We construct an effective single-particle Hamiltonian K_eff from Monte Carlo–averaged matrix logarithms of the imaginary-time propagator in determinant quantum Monte Carlo (DQMC). The logarithm maps the multiplicative sign problem into an additive framework where the central limit theorem guarantees convergence, rendering K_eff sign-problem-free: both sign sectors yield identical dispersions to <1%. K_eff captures the exact correlated single-particle spectrum, incorporating all self-energy effects non-perturbatively. Applied to the Hubbard model (t'/t = -0.30, U/t = 4), K_eff reveals a d-wave pseudogap with strong nodal-antinodal dichotomy below a computational phase transition at T^*. Three sign-free observables provide evidence consistent with spin-fluctuation pairing: (i) the gap ratio R_g > 1 confirms d-wave symmetry – a temperature-independent property of the correlated band structure that provides the medium for pairing; (ii) the superfluid stiffness ρₛ forms a dome across doping at L = 8, 10, and 12, exceeding the Berezinskii-Kosterlitz-Thouless threshold by 5-7× at the dome peak; (iii) S(π,π) is approximately flat across doping, establishing that the dome originates from Fermi-surface geometry responding to uniform spin-fluctuation glue. The pseudogap grows monotonically toward half-filling while ρₛ forms a dome, mirroring cuprate phenomenology where T_c is limited by the superfluid density (Uemura relation). Vertex corrections remain to be quantified.