Agentic AI systems act through tools and sub-agents, yet the controls meant to bound their financial and environmental cost still sit on dashboards evaluated beside or after execution. Green SARC applies the SARC governance-by-architecture framework -- four enforcement sites in the agent loop -- to FinOps and GreenOps, contributing the theory of what to enforce and how to predict it. We report four policy-independent results. (i) The unconstrained "State Snowball" is $Θ(n^2)$ in loop depth; on 3,000 real multi-step plans (SWE-rebench) it holds on 100%, with median curvature $\hat{c}_2=216$ exceeding the linear-accretion prediction $p/2=134$ -- real plans accrete faster than the model. (ii) On real residuals the Normal-$σ$ gate under-covers (92% at nominal 95%); split-conformal calibration holds (95.2%). (iii) A soft Lagrangian penalty tuned to the budget in expectation breaches it on 91.5% of seeds; the architectural gate breaches 0%. (iv) Under binding budgets the gate's over-budget incidence is 0% on synthetic and real (BurstGPT) arrivals. End-to-end token/USD/carbon savings (47--55%) are real but policy-dependent in magnitude -- set by a scope-cap knob, not by gate rejections. The library is open-source, dependency-free, and ships a regeneration script for every cited number.