Four-Quadrant Ontology, Categorical Structure, and the Recovery of Eight Theoretical Sectors in PODA: Symmetry, Gauge Covariance, Causality, and Common Quantum–Gravitational Dynamics

An experimental record depends both on the state of the system and on the conditions under which it is observed. Physical–Observation Dual-Axis (PODA) theory retains these as distinct arguments of the record law. The four-quadrant ontology identifies their roles; equivalence of complete responses gives a canonical, behaviorally minimal representation. We prove that a fixed observation section recovers a reduced theory precisely when the full evolution is tangent to that section and the constraints, boundary data, transport, and records agree. With the continuous, coherent, and probabilistic structures specified here, this criterion recovers general relativity, Maxwell and Yang–Mills fields, Dirac and Schrödinger dynamics, and quantum and classical information. Symmetries arise as automorphisms of the record diagram, gauge covariance from changes of response frame, and causal boundaries from finite propagation together with realizable histories. For gravity, a single action governs the response projector, connection, matter, and metric. Its variation relates response stress to Ricci curvature. About a stable flat background, the quadratic metric action admits a finite-mode quantum representation with two transverse-traceless polarizations. Exact recovery, the nonrelativistic limit, and the weak-field expansion are treated separately: the first gives a correspondence of full solutions, while the latter two hold to their stated orders.