Consistency constraints on mathematical theories of phenomenal consciousness

A mathematical theory that assigns a continuous 'phenomenality score' to physical systems cannot produce a sharp yes/no classification of consciousness without a discontinuity somewhere. Any such scoring function that varies smoothly must take on intermediate values between zero and a positive threshold. Under certain smoothness conditions, the extreme scores of 0 and 1 are impossible to achieve. These results show that descriptive mathematical models of consciousness can identify boundaries but cannot explain why or how consciousness arises, consistent with the idea of an explanatory gap.

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Abstract

We state clean consistency constraints on any descriptive (structural) mathematical theory of phenomenal consciousness. Let the configuration space \(U\) of physically realizable systems be connected (under admissible deformations) and let a descriptive theory deliver a symmetry\-invariant scalar \(p:U [0,1]\) (a ``phenomenality score'') together with an optional crisp classifier \(P= {1}\{p c\}\). We prove: (i) any nontrivial crisp \(P\) must be discontinuous somewhere (Triviality of continuous crisp predicates); (ii) nonzero scores between 0 and a positive value are unavoidable along any continuous deformation (Intermediate\-value necessity); (iii) under mild differential conditions (submersion), endpoints \(0,1\) are excluded from the image of \(p\) even when \(U\) carries symmetries (Endpoint exclusion under submersion). We also note computability barriers (Rice's theorem) and give templates (order parameters, topological invariants) that realize phase boundaries without explanatory import. Conceptually, such a theory is descriptive but not explanatory, consonant with the explanatory gap literature. {Levine 1983}{https://www.informationphilosopher.com/solutions/philosophers/levine/Explanatory_Gap.pdf}, {Chalmers 1995}{https://consc.net/papers/facing.pdf}