Pipeline: lines (FSc), correction (corrc), last bit (XLc) compose to the row addition RAc: rixT XLc (rixT…

Pipeline: lines (FSc), correction (corrc), last bit (XLc) compose to the row addition RAc: rixT XLc (rixT corrc (rixT FSc zs)) = rixT RAc zs.

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

theorem NLogN.KFL.pipeline_rixT {K ρ f : ℕ} (hK : 1 ≤ K) (hρK : ρ < K) (hf : 1 ≤ f) (g : Geo) (hg : g.Pos)
    (hQe : g.Q = 2 ^ (f * K)) (zs : List ℤ) :
    rixT g (XLc K ρ f) (rixT g (corrc K ρ f) (rixT g (FSc K ρ f) zs)) = rixT g (RAc K ρ f) zs

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.