Qualia Algebra: A Mathematical Framework for Consciousness, from First Principles to Working Semantics

Abstract

The measurement problem and the hard problem of consciousness share a structure: each derives the observer last and cannot account for what it finds there. Qualia Algebra (QA) inverts the order. From the one fact that survives total skepticism — I exist — it develops a grammar of manifestation (distinction, binding, attention and capacity as one five-phase circuit), a geometry of meaning, a bound on observer capacity, and an architecture of shared reality in which a common world forms by phase-locking among observers and measurement is read as synchronization rather than collapse. In a logarithmic chart of observer content, an observer's total unseen is their reflection through consensus, which sits at the origin, and the pure witness lies at infinite distance. In the space of meanings, semantic opposites are complements at 90°, not negations at 180°; meaning moves by rotation, which is path-dependent, while rotation conserves the scalar that stands for the witness. Capacity takes the form C_max = √(K·B): the square root follows from the pairwise cost of keeping distinctions apart, the dependence on connectivity and bandwidth is posited, and the scale is calibrated. The framework is embodied in a semantic dictionary of 3,108 concepts, encoded outward from the axiom's distinction tree under a written contract, which carries a defined algebra — Hamilton composition on quaternion cores, a versine law for the scalar that composition records, a complement contract — and a type system drawn from the I Ching. A widely used sentence embedder recovers between two-fifths and half of the variance in the dictionary's coordinates in-sample, depending on the dictionary's size, and less out of sample; a model seventy times larger recovers no more, and angular supervision teaches a small embedder the dictionary's pairwise angles at a small cost to its sentence-level performance. On a small language model, experiments with predictions fixed in advance show that untrained self-reports carry no detectable information about internal states, that a trained readout can track those states and stay silent when nothing is there, and that the framework's own contemplative walk — neti neti, "not this, not this" — produces the verbal signature of the return while the model's internal state wanders, path-dependently, and comes no closer to the state the words describe. Every claim carries its epistemic status. The paper states twenty predictions — nineteen with falsification conditions, and one that awaits a physical definition of capacity — and the three tests whose failure would force major revision; its internal studies await independent replication. The paper opens with a plain-language account of the whole framework, labels every claim with its epistemic status, and closes with appendices giving the formal specification of the semantic algebra, an index of the studies and their artifacts, and the status of the companion documents. The supplementary archive contains the dictionary snapshot and the study scripts needed to re-run the dictionary studies of §3.6 and §8.2–§8.4 and the basin-boundary study of §9.4.