe.
3. Contextual observational rewriting. If the exact suffix responses satisfy H_t = lambda H_u, a background addition can be retargeted with matching coefficient and regenerated inverse cleanup. The concrete gate A[645], 31 += (1/2)*240, becomes 245 -= 240 because H_31 = -2 H_245 != 0. Every frame chain is unchanged; this is a nonzero-response rewrite, not dead-store deletion. It saves four counted local scalar primitives but has no independent recursive-moment gain.
4. Finite supplier-registry gluing. The filtered and retargeted concrete words are selected in36 complete calls each. Their entire normalized endpoints, including paid helper residual operators, agree under the actual charts. Scalar dirty restoration is distinguished from physical residual action. A common pending-operator normal form proves legal contextual assembly and the exact mass ledger; every selected program/hash/frame/width is bound per call.

Full sufficient proofs and construction witnesses are in proof/. These are newly written arguments for this composition, not claims that contextual equivalence, concavity or matching decomposition are globally new mathematics. The first two produce the exponent gain; the latter two are exercised concrete program-calculus extensions. No new Lean certification or four independent numerical gains are claimed.

Direct sources and credits

• Direct delivery with 504 helpers for conditional κ = 0.001517760977820331 #531, head c609855ff92acc8cae3aa0808a4ecb624165f548:504-helper direct delivery, ordered112-port factory,72 gauges, frame-coalescing mechanism, data controller, exact phases and global invoice. This is the direct comparison baseline and the main inherited construction.
• Punctured Hamming ports for conditional κ = 0.001457945761723368 #523, head a900f98be8ff2c09d3292b5112d4731dbd0c8438:112-port restriction, data geometry, phase cache and retained finite allowance.
• Conditional kappa 0.001391523459591829: partial-own factory, 92 pivots and asynchronous whole owners #514, κ = 0.001363614743745701: partial-own factory with 84 kernel pivots #507 and 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: Hamming/scalar factory, entrance gauges, shared-donor response-kernel construction, whole-owner and dirty-wrapper framework. Conditional kappa 0.001402213284556683: rank-factor coordinate totals merge 56 recursive pairs #522: coordinate-total/rank-sector viewpoint and previous response-space exploration. Open asynchronous whole-owner allocator and a delivery-layer map of the complex word #520: delivery and explicit allocation perspective.
• Asynchronous whole-owner replicas: kappa 0.001357701757628940 (+1.9236% over #496) #505, Conditional kappa 0.001372554837726243: coordinated gauge collapse with retimed asynchronous whole owners #508 and Conditional kappa 0.001393662920776407: blockwise pair exposure, 99 pivots and whole-owner allocation #518: complete-invocation scheduling and composition ancestry. Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493, head c3a9a436b84fe44fdc223a5784d84736d1c2d3c7: unchanged bit supplier, reused as a pinned dependency and repriced without redundant operator reconstruction.
• Conditional κ = 0.001274032589159993: complex kernel with paid residual adapters #458, Split centres and source-basis reuse for conditional κ = 0.001120864952258278 #433, κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425 and κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370: normalized residual wrappers, scalar/frame transport and exact phase/compiler ancestry. Conditional κ = 8.04849251275843·10⁻⁴: post-Design-T kernel, retiming and reorders on #357's reordered base word with all 471 twins (+0.0679 % over #359) #361 and Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315: the two exact moments and outer assembly engines.
• [Lean] Connect complete-bank endpoints to PR516 realization #528, head ba5e50abb80c6b854b127ccbe37d524e11c6db72: complete-bank full-state interface informed the registry proof. Its physical realization hypotheses are retained, not silently discharged.
• [Lean] Bound automatic-width counting by 12n+4 #526 and [Lean] Compose prepared-input counting within 18n+12 #530: recent machine/counting compositional results read; no numerical saving imported. Conditional kappa 0.001467289440740487: rank-sector factory on112 ports #527 and Conditional κ = 0.001469603966017302: rank-sector102 on112 ports, fully rebuilt #529: recent comparable numerical constructions read and superseded by the Direct delivery with 504 helpers for conditional κ = 0.001517760977820331 #531 baseline; their gains are not stacked.
• Structural sources: Mathlib's Real.concaveOn_rpow; OpenAI's MatchingCount.UnitFlow, PerfectMatching.Main, WLIdentification.FiniteRefinement and GroupRing.WordResponse; and prior tensor-isomorphism dirty-workspace composition work. These motivated the abstractions rather than supplying an unproved imported algorithm. Module/source hashes and precise roles are recorded in research-notes/.

The public snapshot is frozen through #531. SOURCES.json and the retained ancestry pin relevant heads. No live refresh was performed during packaging.

Reproduce and evidence

From research/filtered-supplier-registry, with Python3.11+ and NumPy:

python -B verify.py --work /absolute/path/to/a-new-output-directory

Offline, with no optimizer or search during verification. All 18 stages passed from a fresh CRLF copy, including:

• deterministic rational source reconstruction and the exact observational rewrite;
• both728-column orientations for both variants, actual frame chains, phase-correct residual wrappers and13 local mutation controls per variant;
  -2,934 directed phase edges per variant, including120 freshly lowered changes on the filtered source; unchanged edges are explicitly source-bound to the public cache;
• independent owner chronology, all charts,11 rejected physical/registry corruptions and the separate exact resource audit with5 controls;
• actual scalar/Horner and finite invoice:225 prefix bits required versus512 charged; row coefficient16467 versus20161 retained;
• two exact rational moments with adjacent-endpoint rejection, six bootstrap steps and all47 strict outer inequalities.

Template SHA-256:

f75929eadcdeb5f242706c409afd4e1e71efea2297961f15ec3365245f4de00b

The first packaging test caught elapsed time in phase-receipt fingerprints. Only that nonmathematical field is excluded from the stable registry fingerprint; mathematical source/recipe/compiler hashes and global-phase checks remain. Other artifacts retain raw hashes. Gzip level/header and LF serialization are pinned; the final outer-driver explicit level9 setting was byte-compared with the full replay. Mandatory manifest/expected-result checks and five transport corruption controls reject before output creation. Upload batches are reassembled byte-for-byte. Local execution was Windows Python3.11/NumPy2.4.6; no unexecuted platform test is claimed.

Preserved alternatives and scope

The notes retain the failed55-call allocation, the smaller positive46-call cross-call transfer, the later dominating72-call saturation, two256-mask response hybrid screens, projective/suffix/protected-column rewrites and bounded neutral searches. Failure on a bounded family is not an impossibility theorem. The package does not claim a global optimum.

This remains conditional on the project's inherited all-size compilation, common-label/mutable-basis transport, normalized complete-wrapper realization, restored-row, precision/tape and analytic-transfer contracts. It is a checked improved finite construction under those interfaces, not an unconditional integer-multiplication theorem. Nothing was published automatically.

[Mister-Meeseeks]

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

Your filtered supplier, resource calculus, observational rewrite and finite supplier registry are integrated. Both concrete variants, physical registry/owners, exact resource checks and assembly replay reproduce κ = 0.001538356846647141. The moment-neutral rewrite and registry demonstration are credited separately from the numerical gain; bounded negative experiments and research notes are retained.

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.