The native chunk pair on every axis (round six, `chunk_native`)

Proved

J6Seg.PJ.chunk_native

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

complexity-theoryinteger-multiplicationkappa-improvement

The native chunk pair on every axis.

The following holds (Lean statement, in the scope of module Solutions.J6SegILPreJC15):

theorem J6Seg.PJ.chunk_native (P : Prm) (hP : P.Ok) (pl : Plan6) {Zn : Fin HS} {oN : Fin 19 → Fin HS}
    (hZ : Zn ∉ rg) (hZ5 : Zn.val < 56) (hZT : Zn ≠ sTmp) (hZC : Zn ≠ cRdC) (hZ2 : Zn ≠ sT2)
    (hZx : Zn ≠ J6Seg.PJ.sSt ∧ Zn ≠ sX ∧ Zn ≠ sOut ∧ Zn ≠ sLn ∧ Zn ≠ cLC ∧ Zn ≠ 13 ∧ Zn ≠ 51 ∧ Zn ≠ 53 ∧
      Zn ≠ 54 ∧ Zn ≠ PreT.Cu)
    (hinj : Injective oN) (hlt : ∀ i, (oN i).val < 56)
    (hx : ∀ i, oN i ≠ sX ∧ oN i ≠ sLn ∧ oN i ≠ sOut ∧ oN i ≠ sT2 ∧ oN i ≠ cLC)
    (hZo : ∀ i, oN i ≠ Zn) :
    ∃ n₁ : ℕ, ∀ n, n₁ ≤ n → ∀ src : ℕ → ℕ, SrcOK6 P n src → ∀ D : ℕ, dP6 P n = D + 1 →
    ∀ k ≤ D, ∀ A' : ℕ, ∃ cU cD : ℕ,
      (∀ s0, Sp (eCn pl Zn oN) (encK s0 (F0 Zn (descAx6 P n src D k)))
        (encK s0 (extend oN (chunkRegs6 P n src D k pl) (F0 Zn (descAx6 P n src D k)))) cU) ∧
      (∀ s0, Sp (tearP oN) (encK s0 (extend oN (chunkRegs6 P n src D k pl)
        (F0 Zn (descAx6 P n src D k)))) (encK s0 (F0 Zn (descAx6 P n src D k))) cD) ∧
      ((cU + 2 : ℕ) : ℝ) ≤ 2 * (cU6 P n A' / (D + 1)) ∧
      ((cD + 2 : ℕ) : ℝ) ≤ 2 * (cU6 P n A' / (D + 1))

Context. B5 chunk side, part 15: the native chunk pair on every axis, within budget

chunk_native: eventually, on every axis, eCn builds chunkRegs6 from the descriptor and tearP clears it, each at cost + 2 ≤ 2 · cU6 P n A' / (D + 1).

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_R6_Defs71

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg J6Seg.PJ
set_option maxRecDepth 100000
open IntMul NLogN NLogN.KIL NLogN.RS Function IntMul.Kappa.R6 J6Seg.Side

Formal statement

theorem J6Seg.PJ.chunk_native (P : Prm) (hP : P.Ok) (pl : Plan6) {Zn : Fin HS} {oN : Fin 19 → Fin HS}
    (hZ : Zn ∉ rg) (hZ5 : Zn.val < 56) (hZT : Zn ≠ sTmp) (hZC : Zn ≠ cRdC) (hZ2 : Zn ≠ sT2)
    (hZx : Zn ≠ J6Seg.PJ.sSt ∧ Zn ≠ sX ∧ Zn ≠ sOut ∧ Zn ≠ sLn ∧ Zn ≠ cLC ∧ Zn ≠ 13 ∧ Zn ≠ 51 ∧ Zn ≠ 53 ∧
      Zn ≠ 54 ∧ Zn ≠ PreT.Cu)
    (hinj : Injective oN) (hlt : ∀ i, (oN i).val < 56)
    (hx : ∀ i, oN i ≠ sX ∧ oN i ≠ sLn ∧ oN i ≠ sOut ∧ oN i ≠ sT2 ∧ oN i ≠ cLC)
    (hZo : ∀ i, oN i ≠ Zn) :
    ∃ n₁ : ℕ, ∀ n, n₁ ≤ n → ∀ src : ℕ → ℕ, SrcOK6 P n src → ∀ D : ℕ, dP6 P n = D + 1 →
    ∀ k ≤ D, ∀ A' : ℕ, ∃ cU cD : ℕ,
      (∀ s0, Sp (eCn pl Zn oN) (encK s0 (F0 Zn (descAx6 P n src D k)))
        (encK s0 (extend oN (chunkRegs6 P n src D k pl) (F0 Zn (descAx6 P n src D k)))) cU) ∧
      (∀ s0, Sp (tearP oN) (encK s0 (extend oN (chunkRegs6 P n src D k pl)
        (F0 Zn (descAx6 P n src D k)))) (encK s0 (F0 Zn (descAx6 P n src D k))) cD) ∧
      ((cU + 2 : ℕ) : ℝ) ≤ 2 * (cU6 P n A' / (D + 1)) ∧
      ((cD + 2 : ℕ) : ℝ) ≤ 2 * (cU6 P n A' / (D + 1)) := 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 `J6Seg.PJ.chunk_native`, Solutions/J6SegILPreJC15.lean (lines 114–346) of the round-six development.

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