Remark. Symmetry of a marginal and restriction of a precision [fdip-0058]

✍️sourceAGENTDRAFTED

The sector laws use an ordinary orthogonal finite-group action on a real field space and a strictly positive precision. These hypotheses allow independent Gaussian factors and equivariant elimination. They do not make every stable retained subspace a reducing subspace for the precision. A stable vertex set says that its coordinate projection commutes with the group action; cycle decimation shows that it need not commute with \(Q\). The resulting Schur precision can therefore differ from the compression \(Q_{RR}\). Commuting with the group action and commuting with \(Q\) are distinct conditions.

Three other representation-theoretic questions need additional data. Projective representations and twisted group algebras record a multiplier rather than an ordinary group action; the projective obstruction explains its role. Restriction to a normal subgroup and Clifford's homogeneity theorem concern constituents under a specified subgroup inclusion; this is Clifford theory of group representations, distinct from Clifford algebras. In positive characteristic, the averaging denominator or Gaussian measure used here may be absent. None of these is supplied by real isotypic factorization alone.