Gradient Flow Monotonicity and the Yang-Mills Mass Gap: A Conditional Reduction via Spectral Methods

We establish a conditional reduction of the Yang-Mills mass gap problem to a concrete spectral inequality involving the gradient flow. Main result (informal): for pure SU(N) Yang-Mills theory, if the gradient flow beta-function satisfies a uniform strict asymptotic freedom condition |beta_GF(g)| >= delta g^3 for large g, and a Tauberian regularity condition holds for the spectral density, then: in d = 3 the theory has a mass gap Delta > 0; in d = 4 the infrared trace anomaly vanishes, a_IR = 0, ruling out a conformal infrared fixed point -- combined with the phase exclusion of the companion paper, this reduces the mass gap to explicit spectral conditions. However, the spectral argument is marginal in d = 4 and requires additional non-perturbative input. The proof uses a spectral representation of the gradient flow energy E(t) with the monotonicity identity R'(t) = -2 Var_t(lambda) <= 0, the Komargodski-Schwimmer a-theorem, and a gradient flow Poincare inequality connecting functional inequalities to exponential clustering. We verify all perturbative inputs: the free-field calibration gives R_free(t) = 2/t in d = 4 and the one-loop correction has the correct sign. We identify the indefiniteness of the Weitzenbock curvature term as the precise technical barrier in d = 4. v2 (no v1 numbered statement is changed): the fixed-measure spectral representation is retagged as an explicit hypothesis (H0) for the nonlinear interacting flow (equivalent to complete monotonicity of E(t), proven only for the linearized flow; the status table is retagged accordingly); an unresolved citation is repaired; the Holley-Stroock route claimed in v1 to give the Poincare inequality unconditionally at finite beta is corrected (its constant is exponential in the volume, so the L-uniform statement remains conditional outside the two controlled regimes); the a_IR = 0 phase classification behind the d = 3 mass-gap corollary is retagged as imported physical input; the Karamata attribution is replaced by the elementary direct bound actually used; and a verification suite adds numerical evidence: exact spectral machinery, lattice free-field calibration, the quantified d = 3 vs d = 4 dichotomy on synthetic spectral densities, and a first in-framework SU(2) Wilson-flow Monte Carlo diagnostic (8^4, beta = 2.4: c(t) = t^2 rises 42 percent across the measured window and R(t) < 2/t pointwise, i.e. beta_GF < 0 throughout -- numerical support for Hypothesis B' in a controlled lattice window, evidence not proof).