The Origin of Spin: A New Perspective on the Retarded Green’s Function

Two postulates—the massless wave equation \square \phi = 0 and Minkowski spacetime—determine a reproducing- kernel Hilbert space (RKHS) on the celestial sphere S^2 whose evaluation functional \mathrm{ev}_0 at the null- cone coincidence point categorically encodes the physics of all massless fields. We embed this structure in the six- dimensional null cone of \mathbb{R}^{4,2} , where \mathrm{SO}(4,2) acts linearly and the Aldaya obstruction disappears. We prove the exact partial- wave decomposition of \delta (\sigma^2) : the spectral function is f(\ell ;u_0) = P_\ell (u_0) with u_0 = (r^2 + r'^2 - \tau^2) / (2rr') , giving P_\ell (1) = 1 on the null cone—the Isometric Sampling Condition (ISC). The proper- time spectral sum \Sigma^{(4)}(t) = \cosh (t / 2) / [2\sinh^2 (t / 2)] has Laurent coefficient ratio c_{- 2} / c_0 = 24 = \chi (K3) . Through the identity B(1 - \alpha_t,1 - \alpha_s) = \int_0^\infty dr e^{- (1 - \alpha_t)r}(1 - e^{- \tau})^{- \alpha_s} , we establish that the Veneziano amplitude is the Laplace transform of the null- cone heat kernel, with the ISC emerging at \alpha_s = 1 where the Pochhammer weights (1)_n / n! = 1 . We identify this structure with the observer encoding map \hat{V} of Harlow, Usatyuk, and Zhao for quantum gravity in a closed universe, where the one- dimensional Hilbert space corresponds to P_\ell (1) = 1 and the observer's bandwidth L grows a Hilbert space of dimension (L + 1)^2 . We establish a 25- entry dictionary with Nagano's K3 lattice theory. The \mathrm{SL}(2,\mathbb{C}) holonomy R(2\pi) = -\mathbb{I} at the null- cone branch point yields the spin- statistics theorem. All numerical results are verified to precision < 10^{- 13} .