Replica cloud contact geometry and coherence on real Hilbert spaces

A replica cloud describes one physical identity through distributed alternatives whose coherence depends on contact. Coherence relative to a common reference determines a response operator, while the full contact covariance contains additional relative couplings that determine the real statistical generator. For any real dephasing channel in finite dimensions, the closest orthogonally covariant channel is obtained from an exact reduced semidefinite program, and an optimizer lies on one edge of the simplex of covariant channels. Geometries with transitive permutation symmetry admit a closed spectral solution. Two contact groups of arbitrary sizes give a second exact family in which unequal groups make orthogonal averaging strictly suboptimal whenever intergroup coherence decays. Separately, a contact Gram constraint for four modes uses five calibrated coherences to bound the allowed sixth coherence under a law of commuting contacts in a common basis. A version for finite counts includes monitored loss and bounded contrast drift and is evaluated with synthetic data. For each specified dephasing channel, the channel distance gives its best approximation error within the orthogonally covariant class; the Gram protocol tests the hypothesis of commuting contacts in a common basis.