fredA_runs (Schönhage–Strassen in the item language: stack layout and Fermat-ring arithmetic)

The following holds (Lean statement, in the scope of module NLogN.SSBase):

theorem NLogN.fredA_runs {X : Fin NS} (h1 : X ≠ RN) (h2 : X ≠ T1) (h3 : X ≠ T2) (s : IState NS)
    (z : ℤ) (n : ℕ) (L LN : List Item) (hX : s.G X = .int z :: L)
    (hN : s.G RN = .int (2 ^ n) :: LN) (hz : z.natAbs < 2 ^ (2 * n + 1)) :
    IRuns I iprog (fredA X) s (s.upd X (.int (fredW n z) :: L)) (2000 * (n + 2))

Context. The lemma belongs to the part Schönhage–Strassen in the item language: stack layout and Fermat-ring arithmetic: The program uses NS item stacks with fixed roles, so that stack (in)equalities are decided by simp. The current ring ℤ/(2^n+1) is given by the ruler 2^n on top of RN and the modulus 2^n+1 on top of RF.

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_Defs18.