General Equilibrium in Hilbert Space: Competitive Equilibrium as an Orthogonal Projection of the Adaptive State

What is a market equilibrium, the state where every plan agrees and nothing moves? This paper argues it is the still shadow of a restless, adaptive economy. By placing the whole economic state, agents, firms, machines, and prices, in a single geometric space where closeness means statistical correlation, it shows that competitive equilibrium is the projection of the economy onto its unchanging part, and the living economy is what is left over. One geometric step recovers three classical ideas at once: equilibrium, rational expectations, and the welfare benchmark. The leftover part measures an economy's ignorance and the information an agent must pay to describe it. The work is theory, with a worked example for illustration and no empirical test, careful that a unique projection is not a determinate equilibrium.