Proposition 14 around any IStmt C-layer, with one-pass phases

Proposition 14 around any IStmt C-layer, with one-pass phases.

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

theorem NLogN.KFM.p14Ph_iruns (hρ : ρ < K) (hD : 1 ≤ D) {r p w : ℕ} (hr : 1 ≤ r) (hS : 1 ≤ S)
    (hP : 1 ≤ P) {Cc : ℕ → IStmt HS Pp} {E dC : ℕ} {kC : ℕ → ℕ}
    (hCimpl : SImplI Isym iprog r (fun i : Fin D => KFH.stT K D S ρ i)
      (vol (Butterfly.layerShape P K D S)) (Butterfly.tens Cm : Butterfly.Op (Fin D)) Cc E dC kC)
    (s : IState HS) (f : Fin HS → List ℤ) (zs : List GI) (l : List ℤ) (xs : List (Fin r → ℂ))
    (hx : RecRep r p zs xs) (hl : xs.length = vol (Butterfly.layerShape P K D S))
    (h0 : f 0 = gflat zs ++ l) (hC : ClN f) (hb : NLogN.KFB.GB w zs)
    (hdisk : ∀ j (h : j < xs.length), ‖xs[j]‖ ≤ 1) :
    ∃ zs', RecRep r p zs' (Butterfly.prop14Proc (P := P) (S := S) hρ p
        (Butterfly.tens Butterfly.Cm : Butterfly.Op (Fin D)) xs) ∧
      NLogN.KFB.GB (p + 1) zs' ∧
      (∀ j (h : j < (Butterfly.prop14Proc (P := P) (S := S) hρ p
          (Butterfly.tens Butterfly.Cm : Butterfly.Op (Fin D)) xs).length)
        (h' : j < (Butterfly.layerH P K D S ρ hρ xs).length), ∀ c,
        ‖(Butterfly.prop14Proc (P := P) (S := S) hρ p
          (Butterfly.tens Butterfly.Cm : Butterfly.Op (Fin D)) xs)[j] c -
          (Butterfly.layerH P K D S ρ hρ xs)[j] c‖ < Real.sqrt 2 * 2⁻¹ ^ p) ∧
      IRuns Isym iprog (p14Ph P K D S ρ r E dC Cc w) (encK s f)
        (encK s (Function.update f 0 (gflat zs' ++ l)))
        (kC w + cWrapPh P K D S r E (zsz ((16 * (vol (Butterfly.layerShape P K D S) * r) : ℕ) : ℤ))
          w dC)

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.