twCU_runs (KFM U2: uniform root and chirp tables)

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

theorem NLogN.KFM.twCU_runs (rn rk rj X : Fin HS) (n k j : ℕ) (hk : 0 < k)
    (hn28 : rn ≠ 28) (hk28 : rk ≠ 28) (hk29 : rk ≠ 29) (hj28 : rj ≠ 28) (hj29 : rj ≠ 29)
    (hj53 : rj ≠ 53)
    (hX51 : X ≠ 51) (hX52 : X ≠ 52) (hX28 : X ≠ 28) (hX29 : X ≠ 29) (hX53 : X ≠ 53)
    (s : IState HS) (Ln Lk Lj : List Item)
    (hn : s.G rn = .int (n : ℤ) :: Ln) (hkk : s.G rk = .int (k : ℤ) :: Lk)
    (hj : s.G rj = .int (j : ℤ) :: Lj) :
    IRuns I iprog (twCU rn rk rj X) s
      (s.upd X (.int (HvdH.twiddle n k j).1 :: .int (HvdH.twiddle n k j).2 :: s.G X))
      (cTwU n k j)

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.