twTab_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.twTab_err (P k j n : ℕ) (hk : 0 < k) (hu : n * (4 * (2 : ℝ)⁻¹ ^ P) ≤ 1) (m : ℕ)
    (hm : m < n) :
    ‖vP P ((twTab P k j n)[m]'(by rw [length_twTab]; exact hm)) -
      exp (2 * Real.pi * I * j / k) ^ m‖ ≤ 4 * n * (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.