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Conditional kappa = 0.001591975256487475: five-copy remainder pooling and exact carrier epochs - #560
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Thank you @Dugongue. This contribution at Five-clone pooling replaces five rank-39 remainders by four rank-48 and one rank-3 remainder; rank mass is unchanged. Concavity improves the moment, while actual owner chronology and all 360 calls remain charged. Preserved implementation and research: The family review and individual disposition record the scope, evidence and attribution. Parallel and intermediate contributions remain credited even where a later composition has a stronger bound. The published multiplication bound remains conditional on the documented compiler, realization, hosting and analytic assumptions; these finite checks are not a full Lean multiplication proof. |
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This adds research/pooled-clone-epochs, with conditional
kappa = 0.001591975256487475 = 63679010259499 / 40000000000000000.
This is a small, exact increase of 0.000399106673% over the equal kappa reported in #551 and #552, at the public snapshot frozen through #555. The numerical gain comes from a new finite five-copy whole-owner pooling construction. A separate exact carrier-basis rewrite removes 28 local scalar instructions; its paid rank profile is unchanged.
The certified complex saving is b = 0.001594513682816650764990. The cited #555 bit saving is c = 0.001733052823330162099157, pinned at 65a6bc1. Complex remains binding. Both endpoints are composed through the complete profiles, fresh finite invoice and all 47 strict outer inequalities; gains are not added to earlier kappa values.
Numerical mechanism: turn a fractional remainder pattern into a finite controller
The previous complex allocation leaves one rank-48 pending child containing one width-9 residual, hence a paid rank-39 remainder. A capacity relaxation suggested concentrating five such residuals together. The construction makes five copies of the finite source controller and interleaves the five corresponding complete calls at each original call time. Each local call still has ambient dimension 73; the local supplier is not enlarged.
Exactly five width-9 helper occurrences are redirected into disjoint offsets 0,9,18,27,36 of one rank-48 pending family. Each helper stays whole in one family. The selected owner is idle through all five calls, is never an active data family there, and spectators are restored at each complete-call boundary. Four former owners become entirely unused pending children. The resulting exact change relative to five copies of the old complete profile is
Total rank mass is unchanged. For every 0 < a < 1, strict concavity gives
This proves a strictly better recurrence moment, but the numerical bound additionally pays the enlarged finite controller and prefix. Live stock is 35,840, physical stock is 36,354 and deficit is 12,800. The literal witness contains 181,440 whole helper roles over 360 complete calls. Measured prefix 801 is charged at 1,024; physical row coefficient 17,495 is below the retained allowance 20,161.
The complete complex histogram is
Scalar mechanism: a paid, reversible carrier epoch
Within the source-pinned A epoch, helpers 571 and 585 are read but never written. Both have the common boundary frame [480]. Introduce r' = r+s, rewrite every read t += qr as t += qr' - q*s, combine only commuting equal-frame shears, and undo the basis change at the end. Both boundary operations are explicit and charged.
The exact map on all input coordinates is unchanged. A decreases from 2,007 to 1,993 gates, and inverse background cleanup from 2,847 to 2,833: 28 fewer scalar instructions per invocation. Emitted events decrease from 21,483 to 21,455 in each orientation. This is a circuit simplification, not an additional paid-moment saving.
Verification
The changed complex supplier passes fresh transport, phase, adapter, emitted-local and arbitrary-dirt checks. All 728 formal columns are checked in both orientations, with all 504 dirty helpers restored. Phase admission covers 2,929 directed edges and 224 endpoints, using explicitly hash-bound cache reuse where appropriate. An independent multiprecision-dyadic C++ audit also compares all 529,984 coefficients of the entire main transfer matrix. Its terminal-inverse and transformed-reader omission controls change 4 and 8 target coefficients respectively.
The new controller is independently checked against its actual source-bound 73-column charts, complete idle intervals, active-data exclusions, residual widths, disjoint pending coordinates and every paid remainder. Seven owner corruption controls are rejected. A fresh scalar/prefix invoice, two rational moment engines with next-grid rejection, six bootstrap steps, all 47 outer inequalities and four assembly controls pass. The unchanged #555 physical bit operator is cited and numerically repriced, rather than redundantly rebuilt here. Remote GitHub CI status is separate from these local receipts.
Replay, with a new work directory outside the package:
Keep assertions enabled. Source and generated-word pins use canonical content where compression or line-ending differences would otherwise be incidental.
Direct lineage and evaluated related work
Existing upstream notices, licenses and assistance disclosures remain with the vendored sources.
Retained research and scope
The research notes preserve the sufficient epoch identity, the whole-owner cloning/pooling proof and unsuccessful finite searches: cumulative rank-four retirement and mixed providers; PR554 direct/meet/join frame components; paid two-delivery windows; coherent read-only epochs; generalized producer/reader gauges with exact suffix-observability guards; and bounded frame reoptimization. The epoch lane screened 128 paired windows, 1,052 maximal epochs and 22,394 generalized windows. The fixed-capacity optimization nominated the denominator-five pattern; no floating solver result is presented as an exact global optimality theorem.
A useful next direction is to lift other small-denominator resource patterns into explicit complete-call schedules, or change the local residual inventory before repeating the allocation search. Each proposal still needs actual owner chronology, all dirty/phase checks and the full finite invoice.
The result remains conditional on inherited all-size compiler realization, common-label/mutable-source transfer, whole-call hosting, precision/tape and analytic-transfer premises, including the bit supplier's stated interfaces. It is not an unconditional integer-multiplication theorem, a complete Lean formalization, a measured practical speedup, or a claim of a large breakthrough.