A Gauss–Hermite Hierarchy of Positive Chernoff Approximations of the Heat Semigroup

For the heat semigroup T(t) = exp(t d²/dx²) on R the q-point Gauss–Hermite Chernoff functions C_q(t)f(x) = Σ_j p_j f(x + 2√t z_j) form a hierarchy of positive contractions with the sharp rate ||C_q(t/n)^n f − T(t)f||_∞ ≤ t^q n^{−(q−1)} ||f^(2q)||_∞ / (2q−1)!!, whose 3-point member coincides with the fast Chernoff function S of Vedenin, Voevodkin, Galkin, Karatetskaya and Remizov. The price of order is the dimension r = floor(q/2) of the shift lattice: after n steps the iterate is an exact atomic measure whose evaluation costs N_supp = Θ(n^r) readings of the datum and whose heaviest atom obeys the proved local-limit law max_k W_k ~ c_q n^{−r/2} with explicit constants (c_5 = 5/π), verified to 0.5% at n = 1024. A cost-fair benchmark yields a hybrid verdict with an explicit decision rule: C_5 wins on spectrally narrow data at sup-norm targets ≤ 1e−5 (up to 6.7× fewer datum evaluations), C_3 = S wins on broadband and kinked data, and C_4 is never the cost-fair choice in the tested sup-norm benchmarks; in weak (mollified) metrics the full order n^{−(q−1)} returns for arbitrarily rough L¹ data by duality, so the nonsmooth ceiling is a sup-norm phenomenon. Ancillary studies are collected in a supplement; every number cited in either text is checked against the archived data by an automated verifier (152 claims).