ss_final (Correctness of one Schönhage–Strassen level)

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

theorem NLogN.ss_final (N k M : ℕ) (a b : ℤ) (hN : 1 ≤ N) (hKM : 2 ^ k * M = N)
    (ha : 0 ≤ a ∧ a ≤ 2 ^ N) (hb : 0 ≤ b ∧ b ≤ 2 ^ N) :
    (2 ^ (2 ^ k * M) + zev (2 ^ M) (carZ M (ncZ (2 ^ k) (pcs M (2 ^ k) (fold N a))
      (pcs M (2 ^ k) (fold N b))) 0).1 - 2 ^ N -
      (carZ M (ncZ (2 ^ k) (pcs M (2 ^ k) (fold N a)) (pcs M (2 ^ k) (fold N b))) 0).2) %
        (2 ^ N + 1) = (a * b) % (2 ^ N + 1)

Context. The lemma belongs to the part Correctness of one Schönhage–Strassen level.

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.