The child step on the encoding, given the blockwise semantics of the called program

The child step on the encoding, given the blockwise semantics of the called program.

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

theorem NLogN.KFB.rep_child {W m n : ℕ} (w : Fin W) (h : Fin m) (U : Butterfly.Op (Fin n))
    (F : List GI → List GI) (hF : BlkT U F) (zs : List GI)
    (x : Fin W × (Fin m × Fin n → Fin 2) → ℂ) (hx : Rep zs x) :
    Rep (chL (2 ^ (m * n)) (w.val * 2 ^ (m * n)) (2 ^ n) (2 ^ (n * h.val)) F zs)
      ((P8Lift.RStep.childR w h U).mulVec x)

Context. The lemma belongs to the part KFB: the child step (a recursive call on one slot of one role): Role w's segment, as Gaussian indices (hi, s, lo) (s = the n bits of slot h), is rearranged to (hi, lo, s) by KFC's fwdF (q = 2^n, the real/imaginary pair kept as the last field), so that slot h becomes 2^{n(m-1)} contiguous blocks of 2^n entries; a program PT computing an operator T blockwise runs on the segment; bwdF moves the field back.

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.