Residual of the elimination (round six, `geSolve_resid`)

Proved

J6Seg.geSolve_resid

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

complexity-theoryinteger-multiplicationkappa-improvement

Residual of the elimination. If A/2^P is within η of the identity in row sums on L, η + |L|²/2^P ≤ 1/2, and the computed solution is bounded by Z on L, then every residual (A z - b)_i / 2^P (i ∈ L) is at most (Z+1)(|L|+1)²/2^P.

The following holds (Lean statement, in the scope of module Solutions.J6SegFix):

theorem J6Seg.geSolve_resid : ∀ (L : List ℕ) (A : ℕ → ℕ → ℤ) (b : ℕ → ℤ) (η Z : 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) →
    (∀ k ∈ L, |(geSolve P L A b k : ℝ) / 2 ^ P| ≤ Z) →
    ∀ i ∈ L, |∑ k ∈ L.toFinset, (A i k : ℝ) / 2 ^ P * ((geSolve P L A b k : ℝ) / 2 ^ P) -
      (b i : ℝ) / 2 ^ P| ≤ (Z + 1) * ((L.length : ℝ) + 1) ^ 2 / 2 ^ P

Context. Segmented inverse (S2): fixed-point primitives and elimination without pivoting

Fixed-point numbers of precision P are integers a standing for a / 2^P.

rq a b = ⌊a/b + 1/2⌋ (rounded quotient), rs P a = rq a (2^P), fmul P x y = rs P (x*y) and fdiv P x y = rq (x 2^P) y (errors ≤ 1/2 ulp).
geSolve P L A b: Gaussian elimination without pivoting in the order of the index list L, written as forward elimination of the head pivot followed by back substitution.
schur_row (report L9): if every row of A - I has absolute sum ≤ η ≤ 1/2, the same holds for the Schur complement after eliminating one pivot; hence for every Schur complement, and no pivoting is ever needed (any order, any sparsity pattern, cyclic borders included).
near_id: ‖A - I‖_∞ ≤ η < 1 and ‖A z - r‖ ≤ ρ give ‖z‖ ≤ (‖r‖ + ρ)/(1 - η).
geSolve_resid: the computed solution has residual ≤ (Z+1)(|L|+1)²/2^P; geSolve_bound: and |z| ≤ 4B + 1.

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_Defs3

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg
open Finset
variable (P : ℕ)

Formal statement

theorem J6Seg.geSolve_resid : ∀ (L : List ℕ) (A : ℕ → ℕ → ℤ) (b : ℕ → ℤ) (η Z : 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) →
    (∀ k ∈ L, |(geSolve P L A b k : ℝ) / 2 ^ P| ≤ Z) →
    ∀ i ∈ L, |∑ k ∈ L.toFinset, (A i k : ℝ) / 2 ^ P * ((geSolve P L A b k : ℝ) / 2 ^ P) -
      (b i : ℝ) / 2 ^ P| ≤ (Z + 1) * ((L.length : ℝ) + 1) ^ 2 / 2 ^ P := 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.geSolve_resid`, Solutions/J6SegFix.lean (lines 186–426) of the round-six development.

View graph