renorm_rep (KFH: the Proposition 14 wrapper around an arbitrary `C^{⊗D}` implementation)

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

theorem NLogN.KFH.renorm_rep {r F Δ : ℕ} {zs : List GI} {xs : List (Fin r → ℂ)}
    (h : RecRep r (F + Δ) zs xs) (hd : ∀ z ∈ zs, (2 : ℤ) ^ Δ ∣ z.1 ∧ (2 : ℤ) ^ Δ ∣ z.2) :
    RecRep r F (zs.map (gtr Δ)) xs

Context. The lemma belongs to the part KFH: the Proposition 14 wrapper around an arbitrary C^{⊗D} implementation: The recursion of §5.3 replaces the D individual C kernels by the batched network. The interface between the two is LImpl … (tens Cm) Cprog D dC kC: any fine IL code that maps a record array at scale F to the C-layer at scale F + D (exact, the D halvings of the kernels (25) kept as scale), widths +dC, cost kC w.

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.