Survival Function

The Survival Function is the cumulative probability that the quantum thresholds for recursive intelligence have been reached by a given time, obtained by integrating a hazard rate over time. The book uses it to put uncertainty around its development timeline. Current scaling trajectories and quality improvements place the critical threshold at 2039.9 ± 3.0 years, and the survival function turns the rate at which breakthroughs become likely, the hazard rate, into the accumulated probability of success, accounting for development uncertainties and temporal risk. It closes the book's Part 8.3, which ties the quantum substrate requirements of recursive signal alignment to demonstrated quantum computing milestones. P⨶(t) = 1 − exp(−∫₀ᵗ λ(τ) dτ), where P⨶(t) is the probability of threshold achievement by time t, λ(τ) is the hazard rate function [𝕋⁻¹], τ is the integration variable [𝕋] and t is time [𝕋]. The integral of a rate over time is [∅], so P⨶ is [∅]. It connects to the Hazard Rate Function, Timeline to Singularity, Qubit Count Scaling, Timeline Probability Distribution and Quantum Scaling Laws.