Feasible Spectral Region of Saturated Order Defect and Iterative Persistence

For two orthogonal projections P and Q, define the overlap operator T as the compression of PQP to the range of P. By setting sigma = 2P − I and nu = Q, the pair is embedded into an involution–idempotent algebra associated with the mathematical formulation of the Superposition Principle. The Hermitian residual order-defect operator is defined by R = i(nu sigma − sigma nu), and it satisfies the identity P R² P = 4 T(I − T) on the range of P. This identity places Jordan blocks, local noncommutativity in the CHSH operator, and persistence under von Neumann alternating projections under a common spectral structure. Two global invariants are introduced. The maximal residual order defect is delta(T) = 2 sqrt(||T(I − T)||), while the persistence endpoint rho(T) is the largest nonunit spectral value of T. In finite dimensions, the feasible pairs (rho, delta) are exactly characterized as follows: For 0 ≤ rho ≤ 1/2, delta = 2 sqrt(rho(1 − rho)). For 1/2 ≤ rho < 1, 2 sqrt(rho(1 − rho)) ≤ delta ≤ 1. Thus defect and persistence are constrained to a single curve in the fast-convergence regime but can vary independently in the slow-convergence regime. The paper also derives the interval of CHSH operator norms compatible with prescribed local persistence endpoints. In particular, maximal CHSH capability, reaching the Tsirelson value 2 sqrt(2), can coexist with arbitrarily slow alternating-projection convergence. The results show that a single defect norm does not determine long-term dynamics and motivate the use of a joint defect–persistence spectral profile.