Coherence Before Symmetry: Positive Kernels, Random Frames, and Compatible Perspective Transformations

When do positive comparisons between perspectives become a system of unitary transformations? For normalized positive block kernels, maximal Hilbert–Schmidt overlap forces exact composition and descent to one shared Hilbert space. This pair-groupoid representation is pure gauge. A sharper result holds when a group organizes the perspectives: a normalized positive-definite function with maximal overlap must itself be a unitary representation. The group is input; the multiplicative realization on local data is output. A gauge-covariant formulation classifies saturated equivariant data by representations up to conjugation. Defect bounds, finite-group rounding, and a ℤ₃ example connect the reconstruction to approximate representations and Bargmann invariants. A random-frame Gibbs model converges to an explicit invariant law on the saturated locus, with a leading mean deficit given by a Morse–Bott calculation. In rank one its energy is the quartic complex-projective spin interaction. A Lean 4/mathlib companion verifies the core reconstruction directly from finite block-matrix positivity in arbitrary dimension, including the finite-group representation law. The proposed contribution is a conditional framework connecting positive quantum comparison, equivariant descent, and statistical selection, with the selection principle stated as an assumption.