The scale pad implements the identity with scale shift `e` (round six, `sImplQ_scale`)

Proved

IntMul.Kappa.R6.IL3H.sImplQ_scale

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

complexity-theoryinteger-multiplicationkappa-improvement

The scale pad implements the identity with scale shift e.

The following holds (Lean statement, in the scope of module Solutions.R6IL3Pad):

theorem IntMul.Kappa.R6.IL3H.sImplQ_scale {Isym : List Sym} {ι : Type*} [Fintype ι] [DecidableEq ι] {lo hi r : ℕ}
    {st : ι → ℕ} {n : ℕ} (hr : 0 < r) (hn : 0 < n)
    {σ : Fin HS → Option ℤ} (hlt : ∀ i v, σ i = some v → i.val < 56) (hoff : OffG σ)
    {xL xQ ρ ρE : Fin HS} (hL : σ xL = some ((2 * (n * r) : ℕ) : ℤ))
    (hQ : σ xQ = some ((2 * (n * r) : ℕ) : ℤ)) {e : ℕ} (hR : WReg σ ρ ρE (n * r) e) (Z : ℕ)
    (hZ2 : zsz ((2 * (2 * (n * r)) : ℕ) : ℤ) ≤ Z) (hZ : zsz ((2 * (n * r) : ℕ) : ℤ) ≤ Z) :
    SImplQ Isym iprog (fun s => Agree σ s) lo hi r st n 1 (scaleQ xL xQ ρ ρE) e ((e + 2) * 2)
      (fun w => 400 * (cPadC (2 * (n * r)) (2 * (n * r)) Z w ((e + 2) * 2)
        (cWrapQ 2 (n * r) Z e 0 (fun _ => 1)) + 0) + cWrapQ 2 (n * r) Z e 0 (fun _ => 1) w)

Context. Round six, item language (I3): the scale pad

scaleQ xL xQ ρ ρE = padCU xL xQ (wrapQ 2 1 ρ ρE .skip): append n zero records, swap the zero half to the front, level the data half by 2^e, swap back, drop the zeros.

wrapQ_zero: the wrapper with U = skip on a zero role (zeros sit at every scale);
sImplQ_scale: scaleQ implements the identity with scale shift e.

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_Defs21

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6 IntMul.Kappa.R6.IL3H
open IntMul NLogN NLogN.KIL NLogN.KFD NLogN.KFB NLogN.KFG NLogN.KFH NLogN.KFN NLogN.KFW Complex P8Lift Butterfly
variable {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp}

Formal statement

theorem IntMul.Kappa.R6.IL3H.sImplQ_scale {Isym : List Sym} {ι : Type*} [Fintype ι] [DecidableEq ι] {lo hi r : ℕ}
    {st : ι → ℕ} {n : ℕ} (hr : 0 < r) (hn : 0 < n)
    {σ : Fin HS → Option ℤ} (hlt : ∀ i v, σ i = some v → i.val < 56) (hoff : OffG σ)
    {xL xQ ρ ρE : Fin HS} (hL : σ xL = some ((2 * (n * r) : ℕ) : ℤ))
    (hQ : σ xQ = some ((2 * (n * r) : ℕ) : ℤ)) {e : ℕ} (hR : WReg σ ρ ρE (n * r) e) (Z : ℕ)
    (hZ2 : zsz ((2 * (2 * (n * r)) : ℕ) : ℤ) ≤ Z) (hZ : zsz ((2 * (n * r) : ℕ) : ℤ) ≤ Z) :
    SImplQ Isym iprog (fun s => Agree σ s) lo hi r st n 1 (scaleQ xL xQ ρ ρE) e ((e + 2) * 2)
      (fun w => 400 * (cPadC (2 * (n * r)) (2 * (n * r)) Z w ((e + 2) * 2)
        (cWrapQ 2 (n * r) Z e 0 (fun _ => 1)) + 0) + cWrapQ 2 (n * r) Z e 0 (fun _ => 1) w) := 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 `IntMul.Kappa.R6.IL3H.sImplQ_scale`, Solutions/R6IL3Pad.lean (lines 175–300) of the round-six development.

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