The Minimal Generative Cycle: Structural Invariance of the G→X→Q→N Sequence Across Physical, Biological, and Cognitive Systems

Three established theoretical frameworks — Prigogine's dissipative structures, Kauffman's self-organization, and Friston's active inference — each describe a specific phase of a recurring structural sequence in complex systems: accumulated potential undergoing a discrete threshold transition to produce stable form, followed by a return with memory. None formalizes the complete four-phase sequence as a unified algebraic object. This paper proposes that the G→X→Q→N cycle — generative potential (G), torsional transition operator (X), manifest stable form (Q), return with memory (N) — is the minimal algebraic structure satisfying four necessary constraints of any generative cyclic process: the existence of a generative potential state, a discrete non-commutative threshold transition, a stable attractor, and a return with memory. A structural argument by elimination demonstrates that no three-element, two-element, or one-element structure satisfies all four constraints simultaneously, and that any five-element extension is formally reducible to the four-element cycle. Apparent counterexamples — Lotka-Volterra oscillators, catalytic cycles, circadian oscillators — are addressed explicitly. A pre-differentiating domain G is proposed: the state anterior to the G/Q distinction in which the category of energy is not yet operative, consistent with the Page-Wootters mechanism and relational accounts of time. The minimality claim is supported by six independent instantiations across biology, neuroscience, and physics, in each of which the mapping was derived from the domain's own formalism: sensory transduction, synaptic memory formation, traumatic memory crystallization, Alzheimer's disease as hypodissolutive failure, collective memory formation, and the vertebrate optical system. Three formal criteria for identifying new instantiations are derived from the cycle's properties.