chirpPU_runs (KFM U2: uniform root and chirp tables)

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

theorem NLogN.KFM.chirpPU_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)
    (L : List (ℕ × ℕ)) (hL : ∀ p ∈ L, p.1 ≤ p.2 ∧ 0 < p.2 ∧ p.2 ≤ T)
    (hu : L.length * (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 (L.length : ℤ) :: LXa)
    (hdS : st.G NLogN.KFM.dS = ints (descA 1 L) ++ RD) :
    IRuns I iprog (chirpPU pM) st
      ((st.upd NLogN.KFM.dS RD).upd pQ1 (ints (gflat (chirpArrL Δ L)) ++ st.G pQ1)) (cChirpPU Δ φ L)

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.