The Kronecker stage (round six, `cKronS_le`)

Proved

NLogN.KFV.cKronS_le

by xuanji · Oct 20, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

The Kronecker stage.

The following holds (Lean statement, in the scope of module Solutions.KFV59CostK):

theorem NLogN.KFV.cKronS_le (p ℓ n BK φ : ℕ) :
    cKronS p ℓ n BK φ ≤ 10 ^ 14 * (p + ℓ + 3) * (2 ^ n * 2 ^ ℓ) + 8000 * 2 ^ n * φ +
      2 * 10 ^ 9 * (BK + 2) ^ 2

Context. KFV §59: cost of the Kronecker stage, the shuttles and the scaling

cRec_le: one record product costs ≤ 10^8 (r+1)(b+Δ+p+10) + 4 φ;
cKronS_le: the Kronecker stage costs ≤ 10^14 (p+3) X + 8000 N φ + 10^9 (BK+2)^2;
cShut_le, cScale_le: ≤ 10^6 (B+2) X + 10^9 (B+2)^2.

Formalization Note. Part of the Lean formalization of the round-six κ-improvement (integer-multiplication-kappa campaign, target IntMul.Kappa.jain_round6_framework). The statement is copied verbatim from the Lean development; the platform node carries the same declaration name. Definitions it mentions live in the IntMul_Kappa_R6_Defs* bundles (on top of the published κ bundles IntMul_Kappa_K_Defs*).

Preamble

-- Round-six κ development (integer-multiplication-kappa), generated by skeleton subtraction.

import Definitions.Def_IntMul_Kappa_K_Defs33
import Definitions.Def_IntMul_Kappa_R6_Defs1
import Mathlib
import Definitions.Def_IntMul_HvdH_M_Defs8
import Definitions.Def_IntMul_MultitapeModel
import Definitions.Def_IntMul_HvdH_Numeric
import Definitions.Def_IntMul_HvdH_ExpApprox
import Definitions.Def_IntMul_HvdH_RealExp
import Definitions.Def_IntMul_HvdH_Twiddle
import Definitions.Def_IntMul_HvdH_MachineRef

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open NLogN NLogN.KFV
open IntMul NLogN.RS NLogN.KIL NLogN.KFD NLogN.KFB NLogN.KFH NLogN.KFC NLogN.KFT

Formal statement

theorem NLogN.KFV.cKronS_le (p ℓ n BK φ : ℕ) :
    cKronS p ℓ n BK φ ≤ 10 ^ 14 * (p + ℓ + 3) * (2 ^ n * 2 ^ ℓ) + 8000 * 2 ^ n * φ +
      2 * 10 ^ 9 * (BK + 2) ^ 2 := by sorry

Source

S. Jain, round-six networks for the integer-multiplication κ-improvement campaign (https://github.com/Swapnil-jain/integer-mult-kappa, commit 5868dc7) and D. Colkitt, integer-mult-bounds (https://github.com/CrocSwap/integer-mult-bounds); target `IntMul.Kappa.jain_round6_framework`. Lean declaration `NLogN.KFV.cKronS_le`, Solutions/KFV59CostK.lean (lines 48–101) of the round-six development.

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