Cost of the uniform chirp table (closed form)

Cost of the uniform chirp table (closed form).

The following holds (Lean statement, in the scope of module Solutions.KFMU2):

theorem NLogN.KFM.cChirpTabU_le (Δ φ AX : ℕ) {d : ℕ} (s t : Fin d → ℕ) (ht : ∀ i, 2 ≤ t i)
    (hAX : ∀ i, cAxU Δ (s i) (t i) φ ≤ AX) :
    cChirpTabU Δ φ (axL s t) (∏ i, t i) ≤
      d * (AX + 4000) + 10 ^ 11 * ((∏ i, t i) + 1) * (φ + Δ + 3)

Context. The lemma belongs to the part KFM U2: uniform root and chirp tables.

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.