The Latent Number in Biology

We establish that three apparently unrelated biological phenomena — neural manifold dimensionality, gene regulatory network stability, and Wright–Fisher population genetic convergence — are governed by a common spectral quantity: the Latent Number \(\rho\). In each domain, the relevant dynamical operator has geometrically decaying eigenvalues, and the universal approximation formula \(N^* = \Theta(\log(1/\varepsilon) / \log \rho)\) determines the effective dimensionality. We prove:

1. Neural Manifold. If the neural covariance matrix has eigenvalue decay ratio \(\rho = \lambda_1 / \lambda_{d+1} > 1\), then \(d = O(\log(1/\varepsilon)/\log \rho)\) dimensions explain \((1-\varepsilon)\) of the total variance, and signal-to-noise, embedding error, and reconstruction quality are all controlled by \(\rho\).

2. Gene Regulatory Networks. For a linearized GRN with degradation rate \(\gamma\) and largest interaction eigenvalue \(\mu_1\), the Latent Number \(\rho = \gamma/\mu_1 > 1\) determines the spectral gap \(\alpha = \gamma - \mu_1\), which governs exponential convergence, noise attenuation, sensitivity, and effective regulatory dimension.

3. Wright–Fisher Diffusion. The Latent Number \(\rho = 1/\lambda_2\) (reciprocal of the second-largest transition eigenvalue) determines the spectral gap \(\Delta = 1 - \lambda_2\), mixing time \(T_\text{mix} = O(\log N / \Delta)\), effective allelic dimension, and the interplay between selection, mutation, and drift.

Additionally, we prove that the eigenvalue growth rate distinguishes convergence classes: Wright–Fisher's quadratic growth gives \(N^ = O(\sqrt{L/t})\), while GRN's linear growth gives \(N^ = O(L/(\gamma t))\), and we characterize the "edge of chaos" where \(\gamma \to 0\) causes both sensitivity and \(N^*\) to diverge.

The proofs are formalized in the proof language: 70 verified declarations (neural manifold), 78 + 51 (GRN dynamics, two complementary formalizations), and 90 (Wright–Fisher), totaling 289 with zero axiom debt. Log-monotonicity and exponential stability properties are derived from bootstrap axioms rather than hypothesized.

Keywords: Latent Number, spectral gap, neural manifold hypothesis, gene regulatory networks, Wright–Fisher diffusion, effective dimensionality, formal verification.