Emergence as Regime Formation I: Law as the Exhaustion of Generative Freedom

This paper develops a structural framework for understanding emergence across physical, mathematical, and complex systems. Emergent laws are reinterpreted not as additional governing principles imposed at higher levels, but as structural residues left behind when a system exhausts its freedom to continue in qualitatively different ways.

A domain-relative notion of generative capacity, Ω, is introduced to track remaining viable continuations, together with a constraint functional Γ governing its contraction. Regime formation is identified with non-smooth behavior in ln Ω, marking points at which further qualitative freedom is no longer available and collective behavior becomes inevitable. A symmetry-independent, dimensionless criterion for regime locking is proposed, applicable across continuous, discrete, and topological domains without reliance on order parameters or energetic minimization.

Within this framework, the quantum-to-record transition is shown to provide the minimal physical instance of regime formation, establishing a direct conceptual lineage with the local generation of time developed in earlier work.

This paper establishes the foundational regime framework. Subsequent papers extend and specialize this structure as follows:

Emergence as Regime Formation II examines incomplete, non-uniform, or path-dependent exhaustion, treating topological defects, anomalous phases, and algebraic instabilities as structural relics of partially exhausted generative freedom rather than symmetry-breaking anomalies.

A subsequent ontological trilogy is then developed:

(i) On Wave Motion interprets oscillatory and wave phenomena as intermediate regimes, in which generative freedom persists under constraint without fully collapsing. Waves are shown to arise from constrained phase evolution, with interference and mode structure understood as mechanisms of continuation pruning rather than energetic cancellation.

(ii) On the Constants of Nature treats dimensionless physical constants as stabilized markers of regime viability, corresponding to phase-locked limits where generative freedom has fully exhausted and structural relations become invariant.

(iii) On Darkness addresses cosmological and gravitational “dark” phenomena as regions in which regime activation fails to reach recordable thresholds, leaving generative structure present but observationally unrealized.

Taken together, these works aim to situate familiar physical laws, constants, and regimes within a single structural account of emergence grounded in the exhaustion of generative freedom.