chirpB_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.chirpB_runs {M : MulSpec} {pM : Fin P} (hM : MulOK I iprog pM M)
    (hdom : ∀ x y, M.dom x y) {w φ : ℕ} (hΦ : ∀ x y : ℤ, zsz x ≤ w → zsz y ≤ w → M.Φ x y ≤ φ)
    (Δ : ℕ) (s : IState HS) (h g c : GI) (R0 RH RK LR : List Item)
    (h0 : s.G 0 = .int g.1 :: .int g.2 :: R0) (hH : s.G pH = .int h.1 :: .int h.2 :: RH)
    (hK : s.G pK = .int c.1 :: .int c.2 :: RK) (hR : s.G pR = .int (2 ^ Δ) :: LR)
    (hg : GBnd w g) (hh : GBnd w h) (hc : GBnd w c) (hg' : GBnd w (gmulT Δ g h))
    (hh' : GBnd w (gmulT Δ h c)) :
    IRuns I iprog (chirpB pM) s
      (((s.upd 0 (.int (gmulT Δ g h).1 :: .int (gmulT Δ g h).2 :: R0)).upd pH
        (.int (gmulT Δ h c).1 :: .int (gmulT Δ h c).2 :: RH)).upd pV
        (.int g.1 :: .int g.2 :: s.G pV)) (cChirpB φ w Δ)

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.