rptXV_runs (KFM U2: uniform root and chirp tables)

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

theorem NLogN.KFM.rptXV_runs {X A C D : Fin HS} (hXA : X ≠ A) (hAC : A ≠ C) (hAD : A ≠ D) (hCD : C ≠ D) (hAK : A ≠ sK)
    (hCK : C ≠ sK) (hCK2 : C ≠ sK2) (hCY : C ≠ uY) (hCB : C ≠ uB)
    (n : ℕ) (T : ℕ → IState HS)
    (hTA : ∀ k, (T k).G A = (T 0).G A) (hTC : ∀ k, (T k).G C = (T 0).G C)
    (hTD : ∀ k, (T k).G D = (T 0).G D)
    (b : IStmt HS P) (cs : ℕ → ℕ)
    (hb : ∀ k < n, ∀ LA LC LD, IRuns I iprog b (cf3 (T k) A C D LA LC LD)
      (cf3 (T (k + 1)) A C D LA LC LD) (cs k)) (LX : List Item) (hX : (T 0).G X = .int (n : ℤ) :: LX) :
    IRuns I iprog (rptX X A C D b) (T 0) (T n)
      ((∑ k ∈ Finset.range n, (cs k + 64)) + 60 + cRpt n + (dupCost (zsz (n : ℤ)) + 17))

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.