Cost of the chirp array: with every axis-table cost ≤ AX and t_i ≥ 2, cChirpP ≤ d·AX + (Δ + 6) + 2 N (4φ +…

Cost of the chirp array: with every axis-table cost ≤ AX and t_i ≥ 2, cChirpP ≤ d·AX + (Δ + 6) + 2 N (4φ + 4·10^8 (Δ + 3)), N = ∏ t_i (the d p^δ row: O(1) products per entry, plus d axis tables).

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

theorem NLogN.KFM.cChirpP_le (Δ φ AX : ℕ) : ∀ (d : ℕ) (s t : Fin d → ℕ), (∀ i, 2 ≤ t i) →
    (∀ i, cAx Δ (s i) (t i) φ ≤ AX) →
    cChirpP Δ φ d s t ≤ d * AX + (Δ + 6) + 2 * (∏ i, t i) * (4 * φ + 4 * 10 ^ 8 * (Δ + 3))

Context. The lemma belongs to the part KFM 9d: the d-axis chirp array — outer products of the axis 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.