The Explanatory Gap as Dimensional Mismatch: Phenomenal Intensity, Qualitative Kind, and the Limits of a Bridging Law

The explanatory gap is standardly treated as an open question, a distance that further theory might one day close. This paper reconstrues it in dimensional terms and, in the reconstrual, puts the apparatus of dimensional analysis to work as more than a metaphor. Phenomenal experience is divided into two parts the standard framing runs together. Its intensive magnitude, the felt strength of a sensation, is a genuine ratio-scaled quantity that carries no physical dimension, and is assigned a fourth base dimension, Ξ, bridged to physical intensity by a coefficient κΞ that fixes its unit. Its qualitative kind, the specific redness as against its strength, is no magnitude at all. The dimensional result is a lemma, that no homogeneous equation can fix the kind from physical quantities, and the explanatory gap is secured only when this lemma is joined to a Russellian intrinsic-nature premise, so that the thesis is conditional, addressed to a realist about felt intensity who declines an a posteriori physical identification. What makes the proposal a contribution rather than a relabeling is that, exactly as in physics, the dimensional treatment does three things beyond naming. It supplies a consistency criterion that rejects, before any experiment, every theory identifying phenomenal intensity with a dimensionless quantity, integrated information among them, on the ground that such an identity is not dimensionally homogeneous. It explains the power-law form of psychophysics as the homogeneous bridge across the dimensional divide that the ratio scales involved permit, holding exactly in the drive and asymptotically in the stimulus, rather than as a curve fitted to data. And it predicts an empirical signature of a single intensity dimension shared across the senses, the multiplicative composition and transitivity of cross-modal matching exponents, which the psychophysical record broadly displays. Read in time, the same bridge fixes the framework's dynamical commitments, locating every temporal feature of felt intensity in the physical drive, denying the phenomenal dimension a time constant of its own, and yielding a further empirical signature together with a boundary result on dissociative phenomena. A closing section turns the meta-problem of consciousness from a threat into an instrument, reading the intuition of a gap as the detection of dimensional mismatch and deriving from that reading a second empirical signature, for which a preregisterable study is specified in full. The framework is accordingly falsifiable, and the observations that would refute it are stated. Read along Russellian lines, Ξ preserves the causal closure of the physical and locates the hard problem precisely in the qualitative kind.