Symbol-based residual certification of reduced and neural surrogates for tempered-fractional CGMY pricing

We certify reduced and neural approximations of European-put price trajectories under the Carr--Geman--Madan--Yor model by residual bounds for the spline Galerkin discretisation of the tempered-fractional pricing equation. The Galerkin matrices are Toeplitz sections of an explicit periodised Fourier symbol; summing the alias series to a stated tail makes the assembled system exact to $10^{-8}$. A Lévy--Khintchine argument gives the parameter-explicit bound $λ_{\min}(S_I)\ge r\,m_{\min}$ for the symmetric part, valid for $Y\in(0,2)\setminus\{1\}$, replacing numerical spectral estimates. A banded comparison norm built from the mass and stiffness symbols yields a rigorous certificate at $O(N)$ cost with explicit effectivity, and empirical interpolation of the symbol makes the operator affine in about 36 terms with a rigorous error transfer, so a reduced model is certified at a cost independent of $N$; a three-stage rule escalates to the full-order bounds only when needed. At the resolution that prices the whole parameter box to one cent ($N=1024$), the full-order certificate costs 6% of a direct solve, and the certified POD--Galerkin model of rank 80 costs 3% of it and settles 99.8% of $10^4$ queries without touching the full space. No false accept occurs on any query with a reference solution (150 inside the parameter box, 264 outside), and outside the box the certificate accepts 193 of the 210 queries on which the reduced model is still within one cent. The neural reduced models trained here miss the one-cent tolerance by a factor of at least 46; the certificate detects this at every query in a few milliseconds and explains it as a trade-off between temporal smoothness and coefficient accuracy. A residual certificate for the American put is proved as an outlook.