chirpTabPU_runs (KFM U2: uniform root and chirp tables)

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

theorem NLogN.KFM.chirpTabPU_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 d : ℕ) (s t : Fin d → ℕ)
    (hst : ∀ i, s i ≤ t i) (ht : ∀ i, 0 < t i) (hT : ∀ i, t i ≤ T)
    (hu : d * (8 * (T : ℝ) ^ 2 + 2) * (2 : ℝ)⁻¹ ^ Δ ≤ 1)
    (st : IState HS) (LR L1 LXa RD : List Item) (hR : st.G pR = .int (2 ^ Δ) :: LR)
    (hD : st.G rDl = .int (Δ : ℤ) :: L1) (hX : st.G aX = .int (d : ℤ) :: LXa)
    (hdS : st.G NLogN.KFM.dS = ints (descT s t) ++ RD) :
    IRuns I iprog (chirpTabPU pM) st
      ((st.upd NLogN.KFM.dS RD).upd pW (ints (gflat (chirpArr Δ d s t)) ++ st.G pW))
      (cChirpTabU Δ φ (axL s t) (∏ i, t i))

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.