Algebraic Collapse, Modular Rigidity, and Sector Exclusion from Contractive Fixed Points on Quantum Markov Semigroups

This paper establishes the structural closure layer of the canonical contractive quantum Markov semigroup (QMS) framework. Working entirely at the operator-algebraic level, we prove that strict contractivity at fixed points forces algebraic collapse at active constraints, enforces modular rigidity (p = 2) via the parallelogram law, yields primitive completeness within the GKSL class, determines a minimal admissible carrier structure, and excludes non-admissible sectors via Osterwalder–Schrader measure positivity. The collapse theorem is derived using the Pinsker inequality and the conditional expectation Pythagorean identity, showing exact stabilization at vanishing free energy. Modular rigidity eliminates non-Hilbert operator geometries within the canonical Schatten family. The GKSL commutant characterization proves that no additional primitive constraints arise beyond the Hamiltonian, Lindblad operators, boundary map, and state. A minimal constraint count is obtained under explicit bidual independence, and higher-arity carriers are excluded by constraint budget analysis. Finally, sector exclusion is established through OS positivity: non-admissible sectors produce destructive phase interference and fail positive reconstruction, leaving only the admissible structural sector. This paper completes the structural closure analysis of the canonical QMS–spectral programme. The universality reduction and gravitational forcing results are developed in a companion capstone paper.