A single transposition contributes rank at most one to the `H × D` block (`rank_hdB_swap_le`)

Proved

IntMul.Kappa.R6.Bit.rank_hdB_swap_le

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

complexity-theoryinteger-multiplicationkappa-improvement

A single transposition contributes rank at most one to the H × D block.

The following holds (Lean statement, in the scope of module Solutions.R6BitNBObst):

theorem IntMul.Kappa.R6.Bit.rank_hdB_swap_le (a b : Fin (m + m)) :
    (hdB ((Equiv.swap a b).permMatrix K)).rank ≤ 1

Context. Round six, bit side: the label route prop11H_labels admits no saving

prop11H_labels (R6BitProp11) derives C.OK from EdgesOK, LabOK, ScalarOK: every gate touches wires with equal frame labels, and every role ends at label swapMat. This file proves that these hypotheses force the total child width to be at least W m:

the H × D block hdB of a frame matrix is subadditive in rank under products (rank_hdB_mul_le), vanishes on lower triangular Bruhat factors and on 1, and has rank m on swapMat;
an edge E₁ σ E₂ therefore has rank hdB ≤ #transpositions ≤ Σ run widths (rank_hdB_edgeMat_le);
along each role wire the labels go from 1 to swapMat, so each of the W (distinct, by ScalarOK) role wires carries edges of total width ≥ m (widths_sum_ge).

Consequently histMoment C.hist m τ ≥ W for every 0 < τ ≤ 1 (labels_no_saving): the hypotheses of prop11H_labels are jointly inconsistent (prop11H_labels_vacuous), and in particular no network with the histogram Jain.bitR6 (rank sum W m − 1344189) satisfies them (no_bitR6_labels).

The real two-stage network escapes this because its data outputs are sums of terms at different frames (B = F x + E y, fixed by the copy correction T F = E), which the single-label invariant cannot express. See R6BitNBFormal for the replacement calculus.

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_Defs5

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6 IntMul.Kappa.R6.Bit
open AModel AModelFac Matrix
variable {K : Type*} [Field K] {m : ℕ}

Formal statement

theorem IntMul.Kappa.R6.Bit.rank_hdB_swap_le (a b : Fin (m + m)) :
    (hdB ((Equiv.swap a b).permMatrix K)).rank ≤ 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 `IntMul.Kappa.R6.Bit.rank_hdB_swap_le`, Solutions/R6BitNBObst.lean (lines 83–130) of the round-six development.

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