The kernel pass represents the kernel (25) of A on selected bit i

The kernel pass represents the kernel (25) of A on selected bit i.

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

theorem NLogN.KFH.ker_rep (hρ : ρ < K) (i : Fin D) {M : M2G} {A : Butterfly.M2} {e : ℕ} (hA : KerOk M A e)
    {r F : ℕ} {zs : List GI} {xs : List (Fin r → ℂ)} (hx : RecRep r F zs xs)
    (hl : xs.length = vol (Butterfly.layerShape P K D S)) :
    RecRep r (F + e) (qopL [M] (stT K D S ρ i * r) (grpG P K ρ i) zs)
      (Butterfly.selOp (Butterfly.layerSel P K D S ρ hρ) (Butterfly.tensF (Butterfly.kerAt i A)) xs)

Context. The lemma belongs to the part KFH: an individual selected-bit kernel as one qop pass on a record array: The layer layout of Proposition 14 is [P] × [2^K]^D × [S] (Butterfly.layerShape), records in lexicographic order. Bit ρ of chunk i has record stride stT = 2^ρ · vol(suffix after chunk i): the address with that bit flipped is j ± stT (layer_partner). On the flat Gaussian list (records of r coefficients) the paired entries are at distance B1 = stT · r, in blocks of 2 B1; this is exactly the two-stream pass KFB.qop [M] B1 G with G = P · 2^{K i} · 2^{K-ρ-1} groups.

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.