cP14_le (KFH: Proposition 14 by individual kernels, as one fine IL program)

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

theorem NLogN.KFH.cP14_le (D N Z w : ℕ) (hD : 1 ≤ D) :
    cP14 D N Z w ≤ 1000000 * (D + 1) * (w + 8 * D + 8) * N + 100000 * (D + 1) * (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.