Knowability Theory: Latent Generator Models and Spectral Pricing in Finance

We present a formally verified mathematical theory of knowability in financial modeling — the framework connecting unobservable latent generators to observable price dynamics. The core construction uses Gaussian latent generators \(L(\mu, \sigma, z) = \mu + \sigma z\) paired with lognormal observables \(O(\mu, \sigma, z) = \exp(L(\mu, \sigma, z))\), establishing an exact exponential bridge. We prove six main theorems covering: (1) the exponential bridge identity; (2) positivity of lognormal observables; (3) crossing step geometry for regime transitions; (4) positive dynamics under amplification and attenuation; (5) spectral pricing kernel properties; and (6) algebraic bounds for price calculations. The proofs are formalized in the Platonic proof language with 28 verified kernel checks, 6 theorems, 2 hypotheses, and 20 structural axioms defining the domain types.

Keywords: Latent generators, lognormal observables, spectral pricing, positive dynamics, formal verification