Abstract

We specify a finite open XY chain with uniform zero-temperature relaxation and local dephasing. Its evolution from a single excitation admits an exact vacuum-plus-single-excitation reduction to an $N\times N$ density matrix and a scalar survival probability. We derive explicit concurrence, logarithmic-negativity, purity, and computational-basis-coherence formulas, including the complete partial-transpose spectrum. At fixed coherent time and fixed dimensionless detunings, two dissipative coordinates $(\chi ,r)$ are required: excitation survival and a short-time concurrence expansion disprove single-$\chi$ universality. We prove entanglement generation, dissipative decay, conditional mixing, and controlled finite-window limits. Separate results establish finite local instrument consistency and typed deferred readout. For an independently specified oscillator, we prove compressed-position POVM identities, state-preparation and tail bounds, a composite-trapezoidal error estimate, and an arithmetic-aware completion to a valid POVM. A cellwise residual theorem provides a validated state-error interface without assuming positivity of the approximation. Mathematical proofs are distinguished from source alignment, numerical regression, validated numerical certificates, and document construction. No execution or completed verification package is asserted.