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

‏The Farran Law of Partial Ambiguity establishes a formal and strictly bounded framework for the attribution of consciousness. It does not approach consciousness as a subjective description or an explanatory abstraction, but as a structure with a determinate domain of existence, defined limits, and an internal order governing persistence, transition, and renewal. Within this law, consciousness is possible only inside a precisely delimited interval. Absolute ignorance eliminates the basis of movement, as a system devoid of any site of disclosure cannot generate questioning, action, or transformation. Absolute knowledge produces terminal closure, as the complete exhaustion of disclosure removes the tension required for continuation. Consciousness exists only between these two limits. Partial ambiguity is not a defect, nor a provisional state to be eliminated, but the necessary structural condition that sustains the possibility of consciousness. The law defines a fixed and invariant sequence through which consciousness unfolds in time: partial ambiguity, curiosity, desire, external event, action, reflection, and the emergence of a new self. This sequence is not a descriptive model of psychological states, but a structural order governing transition, renewal, and the continuous differentiation of the self under determinate conditions. Formally, the law is constructed through a strict derivational architecture, beginning with primitive terms and formal spaces, followed by the Behavioral Representation Postulate Π, definitions, twelve axioms, falsification conditions, structural theorems, and culminating in the governing dynamical equation: dH/dt = Hᵅ(1 − H)ᵝ [−λ(H − H₀) + γE(t) − κℛ(t)], with α < β Falsification is internal to the structure of the law. The attribution of consciousness collapses under precisely defined conditions, including boundary contact, convergence, periodicity, failure of renewal, or other forms of structural breakdown. The Farran Law does not claim to explain qualitative experience. It determines the necessary structural condition under which consciousness may be attributed to a system within a formally examinable domain. Version 3.0 — Canonical Law Edition