Submitted:
10 March 2026
Posted:
12 March 2026
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Abstract
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- 8]. 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 bandlimited two- point function K of the resulting Paley- Wiener space \mathrm{PW}_{\pi /t_{P}} admits the spherical Bessel decomposition
K(x,x^{\prime}) = \frac{\Omega^{3}}{2\pi^{2}}\sum_{l = 0}^{\infty}(2l + 1)j_{l}(\Omega r)j_{l}(\Omega r^{\prime})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 [9, 10]; (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.
Keywords:
bandlimited Green’s function
; Paley-Wiener space
; quantum gravity
; gauge field unification
; celestial holography
; 2D CFT
; light cone
; parameter-free prediction
; COSMICMicrowave Background (CMB)
; spherical bessel functions
; spin representation
; finite energy postulate
Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.
