In this paper, a mathematically rigorous solution overturns existing wisdom regarding New Keynesian Dynamic Stochastic General Equilibrium. I develop a formal concept of stochastic equilibrium. I prove uniqueness and necessity, when agents are patient, across a wide class of dynamic stochastic models. Existence depends on appropriately specified eigenvalue conditions. Otherwise, no solution of any kind exists. I construct the equilibrium for the benchmark Calvo New Keynesian. I provide novel comparative statics with the non-stochastic model of independent mathematical interest. I uncover a bifurcation between neighbouring stochastic systems and approximations taken from the Zero Inflation Non-Stochastic Steady State (ZINSS). The correct Phillips curve agrees with the zero limit from the trend inflation framework. It contains a large lagged inflation coefficient and a small response to expected inflation. The response to the output gap is always muted and is zero at standard parameters. A neutrality result is presented to explain why and to align Calvo with Taylor pricing. Present and lagged demand shocks enter the Phillips curve so there is no Divine Coincidence and the system is identified from structural shocks alone. The lagged inflation slope is increasing in the inflation response, embodying substantive policy trade-offs. The Taylor principle is reversed, inactive settings are necessary for existence, pointing towards inertial policy. The observational equivalence idea of the Lucas critique is disproven. The bifurcation results from the breakdown of the constraints implied by lagged nominal rigidity, associated with cross-equation cancellation possible only at ZINSS. This creates a singular surface inside which the inter-temporal transmission mechanism breaks down and the error terms cancel out. There is a dual relationship between restrictions on the econometrician and constraints on repricing firms. Thus if the model is correct, goodness of fit will jump. 

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