Round six, complex side: cost bound O(V e^σ) for histogram networks

Proved

IntMul.Kappa.R6.Cx.recCostH_le

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

complexity-theoryinteger-multiplicationkappa-improvement

Let $J$ be a round-six complex network with child-width histogram $h$ , $m \ge 2$ , $0 < \sigma \le 1$ , $L_{\mathrm{loc}} \ge 0$ , all widths in $(0, m)$ , and suppose histMoment h m σ < W. Then there is $K$ such that for all $e$ and $V \ge 0$

\mathrm{recCostH}(J, L_{\mathrm{loc}}, e, V) \le K\, V\, e^{\sigma}.

In Lean:

theorem IntMul.Kappa.R6.Cx.recCostH_le (J : Prop8IfaceH W T m h) {Lloc : ℝ} (hL : 0 ≤ Lloc) {σ : ℝ}
    (hm : 2 ≤ m) (hσ0 : 0 < σ) (hσ1 : σ ≤ 1) (hw : ∀ p ∈ h, 0 < p.1 ∧ p.1 < m)
    (hmom : histMoment h m σ < W) :
    ∃ K : ℝ, ∀ (e : ℕ) (V : ℝ), 0 ≤ V → recCostH J Lloc e V ≤ K * V * (e : ℝ) ^ σ

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.recCostH_le (J : Prop8IfaceH W T m h) {Lloc : ℝ} (hL : 0 ≤ Lloc) {σ : ℝ}
    (hm : 2 ≤ m) (hσ0 : 0 < σ) (hσ1 : σ ≤ 1) (hw : ∀ p ∈ h, 0 < p.1 ∧ p.1 < m)
    (hmom : histMoment h m σ < W) :
    ∃ K : ℝ, ∀ (e : ℕ) (V : ℝ), 0 ≤ V → recCostH J Lloc e V ≤ K * V * (e : ℝ) ^ σ := 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.recCostH_le`, Solutions/R6CxCost.lean (lines 98–159) of the round-six development.

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