Factor Rigidity, Spectral Collapse, and A Maximal Gauge Algebra Pattern in Primitive Two-Projection GKSL Systems

We develop a complete rigidity classification for the algebraic structures generated by two orthogonal projections subjected to algebraically generated primitive quantum Markov semigroups (GKSL). Working in separable Hilbert spaces, we combine the Halmos decomposition with the dynamical constraint of primitivity to prove that every invariant irreducible subspace supporting the dynamics must be finite-dimensional with dimension restricted to 2, 3. Both cases are realizable and dynamically stable. For dimension 2 the dynamical closure is the factor (M₂ (C) ). For dimension 3 the static algebra is reducible ( (M₂ (C) C) ), yet the primitive GKSL dynamics generates its irreducible closure (M₃ (C) ) via Burnside’s theorem. We introduce a factor rigidity index ( (M) ) that cleanly distinguishes three phases and identify (n=3) as the universal rigidity threshold. In the three-dimensional sector the derivation algebra is (su (3) ), the stabilizer of the rank-2 projection is (su (2) u (1) ), and a phase-direction uniqueness theorem shows that at most one independent abelian continuous symmetry is permitted. Together these yield the unique maximal gauge Lie algebra pattern (su (3) su (2) u (1) ) compatible with the minimal algebraic and dynamical constraints. This provides a precise structural boundary explaining why this particular gauge pattern is singled out by elementary operator-algebraic principles.