The Four Regimes of Time

This preprint presents a unified mathematical–ontological framework in which the fundamental constantsπ, e, Euler’s formula, and φ are derived not as independent mathematical artifacts, but as unique stable regimes of time arising from minimal dynamical constraints. Within the Quant-Trika framework, time is treated not as a homogeneous parameter, but as a structured dynamical phenomenon emerging from an informational potential. By imposing a small set of explicit axioms—positivity, minimal historical integration (MHIP), scale freedom, stability, and orthogonality—we classify all admissible low-memory temporal dynamics into four and only four stable regimes: π-regime — periodic time without growth or memory e-regime — scale-free growth without memory Euler regime — orthogonal separation of phase and amplitude (spiral dynamics) φ-regime — minimal non-reducible time with memory and stable growth Each constant appears as a negative uniqueness result: under the stated axioms, no alternative invariant can exist within the corresponding class of dynamics. A completeness theorem shows that these four regimes exhaust all stable possibilities under minimal memory constraints, while a depth-escalation lemma demonstrates why uniqueness is lost for higher-order historical dependence. A central conceptual result of the work is the reinterpretation of information: information is not treated as an abstract, context-free quantity, but as a property of specific temporal regimes. In particular, information becomes ontologically meaningful only in time regimes with irreducible memory (φ and beyond), while purely periodic or memoryless exponential dynamics are shown to be informationally degenerate. The framework provides a coherent bridge between mathematics, physics, biology, and cognition, offering new perspectives on: the origin and universality of mathematical constants, the emergence of time and information from a pre-temporal potential, pathological versus non-degenerate temporal dynamics, and the structural conditions required for life, learning, and adaptive systems. This work is intended as a foundational preprint, providing a fully explicit proof layer and clearly stated domain of validity, and is suitable for further extension toward physics, information theory, and artificial intelligence.