We study an exponential summation functional built from graph Laplacian energies with a baseline shift and two nonlinearities: clipping at zero and saturation at a fixed cap. We prove that for explicit absolute constants =20 and =7, it is \#P-hard to approximate the anchored clause-aggregated clipped cut sum ACCS₂₀, ₇ (G) to additive error 1. Our single-graph reduction uses constant integer weights. Moreover, in an equivalent Dirichlet/Kron (spectral) formulation we obtain bounded-degree constant-weight instances (maximum degree 4 in the union graph), and we describe a bounded-degree unweighted simulation. Our reduction is driven by a constant-size integer-gap clause gadget. For a monotone 3-clause, the gadget's Dirichlet-minimized quadratic energy takes the four values 27, 20, 11, 0 depending only on whether the clause has 0, 1, 2, or 3 true inputs. Subtracting the baseline =20 and clipping the negative part collapses this to a perfect 0/7 violation signal. An outer cap at =7 ensures that each Boolean assignment contributes exactly 0 if it satisfies the formula and exactly 7 otherwise, so the full exponential sum equals 7 \#UNSAT (). We present the construction in two equivalent languages. The combinatorial route works directly with weighted cuts (after eliminating auxiliary vertices), while the spectral route uses Dirichlet energies and Kron reduction (Schur complements).
Jacob Oertel (Mon,) studied this question.