S_D in one pass on the layer layout

S_D in one pass on the layer layout.

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

theorem NLogN.KFM.sImpl_SD {P K D S ρ r : ℕ} (hρ : ρ < K) (hr : 1 ≤ r) (hS : 1 ≤ S) (hP : 1 ≤ P) (Z : ℕ)
    (hZ : zsz ((16 * (vol (Butterfly.layerShape P K D S) * r) : ℕ) : ℤ) ≤ Z) :
    SImpl r (fun i : Fin D => KFH.stT K D S ρ i) (vol (Butterfly.layerShape P K D S))
      (SD : Matrix (Fin D → Fin 2) (Fin D → Fin 2) ℂ) (sdP P K D S ρ r) 0 0
      (fun w => cSD P K D S r Z w)

Context. The lemma belongs to the part KFM 2: Proposition 14 with the phase diagonals S_D in one pass each: KFN.p14WithI implements S_D = diag i^{x_1+⋯+x_D} as D kernel passes (kersP … MS), so one simultaneous butterfly round costs Θ(D · N · w) even before the C-layer. Lemma 16 runs ≈ ℓ such rounds with D ≈ d, giving ℓ d V ≈ V log T: a full logarithmic factor, which Ncost (ℓ d^{λ'} for this row) cannot absorb (MACHINE-PLAN.md, item 4, issue (A)).

Formalization Note. This lemma belongs to the item-language (IL) layer of the integer-multiplication-kappa campaign: tape-level programs (register programs over 64 item stacks of the NLogN multitape framework, definition bundles IntMul_HvdH_M_Defs1–IntMul_HvdH_M_Defs8) that implement the data-movement primitives of the OpenAI preprint Integer multiplication below $n\log n$ (field moves, row split/merge, shifts, the uniform Proposition 11 interchange, Lemma 12 with fine $O(1)$-per-element costs, Gaussian record primitives, Proposition 14 pieces) with exact list denotations from the array-model bundles IntMul_Kappa_K_Defs* and explicit cost bounds. All names refer to the definition bundles up to IntMul_Kappa_K_Defs20.