Round-six lemma `entChk_sound`

Proved

R6Chk.entChk_sound

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

complexity-theoryinteger-multiplicationkappa-improvement

An auxiliary lemma of the round-six development, published as a separate node because its proof is long or shared.

The following holds (Lean statement, in the scope of module Solutions.R6ChkCx):

theorem R6Chk.entChk_sound (hbad : bad = BADf n) (hn : n ≤ 24) {kV nH H s : ℕ} (hnk : nH + kV = n)
    (hH : H < 2 ^ (32 * nH)) (hHr : ∀ i < nH, row H i < 2 ^ n)
    {SV : Submodule (ZMod 2) (Fin n → ZMod 2)}
    (hmem : ∀ x, x ∈ SV ↔ ∀ i : Fin nH, vr n H i ⬝ᵥ x = 0) (hV : NondegF SV)
    (hVd : finrank (ZMod 2) SV = kV) (h : entChk bad kV nH H s = true) :
    spanP n (tk (nx s) (f32 s)) (f32 s) ≤ SV ∧
      finrank (ZMod 2) (spanP n (tk (nx s) (f32 s)) (f32 s)) = f32 s ∧
      NondegF (spanP n (tk (nx s) (f32 s)) (f32 s)) ∧
      (spanP n (tk (nx s) (f32 s)) (f32 s) = SV ∨
        ∃ w ∈ SV, w ⬝ᵥ w = 1 ∧ ∀ z ∈ spanP n (tk (nx s) (f32 s)) (f32 s), z ⬝ᵥ w = 0)

Context. Reflective checker for nested nondegenerate F₂ label pairs (complex round six)

Data: a list of groups. A group is one natural number read as 32-bit fields (LSB first): kV, cnt, B (kV rows), H (n-kV), D (kV), E (n-kV), then cnt entries kU, U (kU), EU (kU), R (kV-kU). Rows are vectors of F₂^n (n ≤ 24).

grpChk n bad G : Bool — two packed products for the label V = span B (H·[B;E]ᵀ = [0 | I], D·[H;B]ᵀ = [0 | I]) and two per entry (U·[H;EU]ᵀ = [0 | I], R·[H;U;R]ᵀ = [0 | 0 | I]), plus row range checks.
grpChk_sound: if grpChk n (BADf n) G = true then V is nondegenerate of dimension kV, and for every entry (U, kU), PairOK n U kU V kV: span U ≤ span V, dim span U = kU, span U nondegenerate, and span U = span V or some w ∈ span V has w·w = 1, w ⊥ U (i.e. NetComplexP8.Up (span U) (span V) with Fd = kV - kU; see R6ChkCxBridge).

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_Defs9

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open R6Chk
open Matrix Matrix.Module Module
variable {n bad : ℕ}

Formal statement

theorem R6Chk.entChk_sound (hbad : bad = BADf n) (hn : n ≤ 24) {kV nH H s : ℕ} (hnk : nH + kV = n)
    (hH : H < 2 ^ (32 * nH)) (hHr : ∀ i < nH, row H i < 2 ^ n)
    {SV : Submodule (ZMod 2) (Fin n → ZMod 2)}
    (hmem : ∀ x, x ∈ SV ↔ ∀ i : Fin nH, vr n H i ⬝ᵥ x = 0) (hV : NondegF SV)
    (hVd : finrank (ZMod 2) SV = kV) (h : entChk bad kV nH H s = true) :
    spanP n (tk (nx s) (f32 s)) (f32 s) ≤ SV ∧
      finrank (ZMod 2) (spanP n (tk (nx s) (f32 s)) (f32 s)) = f32 s ∧
      NondegF (spanP n (tk (nx s) (f32 s)) (f32 s)) ∧
      (spanP n (tk (nx s) (f32 s)) (f32 s) = SV ∨
        ∃ w ∈ SV, w ⬝ᵥ w = 1 ∧ ∀ z ∈ spanP n (tk (nx s) (f32 s)) (f32 s), z ⬝ᵥ w = 0) := 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 `R6Chk.entChk_sound`, Solutions/R6ChkCx.lean (lines 162–229) of the round-six development.

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