Proposition 14 (leaf branch), IL run with the paper's guarantees

Proposition 14 (leaf branch), IL run with the paper's guarantees. The output represents prop14Proc … (tens Cm) z = Q_p(H0^{⊗D} z); for disk inputs every output coefficient is in the disk, within √2·2^{-p} of the exact layer, and its numerators have at most p + 1 bits.

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

theorem NLogN.KFH.p14Leaf_iruns {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp}
    (hρ : ρ < K) (hD : 1 ≤ D) {r p w : ℕ} (hr : 1 ≤ r) (hS : 1 ≤ S) (hP : 1 ≤ P)
    (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 : Cl 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 I iprog (fcode (p14Leaf P K D S ρ r)) (encK s f)
        (encK s (Function.update f 0 (gflat zs' ++ l)))
        (cP14 D (vol (Butterfly.layerShape P K D S) * r)
          (zsz ((12 * (vol (Butterfly.layerShape P K D S) * r) : ℕ) : ℤ)) w) ∧
      cP14 D (vol (Butterfly.layerShape P K D S) * r)
          (zsz ((12 * (vol (Butterfly.layerShape P K D S) * r) : ℕ) : ℤ)) w ≤
        1000000 * (D + 1) * (w + 8 * D + 8) * (vol (Butterfly.layerShape P K D S) * r) +
        100000 * (D + 1) * (zsz ((12 * (vol (Butterfly.layerShape P K D S) * r) : ℕ) : ℤ) + 2)

Context. The lemma belongs to the part KFH: Proposition 14 by individual kernels, as one fine IL program: LImpl T code e dw cost: on any clean managed state holding a record array zs (scale F, width w) that represents xs on the layer layout, code leaves an array representing selOp T xs at scale F + e and width w + dw, at cost cost w. Implementations compose (LImpl.comp: matrix product, scales and widths add), and each kernel pass is one (limpl_ker), so A^{⊗D} is D passes (limpl_tens, via Butterfly.tens_eq_prod_kerAt).

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_Defs18.