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Abstract

We classify the coefficient penalties that control point evaluation and absolute coefficient moments of finite arithmetic cutoff certificates. In divisor coordinates, let W_ω = Σ_{r≥2} ω(r)(J₄(r) + 10J₂(r))|v_r|²/180, where ω(r) ≥ 1 is arbitrary. Every fixed integer Bernoulli cutoff value is bounded in this norm if and only if Σ_r ω(r)^(−1) < ∞. For every integer R ≥ 1, the absolute moment Σ_d d^R |c_d| is bounded if and only if Σ_r r^(2R−2)/ω(r) < ∞. Both classifications persist after any fixed finite collection of bounded homogeneous constraints. A full-divisor lower bound supplies necessity and settles the critical logarithmic exponents. Applications give an exact constrained point capacity, a norm-controlled compensated-mean remainder, and an explicit native approximation floor. At the critical moment penalties, the floor gives a quantitative logarithmic lower bound on the energy required for a given native error.