Abstract We present the PPR Theory (Pre-Breakpoint: Proposition of a Canonical One-Dimensional Model under the Reduction Axiom, or The PPR Framework), originating from our monograph in Mathematics, as a mathematical framework to model accumulation, memory, and sudden ruptures in complex systems. We formalize a canonical one-dimensional model based on SDEs with continuous noise and jumps, incorporating nonlinear response functions (f), coupling factors (g), modulation by local entropy ( (C) ), and a Causal Memory term (Mc). Under precise regularity hypotheses (local Lipschitz continuity, sublinear noise, jump intensity with finite moments), we demonstrate results of local existence and uniqueness, state linear stability criteria, and characterize conditions for saddle-node bifurcations; we further present asymptotic estimates of the mean escape time in the small-noise regime (Kramers) and discuss heuristic extensions to heavy-tailed jump processes via large deviations. The manuscript preserves the physical and chemical interpretation of the formalism: operational mappings between (Q) and free energy per volume, (C₀) and effective capacitance, and (Mc) as cycle/fatigue history, formulating conditional conjectures that connect the mathematical theorems to predictions of nucleation and collapse. We include canonical parameterizations, simplified analytically tractable examples, lemmas on structural identifiability, and a technical appendix with pseudocode of the numerical scheme (adaptive Euler–Maruyama with jump handling) and detailed proofs of the main results. The work is conceived as a complete and self-contained theoretical article. Nevertheless, the model architecture and the provided numerical scheme establish the formal basis for future empirical calibrations and experimental validations in specific systems. We thus contribute with (i) a rigorous formalization of dynamical operators incorporating memory and thresholds, (ii) theorems and estimates supporting pre-breakpoint phenomenology, and (iii) a set of conjectures and translation maps to physics and chemistry that guide future validation. The text emphasizes a clear distinction between theorems, lemmas, conjectures, and physical interpretations, ensuring mathematical rigor without losing interdisciplinary motivation. Keywords: PPR; SDE with jumps; causal memory; saddle-node bifurcation; escape time (Kramers) ; identifiability; nucleation; nonlinear models.
Marcelo Barboza Duarte (Mon,) studied this question.