Poincaré and Ze-Theory

Henri Poincaré's famous "illuminations" (sudden insights while boarding a bus, during a geological excursion) are not mysterious—they have a structure. This paper applies Ze-theory as an analogical model to reconstruct the cognitive dynamics of Poincaré's mathematical creativity. In Ze-theory, any information-processing system can be described through the balance of two event types: T-events (incorrect predictions decrease proper time counter τ_Z) and S-events (correct predictions preserve τ_Z). Optimal functioning for an active agent occurs at v* ≈ 0.456 (Tkemaladze, 2026a), where τ is maximized. We propose that Poincaré's creative process—conscious forced analysis → subconscious incubation → sudden illumination—corresponds to S-dominance (v → 0), movement toward v*, and a T-burst (unconscious synthesis exceeding prediction). Three case studies are analyzed: automorphic functions (1881, the prototype Ze-flow), topology and chaos (1890–1900, τ-accumulation through iterative corrections), and relativity (1905, where Ze-flow stopped short). The mechanism of analogy (geological, kaleidoscopic, weaving) is interpreted as τ-transfer between domains. Negative cases (quantum theory, unproven Poincaré Conjecture) define the boundaries of the analogy. Ze-theory provides an operational vocabulary for describing mathematical intuition—explicit, falsifiable via historical records, and connected to a formal theory of adaptive information processing.