<div> Derivative pricing has a clear mathematical target: compute the discounted risk-neutral value of a payoff under a specified model. For European vanilla contracts this target is often analytic or nearly analytic. The difficult cases are path-dependent derivatives — American and Bermudan exercise,&nbsp;barriers, autocallables, coupon memory, hit counts, and structured-product state — where standard methods rely on Monte Carlo simulation, regression continuation estimates, or high-dimensional PDE grids. This paper introduces projected-generator pricing as a finite-dimensional operator method for such contracts. Starting from the infinitesimal generator of the risk-neutral dynamics, the method projects the generator onto a finite basis and propagates continuation values by the resulting matrix semigroup. Path dependence is represented by finite event-state transitions around the same propagation primitive. American and Bermudan exercise become projected semigroup propagation plus pointwise exercise comparison plus re-projection, with no simulated-path regression. The contribution is a pricing architecture rather than a new isolated ingredient. Generator calculus, Galerkin projection, matrix exponentials, spectral bases, and Bellman recursion are combined into a deterministic engine for path-dependent derivatives. When the finite space is invariant under the generator, pricing is exact on that space; otherwise the projection residual measures approximation debt. For analytic payoffs under spectral bases the error decays exponentially, and Greeks are obtained by differentiating the same finite expansion rather than by bump-and-reprice. Benchmarks against Black-Scholes, COS, Longstaff-Schwartz, finite differences, Monte Carlo, and Merton jump-diffusion references show machine-precision European pricing, regression-free American pricing within 1% of published references, deterministic Greeks, finite-activity Poisson-jump support through a nonlocal Lévy generator, and a structured-product example priced without path simulation. The method is strongest when the model has a tractable generator, the path memory is finite state, and the effective dimension is low to moderate. High-dimensional baskets, rough volatility without a controlled lift, and severe discontinuities remain better served by Monte Carlo, finite differences, or adapted bases. </div>