Unique Bandlimited Green's Function as Physical Field: Analytic Unification of Gauge Fields, Quantum Gravity, Fermions, and Dissipative Dynamics on the Light Cone

We prove that \delta^{(4)}(x - y) \notin L^{2}(\mathbb{R}^{4}) is not a legitimate physical Green's function under the quantum- mechanical postulate of finite energy (A1). A fourth postulate of closed sourcelessness (A4)—methodologically analogous to Einstein's postulate of the constancy of the speed of light—is derived as a theorem from the quantum- gravity result \dim \mathcal{H}_{\mathrm{universe}} = 1 [1- 4]. Under three independent postulates A1- A3 together with this result, we derive the unique physical Green's function G = \sin (\Omega \sqrt{-\sigma^{2} - i\epsilon}) / (\Omega \sqrt{-\sigma^{2} - i\epsilon}) , \Omega = \pi /t_{P} . The reproducing kernel K of the resulting Paley- Wiener space \mathrm{PW}_{\pi /t_{P}} admits the spherical Bessel decomposition... We prove: (i) the l = 0,1,2 sectors are precisely the scalar, photon, and graviton propagators; (ii) gauge symmetry emerges as the zero- set geometry of j_{l} ; (iii) restriction to the light cone \sigma^{2} = 0 yields the celestial sphere S^{2} with 2D CFT two- point structure and conformal dimensions \Delta_{l} = l + 1 , parameter- free; (iv) tensor structure \Pi_{l} is not postulated but follows from the SO(4,2) representation theory of massless fields on the six- dimensional light cone [5, 6]; (v) fermions arise necessarily from the spinor representations of SO(4,2) via the tensor- product structure \mathcal{H}_{\mathrm{tot}} = \mathcal{H}_{\mathrm{pos}}\otimes \mathcal{H}_{\mathrm{int}} . All four physical regimes (QFT, quantum gravity, gauge fields, dissipation) are restrictions of the single entire function f(z) = \sin (z) / z to different domains of \mathbb{C} .