Taking conformal invariance and the existence of particle horizons as fundamental principles, we prove that all physical fields on a particle horizon necessarily belong to the same Paley–Wiener space, whose reproducing kernel is the sinc function \frac{\sin\Omega(\eta - \eta')}{\pi(\eta - \eta')}. The cutoff frequency \Omega is uniquely determined by the horizon scale \eta_{\mathrm{max}} via the sampling theorem \Omega \eta_{\mathrm{max}} = \pi. This structure remains form-invariant under conformal transformations, corresponding to the symmetry breaking so(4,2) \to so(4,1). Applying this framework to the black hole horizon, we derive that the Hawking radiation spectrum is strictly zero for frequencies \omega > \Omega = 4\pi^2 T_H, rather than exhibiting exponential decay. This correction arises from the geometric nature of the particle horizon as a natural low-pass filter and leads to an observable prediction: the gamma-ray emission from primordial black holes of mass \sim 10^{15}\,\text{g} should abruptly cut off above 400\,\text{MeV}. This work reveals Hawking radiation as the projected truncation of global quantum fields on the horizon and provides a concrete physical realization of the holographic principle.