Unique Bandlimited Green’s Function as Physical Field: Analytic Unification of Gauge Fields, Quantum Gravity, Fermions, Dissipative Dynamics, and Celestial Holography 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 \linebreak {(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 $\dim\mathcal{H}_{\mathrm{universe}}=1$ \[1-8]. Under three independent postulates A1--A3 together with this result, we derive the unique physical Green's function $G = \sin\!\bigl(\Omega\sqrt{-\sigma^{2}-i\varepsilon}\bigr)/ \bigl(\Omega\sqrt{-\sigma^{2}-i\varepsilon}\bigr)$, $\Omega=\pi/t_{P}$. The bandlimited two-point function $K$ of the resulting Paley--Wiener space $\mathrm{PW}_{\pi/t_{P}}$ admits the spherical Bessel decomposition $ K(x,x') = \frac{\Omega^{3}}{2\pi^{2}}\sum_{l=0}^{\infty} (2l+1)\,j_{l}(\Omega r)\,j_{l}(\Omega r')\,P_{l}(\cos\theta).$ 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 $\Pi_{l}$ follows from the $\mathrm{SO}(4,2)$ representation theory of massless fields on the six-dimensional light cone~\cite{Dirac1963,Aldaya1999}; (v)~fermions arise necessarily from the spinor representations of $\mathrm{SO}(4,2)$ via $\mathcal{H}_{\mathrm{tot}}=\mathcal{H}_{\mathrm{pos}}\otimes \mathcal{H}_{\mathrm{int}}$. 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 $\mathbb{C}$. Bandlimitedness is a theorem, not an assumption. Since all cosmological observables---CMB~($TT$,$TE$,$EE$), large-scale structure, and weak lensing---are recorded along null geodesics ($\sigma^{2}=0$ where $G=1$ exactly, with no dimensional suppression), they jointly probe the same $\Delta_{l}=l+1$ structure on the celestial sphere. Their combined Bayesian posterior $P(\theta|\mathrm{data})\propto\prod_{i}\mathcal{L}_{i}$ compresses the posterior width as $1/\sqrt{N_{\mathrm{datasets}}}$, providing a simultaneous, parameter-free observational test.