The near-identity hypotheses give positive pivots (round six, `pivPos_of`)

Proved

J6Seg.pivPos_of

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

complexity-theoryinteger-multiplicationkappa-improvement

The near-identity hypotheses give positive pivots.

The following holds (Lean statement, in the scope of module Solutions.J6SegLn10):

theorem J6Seg.pivPos_of : ∀ (L : List ℕ) (A : ℕ → ℕ → ℤ)
    (b : ℕ → ℤ) (η : ℝ), L.Nodup → 0 ≤ η → η + (L.length : ℝ) ^ 2 / 2 ^ P ≤ 1 / 2 →
    (∀ i ∈ L, ∑ k ∈ L.toFinset, |(A i k : ℝ) / 2 ^ P - kd i k| ≤ η) →
    (∀ i ∈ L, ∀ k ∈ L, ¬ S i k → A i k = 0) → PatOK S L → PivPos P S L A

Context. Segmented inverse (S2), machine side 10: the solver maps zero to zero

B₀ must send an all-zero line to an all-zero line (the padded entries of the tensor stay exactly zero). The only non-obvious step is the rounded division fdiv P 0 d = 0, true for d ≥ 0: every pivot of the elimination is positive in the near-identity regime.

PivPos, geSolveS_zero: nonnegative pivots give geSolveS … 0 = 0;
pivPos_of: the hypotheses of geSolveS_eq give nonnegative pivots (same induction);
segInvS_zero (hypotheses of segInvS_eq), segInvFullS_zero (parameter hypotheses of segInvFullS_err).

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_Defs61

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg
open Finset Real
open IntMul.HvdH IntMul.HvdH.JPf J6QLU
variable (P : ℕ) (S : ℕ → ℕ → Prop) [DecidableRel S]

Formal statement

theorem J6Seg.pivPos_of : ∀ (L : List ℕ) (A : ℕ → ℕ → ℤ)
    (b : ℕ → ℤ) (η : ℝ), L.Nodup → 0 ≤ η → η + (L.length : ℝ) ^ 2 / 2 ^ P ≤ 1 / 2 →
    (∀ i ∈ L, ∑ k ∈ L.toFinset, |(A i k : ℝ) / 2 ^ P - kd i k| ≤ η) →
    (∀ i ∈ L, ∀ k ∈ L, ¬ S i k → A i k = 0) → PatOK S L → PivPos P S L A := 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.pivPos_of`, Solutions/J6SegLn10.lean (lines 50–176) of the round-six development.

View graph