Σ_{j<L} digit_j (S+1)^j ≤ m · n^(c+1), c = clog_m (S+1): polynomially many calls

Σ_{j<L} digit_j (S+1)^j ≤ m · n^(c+1), c = clog_m (S+1): polynomially many calls.

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

theorem NLogN.KFE.calls_le {m n : ℕ} (S : ℕ) (hm : 2 ≤ m) (hn : 1 ≤ n) :
    ∑ j ∈ range (Nat.digits m n).length, n / m ^ j % m * (S + 1) ^ j ≤
      m * n ^ (Nat.clog m (S + 1) + 1)

Context. The lemma belongs to the part KFE: closed fine cost bound of the Lemma 12 program.

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.