The masked xor as a sum over the mask positions

The masked xor as a sum over the mask positions.

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

theorem NLogN.KFR.xor_mask_sub {K ρ f L Y X : ℕ} (hK : 1 ≤ K) (hρ : ρ < K) (hL : f * K ≤ L)
    (hY : Y < 2 ^ L) (hX : X < 2 ^ L) :
    ((Y ^^^ (X &&& KFH.selMask K ρ f) : ℕ) : ℤ) - Y =
      ∑ u ∈ Finset.range f, (bitZ (Y ^^^ X) (ρ + u * K) - bitZ Y (ρ + u * K)) * 2 ^ (ρ + u * K)

Context. The lemma belongs to the part KFR: bit arithmetic for the packed row addition.

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_Defs22.