Round-six lemma `rchunk_sound`

Proved

R6Chk.Bit.rchunk_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.R6ChkBitDag):

theorem R6Chk.Bit.rchunk_sound {sg : ℕ → ℕ} {x : ℕ} (h : rchunkChk sg x = true) :
    ∀ r ∈ rchunkDec x, sg r.2 = star r.1

Context. Bit round six (h = 23): DAG well-formedness and output supports (kernel checkers)

For a support function sg : ℕ → ℕ (node ↦ bitmask over the 1771 triples; in use sg = supG, a lookup in the support tree):

gchunkChk sg x: every gate record (k, a, b) of the chunk has disjoint operand supports whose union is the support of k (gchunk_sound);
ochunkChk sg x: every output record (c, t₁, t₂, n) has sg n = star c \ (star t₁ ∪ star t₂) (triples through c meeting {c,t₁,t₂} only in c);
rchunkChk sg x: every retained record (c, n) has sg n = star c;
leafChk sg: sg 0 = 0 and sg (i+1) = 2^i (input triple i).

These are the data facts behind JLV = I over ZMod 2 (the scalar word) and behind label nesting (supports grow along every slot chain since sg a, sg b ⊆ sg k).

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 R6Chk.Bit
open R6Chk

Formal statement

theorem R6Chk.Bit.rchunk_sound {sg : ℕ → ℕ} {x : ℕ} (h : rchunkChk sg x = true) :
    ∀ r ∈ rchunkDec x, sg r.2 = star r.1 := 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.Bit.rchunk_sound`, Solutions/R6ChkBitDag.lean (lines 117–118) of the round-six development.

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