One rounded product

One rounded product.

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

theorem NLogN.KFM.step_err (P : ℕ) (z w : GI) (Z W : ℂ) (hZ : ‖Z‖ ≤ 1) (hW : ‖W‖ ≤ 1) {a b : ℝ}
    (ha : ‖vP P z - Z‖ ≤ a) (ha1 : a ≤ 1) (hb : ‖vP P w - W‖ ≤ b) :
    ‖vP P (gmulT P z w) - Z * W‖ ≤ a + 2 * b + 2 * (2 : ℝ)⁻¹ ^ P

Context. The lemma belongs to the part KFM 9b: fixed-point root tables by FM product recurrences — denotations and error bounds: All values are Gaussian numerators at scale 2^P (toC z / 2^P); products are gmulT P z w = Q_P(z · w) (the per-entry operation of KFMPw.gmulP).

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.