On the Number – Theoretic Structure of Extreme Climate Event Recurrence: A Theoretical Framework for Quasi – Periodic Clustering

The prevailing paradigm in analyzing the recurrence intervals of extreme climate events – such as heat-waves, floods, and droughts – relies on stochastic frameworks, including Extreme Value Theory (EVT) and Poisson progress, which presuppose inherent randomness. This paper explores whether low-frequency quasi-periodic climate forcings may introduce weak arithmetic structure into the recurrence statistics of extreme events, superimposed upon an inherently stochastic-chaotic climate background. We propose a novel theoretical model wherein the temporal sequencing of extreme events, under the influence of persistent multi-scale nonlinear forcings (e.g., orbital cycles, ocean-atmosphere oscillations), exhibits hidden number-theoretic patterns. By formulating climate preconditioning as an almost periodic function, we demonstrate that recurrence intervals can cluster around values defined by arithmetical sequences, Diophantine approximations of coupled oscillation periods, and solutions to modular congruences. The core contribution is a formal theorem on the existence of “Arithmetical Recurrence Windows†, providing a semi-deterministic modulation framework for the observed phenomenon of quasi-periodic clustering. This work establishes a pioneering, interdisciplinary bridge between analytic number theory and climate, dynamics, proposing a new diagnostic framework for extreme events timing with potential implications for long-term risk assessment. To render the framework falsifiable, we propose empirical testing against observational and paleoclimate datasets using recurrence interval statistics, surrogate stochastic simulations, and comparisons with Extreme Value Theory (EVT), autoregressive processes, and self-exciting point-process models. The theory predicts statistically enhanced recurrence intervals near denominators of continued-fraction convergents of dominant oscillatory modes, rather than exact deterministic event timing.