The Margin–Timescale Selector

This paper develops a first-passage model for architecture selection under finite-horizon failure risk. A system begins at a dimensionless safety margin m=X0−Xmin⁡σ0τresp,m=\frac{X_0-X_{\min}}{\sigma_0\sqrt{\tau_{\rm resp}}}, where Xmin⁡X_{\min} is the failure boundary, σ0\sigma_0 is a fixed baseline fluctuation scale, and τresp\tau_{\rm resp} is the response timescale. Environmental persistence is represented by ν=τenvτresp,\nu=\frac{\tau_{\rm env}}{\tau_{\rm resp}}, and architectural breadth is represented by b∈[0,1]b\in[0,1], from specialist to generalist. After nondimensionalization, distance to failure is modeled as an absorbing diffusion dYt=a(b,ν) dt+v(b) dWt,Y0=m,dY_t=a(b,\nu)\,dt+\sqrt{v(b)}\,dW_t, \qquad Y_0=m, with failure at Y=0Y=0. The resulting finite-horizon survival probability is available in closed form, allowing the optimal breadth b∗(m,ν;θ)=arg⁡max⁡0≤b≤1S(m,ν,b;θ)b^*(m,\nu;\theta)=\arg\max_{0\le b\le1}S(m,\nu,b;\theta) to be defined as an explicit architecture selector, where θ\theta contains model-specific drift, mismatch, and variance parameters. The model supports specialist, mixed, and generalist survival optima. For linear breadth-dependent drift and variance, the derivative of survival with respect to breadth reduces to a compact stationarity condition involving the Gaussian inverse Mills ratio. An interior mixed architecture is possible only in a specific parameter regime, yielding an explicit timescale feasibility condition for mixed strategies. The paper also derives a drift-dependent variance result. Higher drift always improves finite-horizon survival, but lower variance is not universally beneficial. When the deterministic trajectory remains above the failure boundary over the horizon, variance strictly reduces survival. When deterministic drift would carry the system through the failure boundary, moderate variance can instead increase survival and produce a unique nonzero survival-maximizing variance. This defines a variance-rescue regime in which stochastic excursions can preserve systems that are deterministically doomed. The resulting framework does not assert that low margin or rapid environmental change universally favors generalism. Instead, margin and environmental timescale locate the system within a first-passage selection problem, while domain-specific drift, mismatch, and variance tradeoffs determine the preferred architecture. This version supersedes the earlier Margin–Variance Selector formulation, which treated the (m,ν)(m,\nu) plane as carrying a generic architecture boundary without deriving that boundary from an absorbing-failure model. The present version replaces that claim with an exact first-passage selector and identifies the conditions under which specialist, mixed, generalist, and variance-rescue regimes occur.