The Theory of General Continuity: Continuity as Foundational Law

Abstract

The Theory of General Continuity (TGC) v2 proposes that cumulative inheritance—the capacity of systems to build upon prior states rather than repeatedly reconstructing from zero—depends upon a single foundational enabling condition: monodromic continuity. Monodromic continuity describes the structural preservation of accumulated transformations such that prior states retain sufficient differentiable trace for future recognition, adaptation, and cumulative development.

TGC v2 formalizes this proposition through the primary invariant R(t) ⇒ C(t−1) ≠ ∅, the Continuity Law as the foundational structural enabling condition, the four-layer Continuity Cascade (GC ⊃ MC ⊃ LC ⊃ SC) as a nested architecture of accumulated continuity, the accumulated-product thesis, and the characterization of Special Continuity (SC) as the observer-dependent layer responsible for symbolic modeling, measurement, and deliberate continuity extension.

New in v2 are the formal distinction between monodromic and inert accumulation, explicit differentiation among identity, composition, and continuity as non-equivalent concepts, and the recognition of integration as a fundamental operation alongside differentiation. Within this framework, monodromic continuity is proposed as a necessary, though not sufficient, condition for cumulative inheritance.

The framework is presented as a systems-theoretic architecture rather than a replacement for physics. Physical phenomena operate throughout General, Micro, and Local Continuity (GC, MC, and LC) independently of observers, while Special Continuity (SC) encompasses the symbolic, mathematical, and scientific models developed to describe, measure, and predict those observer-independent phenomena. The Continuity Cascade provides a relational architecture capable of scaling across domains while preserving the distinction between physical reality and its observer-dependent representations.

TGC v2 does not advance a cosmological origin theory, a metaphysical doctrine, or a physical-substance hypothesis. Instead, it proposes a foundational systems framework applicable to biological, ecological, institutional, computational, and synthetic systems in which persistence, recognition, and cumulative inheritance are essential properties.

Keywords: continuity, Continuity Law, monodromic continuity, cumulative inheritance, continuity cascade, relational persistence, observer independence, systems theory