permRecs_recRep (KFR: `permRecs` realises the row operator `Pm i j`, and the statement `Lem13ILP`)

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

theorem NLogN.KFR.permRecs_recRep {sh : Shape} {π : List ℕ → List ℕ} {r F : ℕ} {zs : List GI}
    {xs : List (Fin r → ℂ)} (h : RecRep r F zs xs) (hn : xs.length = vol sh)
    (hπ : ∀ q < vol sh, encode sh (π (decode sh q)) < vol sh) :
    RecRep r F (permRecs sh π r zs)
      ((List.range (vol sh)).map fun q => xs.getD (encode sh (π (decode sh q))) 0)

Context. The lemma belongs to the part KFR: permRecs realises the row operator Pm i j, and the statement Lem13ILP.

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_Defs22.