geoL_err (KFM 9b: fixed-point root tables by FM product recurrences — denotations and erro)

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

theorem NLogN.KFM.geoL_err (P : ℕ) (c : GI) (C : ℂ) (hC : ‖C‖ ≤ 1) {ec : ℝ}
    (hc : ‖vP P c - C‖ ≤ ec) :
    ∀ (n : ℕ) (x : GI) (X : ℂ) (e0 : ℝ), ‖X‖ ≤ 1 → ‖vP P x - X‖ ≤ e0 →
      e0 + n * (2 * ec + 2 * (2 : ℝ)⁻¹ ^ P) ≤ 1 →
      ∀ k (hk : k < n), ‖vP P ((geoL P c x n)[k]'(by rw [length_geoL]; exact hk)) - X * C ^ k‖ ≤
        e0 + k * (2 * ec + 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.