The tail weight bound (`ew_tail`)

Proved

IntMul.HvdH.SPf.ew_tail

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

complexity-theoryinteger-multiplicationkappa-improvement

The tail weight bound: outside the window, e_j ≤ r^{m-1} r^{|j - c|}.

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

theorem IntMul.HvdH.SPf.ew_tail (s t α k m : ℕ) (hα : 0 < α) (hm : 1 ≤ m) (c : ℤ) (hc1 : (c : ℝ) - 1 < (s * k : ℝ) / t)
    (hc2 : (s * k : ℝ) / t ≤ c) (j : ℤ) (hj : m + 1 ≤ (j - c).natAbs) :
    ew s t α k j ≤ Real.exp (-(Real.pi * m / (2 * α ^ 2))) ^ (m - 1) *
      Real.exp (-(Real.pi * m / (2 * α ^ 2))) ^ (j - c).natAbs

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_Defs60

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.HvdH IntMul.HvdH.SPf
open Complex
open IntMul.HvdH

Formal statement

theorem IntMul.HvdH.SPf.ew_tail (s t α k m : ℕ) (hα : 0 < α) (hm : 1 ≤ m) (c : ℤ) (hc1 : (c : ℝ) - 1 < (s * k : ℝ) / t)
    (hc2 : (s * k : ℝ) / t ≤ c) (j : ℤ) (hj : m + 1 ≤ (j - c).natAbs) :
    ew s t α k j ≤ Real.exp (-(Real.pi * m / (2 * α ^ 2))) ^ (m - 1) *
      Real.exp (-(Real.pi * m / (2 * α ^ 2))) ^ (j - c).natAbs := 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.HvdH.SPf.ew_tail`, Solutions/KFMBaseSap.lean (lines 217–263) of the round-six development.

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