axG_runs (KFM 9c: the per-axis root-table program (`chirpL` recurrence in IL))

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

theorem NLogN.KFM.axG_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) (s t : ℕ)
    (hst : s ≤ t) (ht : 0 < t) (hu : (4 * (t : ℝ) ^ 2 + 1) * (2 : ℝ)⁻¹ ^ Δ ≤ 1)
    (st : IState HS) (LR : List Item) (hR : st.G pR = .int (2 ^ Δ) :: LR) :
    IRuns I iprog (axG pM Δ s t) st (st.upd pW (ints (gflat (axTab Δ s t)) ++ st.G pW))
      (cAx Δ s t φ)

Context. The lemma belongs to the part KFM 9c: the per-axis root-table program (chirpL recurrence in IL).

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.