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 $\tau_g$ defines an interval during which fixed photon quanta can be accumulated into a single emission event. We retain a fixed photon quantum, written as $p_{\gamma}=hc$ (a constant), and introduce the dimensionless frequency ratio $\nu^*=\nu/F_q$ and the dimensionless coherence window $\tau_g^*=F_q\tau_g$. The gate admits an integer count of quanta$$N(\nu^*)=\left\lfloor \nu^*\,\tau_g^* \right\rfloor,$$and photoemission occurs when the binary gate-open condition $N(\nu^*)\ge N_0$ is met, where $N_0$ is a receiver-dependent threshold count. Above threshold, the event surplus delivered to the escaping electron state is modeled by the Surplus Law$$E_e(\nu^*)=\eta\,\bigl(N(\nu^*)-N_0\bigr)_+\,p_{\gamma},$$where $\eta\in(0,1]$ captures alignment/polarization coupling and $(x)_+=\max(x,0)$. In the negligible-smear limit, $E_e$ is identified directly with the emitted electron's kinetic-energy descriptor. The resulting staircase in $\nu^*$ is discrete at the event level but averages to an apparently linear trend when $N\gg 1$ and/or when emitter ensembles and coherence jitter smear micro-steps. The step spacing is $\Delta\nu^*=1/\tau_g^*$, 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:$$\varphi_e=\frac{E_e}{e^2},\qquad \psi_e=\frac{E_e}{{e_\mathrm{emax}}^{2}},$$linked by the charge-primitive identity$${e_\mathrm{emax}}^{2}=\frac{e^2}{8\pi\alpha}\quad\Rightarrow\quad \psi_e=(8\pi\alpha)\,\varphi_e.$$The electrostatic scalar $\varphi_e$ corresponds to an electrode-bias readout, while the magnetic scalar $\psi_e$ corresponds to magnetic-primary seat-filling dynamics. The paper reframes the dispute as a control-sensitivity question: does the coherence window $\tau_g^*$ and threshold $N_0$ 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 $\Delta\nu^*$ and $\nu^*_{\mathrm{th}}$ shift under controlled changes of electrostatic preparation versus magnetic shielding/geometry, while separately mapping alignment effects through $\eta$. 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 $\beta_0=1/(16\pi^2)$, yielding $\tau_g^*=\pi/\beta_0=16\pi^3$. This closure fixes $\Delta\nu^*$ and is directly testable in single-emitter regimes.