CALLIOPE · BCOM

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}

Any mathematical theory that scores or classifies consciousness is constrained by basic topology in ways that rule out clean yes/no answers.

The central question is simple: if you want a mathematical theory to tell you whether a physical system is conscious, what must that theory look like? This paper doesn't try to build such a theory — it asks what any such theory must obey just to be internally consistent. The setup is minimal: assume the space of all physically realizable systems can be continuously deformed from one to another (it's "connected" in the topological sense), and assume your theory assigns each system a score between 0 and 1 measuring how conscious it is.

Three results follow from this setup using standard mathematics. First, if you try to turn that continuous score into a crisp yes/no classifier — conscious or not — that classifier cannot itself be continuous. A connected space can't be split cleanly into two parts by a continuous function; the only continuous binary functions on such a space are the trivially constant ones. So any sharp consciousness boundary is, by necessity, a discontinuity — a phase transition, not a smooth cutoff. Second, the intermediate value theorem forces the score to pass through every intermediate value along any path connecting a non-conscious system to a more-conscious one. You can't jump. Third, if the score has no flat spots (no critical points, technically a "submersion"), then the endpoints 0 and 1 are actually unreachable — the image of the score function is an open interval strictly inside (0,1). Perfect zero and perfect one are excluded.

The paper also flags a computability wall: if the space of systems includes universal computers, then Rice's theorem says no algorithm can decide any nontrivial semantic property of their behavior. A universal crisp consciousness detector is not just philosophically fraught — it's provably uncomputable over sufficiently general system spaces.

Importantly, the paper is careful about what this all means. These are constraints on descriptive theories — ones that map structure to structure. They say nothing about why any structural pattern feels like something from the inside. That gap (Levine's "explanatory gap," Chalmers' "hard problem") is explicitly left open. The paper's honest position is that mathematics can characterize the shape of experience — its similarities, transitions, regimes — but cannot explain why those shapes are accompanied by any subjective quality at all. The theorems here are consistency conditions, not a theory of consciousness. They tell you what your score function must look like if it's going to be well-behaved; they don't tell you which score function to pick or why it should track phenomenal experience.