The forward procedure with one chunk per axis (round six, `cFwd_q1`)

Proved

NLogN.KFV.R6P.cFwd_q1

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

complexity-theoryinteger-multiplicationkappa-improvement

The forward procedure with one chunk per axis.

The following holds (Lean statement, in the scope of module Solutions.R6P18Lay):

theorem NLogN.KFV.R6P.cFwd_q1 (hK : 0 < K) (hℓ : 1 ≤ ℓ) (hb0 : bb ℓ K = 0) {W B cCh U : ℕ} (hDW : D ≤ W)
    (hS : 10 ^ 12 * (B + 2) ^ 3 ≤ 2 ^ ℓ) (hBW : B + 2 ≤ 6 * (W + 1)) (hDl : D * ℓ ≤ B)
    (hU : zsz (U : ℤ) ≤ B) :
    NLogN.KFV.cFwd D ℓ K long W B cCh U ≤
      10 ^ 12 * (W + 1) * (2 ^ nB D ℓ K long * 2 ^ ℓ) * (ℓ + 2 * (D + 1)) + ℓ * cCh

Context. Round six, Proposition 18: the layout and round costs with one chunk per axis

At the round-six parameters K = ℓ - 1 (Idea C), so q = ⌊(ℓ-1)/K⌋ = 1 and b = (ℓ-1) mod K = 0. Then

phase 3 of the layout has D steps whose exchange is trivial (xT t = t, cost 1): cLayIn_q1, cLayOut_q1: the layout programs cost ≤ 10^6 (D+1)(W+1) 2^{n+ℓ} (no exchange is ever called, so the swap's exponent does not enter the cost);
there are no prefix leaves: cFwd_q1, cOpp_q1: the Lemma 16 procedures cost ≤ 10^12 (W+1) X (ℓ + 2(D+1)) + ℓ · cCh.

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_SApprox
import Definitions.Def_IntMul_HvdH_DiagApprox
import Definitions.Def_IntMul_HvdH_EApprox
import Definitions.Def_IntMul_HvdH_JApprox
import Definitions.Def_IntMul_HvdH_BApprox
import Definitions.Def_IntMul_HvdH_FPApprox
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 Synth AModel AModelL12 GlueLayout GlueP14 GlueError
variable (D ℓ K : ℕ) [NeZero D] (long : Fin D → Bool)

Formal statement

theorem NLogN.KFV.R6P.cFwd_q1 (hK : 0 < K) (hℓ : 1 ≤ ℓ) (hb0 : bb ℓ K = 0) {W B cCh U : ℕ} (hDW : D ≤ W)
    (hS : 10 ^ 12 * (B + 2) ^ 3 ≤ 2 ^ ℓ) (hBW : B + 2 ≤ 6 * (W + 1)) (hDl : D * ℓ ≤ B)
    (hU : zsz (U : ℤ) ≤ B) :
    NLogN.KFV.cFwd D ℓ K long W B cCh U ≤
      10 ^ 12 * (W + 1) * (2 ^ nB D ℓ K long * 2 ^ ℓ) * (ℓ + 2 * (D + 1)) + ℓ * cCh := 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.R6P.cFwd_q1`, Solutions/R6P18Lay.lean (lines 140–185) of the round-six development.

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