The Delta-Gate Formalism: A Probabilistic Generalization of Threshold Transitions with Interpretable Parameters

We introduce the Delta-Gate formalism, a probabilistic model defined on the logarithmic scale distance Δ = log₁₀(X/X_cr), with transition probability P(R=1|X) = σ(wΔ − τ). In the limit w → ∞, the model recovers the classical hard-threshold rule as a special case. Empirical illustration using LDPE dielectric breakdown data (N=60, RMSE 0.046–0.071) shows fit quality comparable to Weibull models with superior physical interpretability.