Phase products: ‖Q_p(ẽ·x) - e·x‖ ≤ 4hp and ‖Q_p(ẽ·x)‖ ≤ ‖x‖

Phase products: ‖Q_p(ẽ·x) - e·x‖ ≤ 4hp and ‖Q_p(ẽ·x)‖ ≤ ‖x‖.

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

theorem NLogN.KFM.phase_err (p : ℕ) (e et x : ι → ℂ) (he : ∀ i, ‖e i‖ ≤ 1)
    (het : ∀ i, ‖et i - e i‖ ≤ 2 * hp p) (het1 : ∀ i, ‖et i‖ ≤ 1) (hx : ‖x‖ ≤ 1) :
    ‖QA p (et * x) - e * x‖ ≤ 4 * hp p ∧ ‖QA p (et * x)‖ ≤ ‖x‖

Context. The lemma belongs to the part KFM 14: the error chain of the computed pipeline (§9.7.1), abstract form: Arrays are functions ι → ℂ with the sup norm; hp = 2^{-p}. Every computed procedure is an arbitrary function with an error bound on grid inputs (Grid p f: Q_p f = f), so the IL denotations (and P18IL) plug in later.

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_Defs20.