chirpP_runs (KFM 9d: the `d`-axis chirp array — outer products of the axis tables)

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

theorem NLogN.KFM.chirpP_runs {M : MulSpec} {pM : Fin P} (hM : MulOK I iprog pM M) (hdom : ∀ x y, M.dom x y)
    {Δ φ : ℕ} (hΦ : ∀ x y : ℤ, zsz x ≤ Δ + 2 → zsz y ≤ Δ + 2 → M.Φ x y ≤ φ) (T : ℕ) :
    ∀ (d : ℕ) (s t : Fin d → ℕ), (∀ i, s i ≤ t i) → (∀ i, 0 < t i) → (∀ i, t i ≤ T) →
      d * (8 * (T : ℝ) ^ 2 + 2) * (2 : ℝ)⁻¹ ^ Δ ≤ 1 →
      ∀ (st : IState HS) (LR : List Item), st.G pR = .int (2 ^ Δ) :: LR →
      IRuns I iprog (chirpP pM Δ d s t) st
        (st.upd pQ1 (ints (gflat (chirpArr Δ d s t)) ++ st.G pQ1)) (cChirpP Δ φ d s t)

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.