Partner addresses: the selected bit i of record j and the record with it set to b

Partner addresses: the selected bit i of record j and the record with it set to b.

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

theorem NLogN.KFH.layer_partner (hρ : ρ < K) (i : Fin D) {j : ℕ}
    (hj : j < vol (Butterfly.layerShape P K D S)) (b : Fin 2) :
    ((Butterfly.layerSel P K D S ρ hρ).bits (decode (Butterfly.layerShape P K D S) j) i : ℕ) =
      j / stT K D S ρ i % 2 ∧
    ((encode (Butterfly.layerShape P K D S) ((Butterfly.layerSel P K D S ρ hρ).set
      (decode (Butterfly.layerShape P K D S) j)
      (Function.update ((Butterfly.layerSel P K D S ρ hρ).bits
        (decode (Butterfly.layerShape P K D S) j)) i b)) : ℕ) : ℤ) =
      j + ((b : ℤ) - (j / stT K D S ρ i % 2 : ℕ)) * stT K D S ρ i

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.