WHEN BATCH SCHEDULES CANNOT BE CERTIFIED: CAPACITY OF GLOBALLY VALID MONOMIAL CERTIFICATES ON INTEGER LOG-HULLS

A discrete training schedule can be nearly optimal and still be impossible to certify with one reusable global witness. For $J(m_0,m_1)=m_0^{-1}+m_1^{-1}$ and budget $3$, the integer optimum is $3/2$, whereas every globally valid single monomial certifies at most $\sqrt{2}$. We call the best reusable proof endpoint single-monomial certificate capacity; throughout, this term refers only to a monomial lower bound valid on the entire positive orthant, not to arbitrary certificates or true model risk. The capacity gap is the log-ratio to the best integer schedule. For positive reciprocal risk interfaces we show that the strongest reusable monomial is obtained by minimizing $\varphi (z)=\log J(e^z)$ over the convex hull of logarithmic integer schedules. This gives an exact plan-level obstruction, a finite-support bound in terms of covering radius, and a contractive-SGD realization on a strongly convex quadratic loss of the predicted gap, which is $N\gamma \theta (1-\theta )/(2b^3)$ to first order for the log-ratio. Support expansion is therefore a measurable proof resource: it admits a covering-radius upper bound on the remaining finite-domain certificate gap. A product frontier and a controlled noncommuting moment-family audit reproduce the predicted support obstruction without relying on a particular optimizer or data set.