Emergent Time from Contractive Fixed-Point Iterations

This note proposes a minimalist mathematical framework for the emergence of time in dissipative dynamical systems governed by contractive mappings. Starting from the Banach fixed-point theorem, we introduce an emergent time parameter t = n · τ, where n is the discrete iteration count and τ = −1/ln c derives from the contraction constant c ∈ (0,1). This rescaling converts the exponential error decay of fixed-point iterations into the familiar continuous relaxation form e⁻ᵗ/ᵀ.

We prove that this parameter satisfies three essential temporal properties: (1) monotonic progress toward the unique fixed point, (2) dynamical irreversibility due to the dissipative nature of contractions, and (3) a metric-induced temporal ordering of states. No external clock or statistical assumptions are required—time arises purely as a measure of irreversible convergence to equilibrium.

As c → 1⁻, τ diverges, naturally reproducing critical slowing down observed near phase transitions or bifurcations. The framework offers clean analogies to renormalization group flows, open quantum system dynamics (Lindblad evolution), relational quantum mechanics, and the Page–Wootters mechanism.

Numerical examples illustrate exact correspondence in linear contractions and asymptotic validity in nonlinear cases via local linearization. The perspective highlights time not as a fundamental entity, but as a derived quantity encoding dynamical stability in mathematics, physics, and computation.