The Identity of Extremes: A Generative Framework From Fixed Points to Cosmology: Mapping the Invariant Across Disciplines

This preprint presents a generative framework for understanding structural invariants across domains, from mathematics and physics to biology, consciousness, and cosmology. At its core lies a universal fixed point, x=1x=1x=1, representing operational identity situated between the bounded past and the unbounded future. The paper introduces a tiered architecture: Tier I: Rigorous, grounded systems demonstrating energetic and informational costs (1/σ41/⁴1/σ4) required to maintain identity. Tier II: Conjectural bridges linking mathematical structures, physical systems, and biological and cognitive processes, showing how observers impose constraints to stabilize pseudo-identities. Tier III: High-fertility, speculative applications exploring nested pseudo-identities across neural, social, and cosmic scales. Key contributions include: Formalization of the universal fixed point 1= (0↔∞) 1 = (0) 1= (0↔∞). Analysis of paradoxes (Russell, Liar, Sorites, Ship of Theseus) as structural insights into uncollapsed potential. Cross-domain analogies connecting thermodynamic, informational, and cosmological constraints. Multi-scale framework for consciousness, social systems, and cosmic structure as processes maintaining operational identity. This work builds conceptually on prior preprints by the author, offering both intuitive maps and formalization tools for further exploration. It is intended for researchers interested in foundational structures, systems theory, and the intersection of mathematics, physics, and complex adaptive systems.