Error of M̃_C

Error of M̃_C. If M̃r is within Er·hp of the bilinear contraction Mr on grid inputs of norm ≤ 1, then for ‖f‖ ≤ 1, ‖g‖ ≤ 7/10 and Er·hp ≤ 3/10, ‖M̃_C(f,g) - θ⁻·Mr(θ f, θ g)‖ ≤ (Er + 12)hp and ‖M̃_C(f,g)‖ ≤ 7/10 + Er·hp.

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

theorem NLogN.KFM.mc_err (p : ℕ) {Mr Mrt : (ι → ℂ) → (ι → ℂ) → (ι → ℂ)} (hMr : BilC Mr) (Er : ℝ)
    (hMrt : ∀ a b, Grid p a → Grid p b → ‖a‖ ≤ 1 → ‖b‖ ≤ 1 → ‖Mrt a b - Mr a b‖ ≤ Er * hp p)
    {θ θt θi θit : ι → ℂ} (hθ : Phases p θ θt) (hθi : Phases p θi θit)
    {f g : ι → ℂ} (hf : ‖f‖ ≤ 1) (hg : ‖g‖ ≤ 7 / 10) (hE : Er * hp p ≤ 3 / 10) :
    ‖mcT p Mrt θt θit f g - θi * Mr (θ * f) (θ * g)‖ ≤ (Er + 12) * hp p ∧
      ‖mcT p Mrt θt θit f g‖ ≤ 7 / 10 + Er * 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.