The Axioms of Cognitive Geometry: Agential Quadrature in the Sensorimotor Loop

Theories of consciousness invoke internal quantities without specifying their physical invariance. We treat the unfolding critique as requiring both an observation map and an admissible gauge. A realized sensorimotor loop is modeled as two physically identified sectors with first-order dynamics. Three axioms—real graded structure, sector–channel factorization, and balanced graded reciprocity—force the sector factor to the canonical form iσy . On every nonzero channel spectral fiber, the flow is elliptic and the sector signals stand in quadrature. Relaxing balanced gain and strict off-diagonality yields a real negative-loop family whose oscillatory modes obey a constrained phase–frequency–gain relation and terminate at an exceptional point. Factorization makes modal phase independent of channel spectral value, whereas a simple relative delay produces a linear phase slope. After polar normalization, the closed negative cycle has gauge-invariant Z2 holonomy. These results define a falsifiable geometry of a realized perception–action loop, but not by themselves a sufficient criterion for consciousness.