The pipeline error

The pipeline error. F̃ is within Es·hp of the linear contraction F on grid inputs of norm ≤ 1/2; G = conj ∘ F ∘ conj. For grid u, v of norm ≤ 1/4 and Es·hp ≤ 1/100, z = Q_p(F̃u · F̃v) is a grid array of norm ≤ 1/2 and ‖F̃⁺ z - G(Fu · Fv)‖ ≤ (3Es + 2)hp.

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

theorem NLogN.KFM.pipeline_err (p : ℕ) {F Ft : (κ → ℂ) → (κ → ℂ)} (hF : LinC F) (Es : ℝ)
    (hFt : ∀ u, Grid p u → ‖u‖ ≤ 1 / 2 → ‖Ft u - F u‖ ≤ Es * hp p)
    (hEs : Es * hp p ≤ 1 / 100)
    {u v : κ → ℂ} (hu : Grid p u) (hv : Grid p v) (hu1 : ‖u‖ ≤ 1 / 4) (hv1 : ‖v‖ ≤ 1 / 4) :
    Grid p (QA p (Ft u * Ft v)) ∧ ‖QA p (Ft u * Ft v)‖ ≤ 1 / 2 ∧
    ‖plusT Ft (QA p (Ft u * Ft v)) - cnj (F (cnj (F u * F v)))‖ ≤ (3 * Es + 2) * hp p

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.