Coherence-Window Quantization in Photoemission: Dual Electrostatic and Magnetic Gate Laws in QMU

This paper supplies the kinematic mechanism underlying the photoemission gate developed in the companion ``Photoelectric Effect Foundation'' note. The core postulate is that the photoemission receiver is not a continuous energy absorber but a coherent counter: a finite coherence window g defines an interval during which fixed photon quanta can be accumulated into a single emission event. We retain a fixed photon quantum, written as p_=hc (a constant), and introduce the dimensionless frequency ratio ^*=/Fq and the dimensionless coherence window g^*=Fqg. The gate admits an integer count of quantaN (^*) = ^*\, g^*, and photoemission occurs when the binary gate-open condition N (^*) N₀ is met, where N₀ is a receiver-dependent threshold count. Above threshold, the event surplus delivered to the escaping electron state is modeled by the Surplus LawEₑ (^*) =\, (N (^*) -N₀) _+\, p_, where (0, 1] captures alignment/polarization coupling and (x) _+= (x, 0). In the negligible-smear limit, Eₑ is identified directly with the emitted electron's kinetic-energy descriptor. The resulting staircase in ^* is discrete at the event level but averages to an apparently linear trend when N 1 and/or when emitter ensembles and coherence jitter smear micro-steps. The step spacing is ^*=1/g^*, which explains why ordinary macroscopic photoelectric measurements typically appear continuous. A central goal of the paper is to make the ``barrier'' interpretation experimentally decidable without arguing over definitions. The same event energy is parameterized by two conjugate scalars: ₑ=Eₑe², ₑ=Eₑ{eₑmax^2}, linked by the charge-primitive identityeₑmax^2=e²8 ₑ= (8) \, ₑ. The electrostatic scalar ₑ corresponds to an electrode-bias readout, while the magnetic scalar ₑ corresponds to magnetic-primary seat-filling dynamics. The paper reframes the dispute as a control-sensitivity question: does the coherence window g^* and threshold N₀ respond primarily to electrostatic boundary control (E-gate) or to magnetic boundary control (M-gate)? Two experimental protocols are proposed to discriminate these hypotheses by measuring how extracted ^* and ^*ₓ₇ shift under controlled changes of electrostatic preparation versus magnetic shielding/geometry, while separately mapping alignment effects through. An appendix provides an optional parameter-free candidate for the coherence window using a topology-set holonomy slip derived from a constant polarization-rotation angle ₀=1/ (16²), yielding g^*=/₀=16³. This closure fixes ^* and is directly testable in single-emitter regimes.