Spectral Equality for Novikov Integrability: Recursive Criticality and Unbounded Asymptotic Depth

Abstract

We determine the finiteness boundary of the Novikov exponential moment for a one-dimensional constant-volatility mean-reverting diffusion. A localised change of measure cancels the squared drift and leaves a Brownian Feynman-Kac functional with potential $q/2$, where $q=-\lambda'$. If $q(\theta+y)\sim\kappa_\infty y^2$, the exact spectral boundary is $\kappa_\infty\sigma^2T^2=\pi^2$; for cubic drift it becomes $c\sigma^2T^2=\pi^2/3$, and equality is divergent. On the spectral equality surface, the first lower-order transition occurs at power $4/3$, where an explicit coefficient separates the two sides. For symmetric finite pure-power tails, exact tuning generates the recursion $\beta_n=1+3^{-(n+1)}$. We prove that this recursion gives a complete classification of the class. Each individual tail is decided after finitely many comparisons, but the required depth is unbounded: arbitrarily long common critical prefixes can lead to opposite outcomes. The drift-removing stochastic exponential nevertheless remains a true martingale; under the physical law it has no higher moments on the steep-drift class, while the reverse density is essentially bounded.