Variational Backbone and Regime Closures XVI: The Mixed Gate as a Common Backbone Regime of Conservative and Dissipative Readout

This paper should be read as the mixed-gate branch paper: the point is not merely tointerpolate between pure conservative and pure dissipative limits, but to isolate a genuinelyautonomous regime in which persistence, loss, and hidden-mediated restoring effects coexistat leading order on the same retained equation. Part XVI studies this mixed gate within thelow-energy atlas fixed in Part XV: the local overlap regime in which conservative persistence,dissipative loss, and summary-mediated restoring effects remain simultaneously active atleading order on one retained summary equation. The representative-branch conventionsand the licensed summary-instance discipline are therefore inherited from Part XV and arenot restated here in full. Starting from the hidden-reduced mixed equation, we derive theeffective operator Leff, the complex dispersion relationMRω2+iΓRω−Ω2eff(k) = 0,the mixed discriminantand the effective energy law∆mix(k) = 4MRΩ2eff(k)−Γ2R,ddt Emix(t) = −ΓR∥∂tu∥2.These identify underdamped, critical, overdamped, and unstable windows within one retainedequation. In the underdamped window one also obtains the damping scaleτdec = 2MRΓR ,together with a modewise coherence criterion through the mixed discriminant and the qualityfactor Qcoh(k). The formula itself is the standard envelope-decay time of an underdampedlinear mode; what is new here is that the same reduced coefficient set determines, withinone backbone-licensed regime, the damping scale, the oscillatory/non-oscillatory boundary,the coherence factor, and the opening of possible instability bands 9, 11. Lindblad-typestatistical closure and near-horizon irreversible readout are treated only as downstreamdirections of the same mixed branch, not as primitive starting points.