Pairwise Commutator Functionals and Relative Mass in Finite-Dimensional C-Algebras*

Within the framework of finite-dimensional C*-algebras generated by non-commuting orthogonal projections and their centre-breaking extensions, we define a relative mass functional (M_{\rm prim}) — a one-body observable measuring the squared Hilbert–Schmidt distance of a generating set from a block-diagonal reference structure prescribed by a distinguished abelian subalgebra — and a class of pairwise commutator functionals (\Phi) built from two-body commutators. We prove a classification theorem: under natural axioms, any pairwise commutator functional is determined up to three positive block-symmetric coupling constants. The proof relies on the saturation of the adjoint module by the generating set (established via Burnside’s theorem) and on the uniqueness (up to scale) of the invariant bilinear form on (\mathfrak{su}(n)). Both functionals are computed for the minimal irreducible block: (M_{\rm prim} = 2) (constant under standard trace), while the total internal curvature (C) depends on the Halmos angle. Their genuine independence is established by explicit counterexamples. A monotone descent inequality along centre-descent chains provides the quantitative counterpart to the qualitative commutant collapse theorem, and (M_{\rm prim}) is identified as a structural memory functional measuring the progressive erasure of superselection-sector information. The gauge structure is developed in two stages: a rigorous block-diagonal unitary equivalence, and a tentative connection-type deformation proposed as an exploratory direction. The present work should be viewed primarily as a structural classification result in finite-dimensional operator algebras, rather than as a physical dynamical theory. No evolution equation, variational principle, or physical Hamiltonian is assumed. All core results are purely algebraic; physical interpretations are explicitly separated and do not affect the mathematical theorems.