twGU_runs (KFM U2: uniform root and chirp tables)

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

theorem NLogN.KFM.twGU_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) (k j n : ℕ)
    (hk : 0 < k) (hu : (4 * (n : ℝ) ^ 2 + 1) * (2 : ℝ)⁻¹ ^ Δ ≤ 1)
    (st : IState HS) (LR L1 L3 L4 L5 : List Item) (hR : st.G pR = .int (2 ^ Δ) :: LR)
    (hD : st.G rDl = .int (Δ : ℤ) :: L1) (hJ1 : st.G rJ1 = .int (k : ℤ) :: L3)
    (hX : st.G nX = .int (n : ℤ) :: L4) (hJ2 : st.G rJ2 = .int (j : ℤ) :: L5) :
    IRuns I iprog (twGU pM) st (st.upd pW (ints (gflat (twTab Δ k j n)) ++ st.G pW))
      (cTwGU Δ k j n φ)

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.