Round six, complex side: the recursive C-layer computes C^{⊗e}

Proved

IntMul.Kappa.R6.Cx.recOpH_eq

by xuanji · Oct 19, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

For every round-six complex network $J$ (Prop8IfaceH W T m h) and every number of axes $e$ , the operator computed by the recursive C-layer equals the $e$ -fold tensor power of the basic $2\times 2$ operator $C$ :

\mathrm{recOpH}(J, e) = C^{\otimes e}.

Widths $e$ that are not powers of $m$ are handled by splitting $e = m f + r$ .

In Lean:

theorem IntMul.Kappa.R6.Cx.recOpH_eq (J : Prop8IfaceH W T m h) : ∀ e, recOpH J e = tens Cm

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_Defs2

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6 IntMul.Kappa.R6.Cx
open Matrix Finset Complex Butterfly P8Lift
variable {W T m : ℕ} {h : List (ℕ × ℕ)} [NeZero W]

Formal statement

theorem IntMul.Kappa.R6.Cx.recOpH_eq (J : Prop8IfaceH W T m h) : ∀ e, recOpH J e = tens Cm := 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.Cx.recOpH_eq`, Solutions/R6CxRec.lean (lines 124–137) of the round-six development.

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