Moment form (round six, `labels_no_saving`)

Moment form. Under the hypotheses of prop11H_labels, histMoment C.hist m τ ≥ W for every 0 < τ ≤ 1.

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

theorem IntMul.Kappa.R6.Bit.QHNet.labels_no_saving [Nontrivial R] (C : QHNet R W T m) (hm : 0 < m) {q : ℕ} (hq : q.Prime)
    (hruns : C.RunsOK) (hE : C.EdgesOK q) (hlab : C.LabOK q) (hsc : C.ScalarOK) {τ : ℝ}
    (_hτ0 : 0 < τ) (hτ1 : τ ≤ 1) : (W : ℝ) ≤ histMoment C.hist m τ

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*).