The Farran Law of Partial Ambiguity (PACE): A Universal Mathematical Theory of Consciousness

Description

Despite decades of empirical progress, consciousness remains theoretically unresolved. This persistence is treated not as a data problem, but as a structural one. The Law of Partial Ambiguity (PACE) is presented as a universal, domain-independent framework that formalizes consciousness neither as a substance, representation, nor functional module, but as a regulated dynamical process sustained by persistent structural non-closure.

PACE is formulated as an abstract dynamical system defined on a strictly bounded open state space. Its single global state variable represents the degree of partial ambiguity and evolves under explicit admissibility constraints ensuring well-posedness, boundedness, non-monotonicity, and forward invariance. Degenerate limit cases corresponding to complete determinacy or complete indeterminacy are structurally excluded, establishing partial ambiguity as a necessary condition for sustained, non-trivial dynamics. These constraints do not prescribe trajectories or outcomes. They delimit the admissible state space within which multiple, non-unique, and history-dependent evolutions remain possible.

The Partial Ambiguity Cycle Equation specifies the admissible evolution of this state without invoking discrete control sequences, optimal paths, or convergent endpoints. Cyclic organization and recurrence arise as intrinsic relational properties of the dynamics rather than as externally imposed stages.

To render the abstract structure empirically connectable, the Dynamic Ambiguity Index (DAI) provides a minimal structural decomposition of representational non-closure into novelty, structural mismatch, and conceptual distance. These components are aggregated into the bounded state variable without introducing additional state variables, feedback mechanisms, or temporal assumptions.

A physical interpretation treats the evolution law as a self-regulating, non-equilibrium process acting on a macroscopic state variable, while a geometric formulation embeds the dynamics as continuous trajectories in phase space. Recurrence, cyclic organization, and non-monotonic behavior appear as structural properties of the admissible dynamics, not as predictive guarantees.

The formal architecture is completed by introducing the Self (Σ) as a dynamic internal reference frame specifying evaluative conditions without constituting an agent, representational entity, or phenomenological subject. Structural closure is achieved without explanatory reduction.

PACE provides a mathematically well-defined, empirically connectable, and computationally implementable theory of consciousness. No claims of determinism, convergence, or prediction are entailed. The framework specifies conditions of persistence rather than outcomes. Consciousness is reframed as a bounded, recurrent, and self-renewing process sustained by the lawful regulation of partial ambiguity.

Version 2.0 — Canonical Reference Edition