Operational-Manifold Redshift from a Six-Internal-Coordinate Schrödinger Operator

This report develops a detailed operational-manifold derivation of cosmological redshift from a six-internal-coordinate Schrödinger-type operator with three internal imaginary generators. The aim is not to replace the empirical success of Maxwell electrodynamics, the Schrödinger equation, quantum field theory, or the standard cosmological model. Instead, the report formulates a mathematically explicit research hypothesis: observed redshift may be represented as an accumulated phase-holonomy mismatch between the observable electromagnetic readout and a deeper internal-phase carrier defined on an operational multidimensional manifold. The central result is a compact redshift law, ln(1+z) = integral kappa_op(s) ds, which reduces to ln(1+z) = s/R_op for an approximately constant operational curvature radius. With H0 = 67.4 km s^-1 Mpc^-1, the corresponding R_op = c/H0 is about 14.51 billion light-years. The report gives the algebra of the three internal imaginary units, the covariant operator, the TERRA SPACE / I/Z0 closure-readout rule, a matrix differential derivation of the phase law, and a list of testable consequences and possible explanations of selected cosmological and quantum-foundation paradoxes