One Parameter: How ρ Unifies Computation, Structure, and Safety

We present a unified framework in which a single parameter — the analyticity radius \(\rho\) — simultaneously controls five properties of smooth systems: (1) structural complexity (the number of interaction grades needed), (2) computational cost (eigenvalue conditioning reduces dimension from \(n\) to \(K_{\text{eff}}\)), (3) convergence rate (exponential in \(\rho\)), (4) AI safety (certified adversarial radius amplified by factor \(I = \lambda_{\max}/L_{\text{eff}}\)), and (5) verifiability (the machine-verified science pipeline achieves 100% verification on systems with \(\rho > 1\)). The phase transition at \(\rho = 1\) is sharp: all five properties hold when \(\rho > 1\) and none hold when \(\rho \leq 1\).

The framework integrates five previously independent results into a single chain:

\[\text{Latent} \xrightarrow{\text{represents}} \text{Grade} \xrightarrow{\text{decomposes}} \text{EC} \xrightarrow{\text{bridges}} \text{Safety} \xrightarrow{\text{validates}} \text{Pipeline}\]

Each arrow is a machine-verified mathematical bridge. The Latent theorem provides the finite representation. The Grade equation decomposes it into interaction orders. Eigenvalue conditioning (EC) diagonalizes the grade-2 component. The EC-safety bridge shows that conditioning amplifies adversarial robustness. The verification pipeline validates the entire chain.

We demonstrate the framework with 150 machine-verified theorems across 10 proof files, producing 619 kernel-level type checks with zero errors. The theorems span eigenvalue conditioning (43), grade decomposition (53), pipeline properties (18), bridge methodology (20), and this paper's unification (16). The framework applies to any smooth system — from epidemics to turbulence to neural networks — and the diagnostic is always the same: compute \(\rho\).