The last-bit step on the geometry: with j = ρ + (f-1)K, D = 2^j, Q = 2^{fK} = 2 D E, the step computes…

The last-bit step on the geometry: with j = ρ + (f-1)K, D = 2^j, Q = 2^{fK} = 2 D E, the step computes rixT g (XLc K ρ f).

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

theorem NLogN.KFL.xorStep_geo {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp}
    {K ρ f : ℕ} (hρK : ρ < K) (hf : 1 ≤ f) (g : Geo) (hg : g.Pos)
    (hQe : g.Q = 2 ^ (f * K)) {Bb : ℕ} (hB2 : 2 ≤ Bb) (zV : zsz (g.V : ℤ) ≤ Bb) (w : ℕ)
    (t : IState HS) (xs : List ℤ) (R : List Item) (hlen : xs.length = g.V) (hbw : Bnd w xs)
    (h0 : t.G 0 = ints xs ++ R) :
    IRuns I iprog (xorStep (g.N1 * 2 ^ (f * K - (ρ + (f - 1) * K) - 1)) g.Q (ρ + (f - 1) * K)
        (g.N0 * g.Q * g.N1 * 2 ^ (f * K - (ρ + (f - 1) * K) - 1)) (2 ^ (ρ + (f - 1) * K) * g.S1)) t
      (t.upd 0 (ints (rixT g (XLc K ρ f) xs) ++ R))
      (cXor (g.N0 * g.Q * g.N1 * 2 ^ (f * K - (ρ + (f - 1) * K) - 1)) (2 ^ (ρ + (f - 1) * K) * g.S1)
        w Bb)

Context. The lemma belongs to the part KFL §8: the last selected bit (two-bit XOR): xorStep: table b(c₀) = bit j of c₀ (j = ρ + (f-1)K, xorGen), then rotF on the view [N0·Q·N1·E, 2, D·S1] (D = 2^j, Q = 2 D E): the bit j of the y chunk is flipped iff bit j of x is set. xorStep_geo: the result is rixT g (XLc K ρ f).

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.