exSm_bnd (KFM Sm: bit bounds for register expressions from bounds on their atoms)

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

theorem NLogN.KFM.exSm_bnd (A B : ℕ) (s : IState HS) (hA : 2 ≤ A) :
    ∀ e : Ex HS, exSm A s e → A ^ exDg e < 2 ^ B → e.Bnd s B

Context. The lemma belongs to the part KFM Sm: bit bounds for register expressions from bounds on their atoms: exSm A s e: every register, literal and 2^a that e reads is at most A in absolute value, and every divisor is positive. exDg e is a degree (sums add one, products add). Then |e.eval s| ≤ A^{exDg e} (exSm_abs), and A^{exDg e} < 2^B gives e.Bnd s B (exSm_bnd), which is the hypothesis of the RS cost lemmas.

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.