Effective Quantum and Gravitational Structure from the Compact Fiber: Cross-Frame Projection, Hilbert Representation, Born Rule, and Coordinate-Time Metric

This paper develops an effective quantum and gravitational representational layer from the compact fiber Z/4Z × Z/4Z × Z/8Z and its degree-2 covering map, previously derived from discrete scalar-motion postulates. It argues that core structures usually treated as primitive in standard quantum mechanics—complex amplitudes, inner product, Born-rule probabilities, effective Hilbert space, superposition, entanglement, Bell correlations, and the measurement projection structure—can instead be derived as representational consequences of the compact carrier and cross-frame projection. The paper also develops a parallel gravitational program in which an effective coordinate-time metric is obtained from reciprocal partition of gravitational potential, reproducing the first-order PPN values beta = gamma = 1, the equivalence principle, and the four classical Solar System tests, while identifying second-order departures from general relativity as empirical discriminators. The manuscript explicitly audits which ingredients are derived, which are imported mathematical tools, which are interpretive identifications, and which remain open formalization tasks. A quantitative anchor is provided through the 87Rb Rydberg Förster-resonance regime, where the framework reproduces the observed R^-3 interaction scaling and the reported C3 scale to within the stated accuracy.