Morphogenesis as Spectral Selection

We apply the Latent framework to reaction-diffusion systems that exhibit Turing pattern formation, revealing that pattern selection, stability, and convergence are governed by a single spectral quantity: the Latent Number \(\rho\) of the linearized reaction-diffusion operator. We prove 36 theorems in the Lean 4 covering six aspects of morphogenesis: diffusion operator properties, Turing instability conditions, pattern selection via spectral maximum, convergence to pattern via exponential decay, phase transition between homogeneous and patterned states, and structural transfer from Navier-Stokes PDE theory. Numerical validation on three canonical systems (Schnakenberg, Gierer-Meinhardt, Brusselator) confirms all theoretical predictions. The Latent Number \(\rho\) determines the effective dimension \(N^ = \lceil \log(1/\varepsilon)/\log\rho \rceil\) in the Latent compression bound — the number of Fourier modes sufficient for \(\varepsilon\)-accurate prediction in the reported numerics — and the spectral gap \(\Delta = \sigma(k^) - \sigma_{\text{next}}\) controls both pattern clarity and convergence rate.

Keywords: morphogenesis, Turing patterns, reaction-diffusion, Latent Number, spectral gap, pattern selection