The Retarded Green's Function: Construction of Yang-Mills Theory on the Null Cone

We construct a quantum Yang--Mills theory with gauge group G (any compact simple Lie group) on four-dimensional Minkowski spacetime \( M^{3,1} \), starting from the retarded Green's function \( G_{\mathrm{ret}} = (2\pi)^{-1}\delta(\sigma^2)\theta(\Delta t) \) and a compact simple Lie group G with \( C_2(G) > 0 \). The construction proceeds entirely on the null cone. The partial-wave decomposition on the celestial sphere S2 yields the Isometric Sampling Condition (ISC) \( P_\ell(1) = 1 \) for all \( \ell \). The Whittaker cardinal-function theorem together with the commutation relation \( \lbrack L^2, T^a\rbrack = 0 \) guarantees that the ISC holds for arbitrary coupling g. We replace the path integral by a discrete sum over the reproducing-kernel Hilbert space (RKHS) on S2, where the interacting propagator is the unique solution of a Fredholm integral equation of the second kind. Complete monotonicity of the spectral measure is established through a rigorous proof chain: self-adjointness of \( H = H_0 + gW \) via the Kato–Rellich theorem (using the Hilbert–Schmidt property \( \|K_0 V\|_{\mathrm{HS}}^2 \approx 4.73 \)), the spectral theorem for self-adjoint operators, and the Bernstein theorem. The ISC determines the conformal weights \( \Delta_\ell = \ell + 1 \) and, via the Hurwitz-zeta evaluation, yields the exact one-loop \( \beta \)-function coefficient \( b_1 = 11C_2(G)/(12\pi) > 0 \)—asymptotic freedom—for all non-Abelian gauge groups. We prove that for \( g > 0 \) and \( C_2(G) > 0 \), the Yang–Mills self-interaction forces information off the null cone into the timelike interior (verified numerically via the convolution \( \delta(\sigma^2)*\delta(\sigma^2) \)), activating the angular spectral gap \( E_0 = 1/2 \) inherent in the SO(3) representation theory. The Aldaya–Calixto–Cervéro obstruction theorem establishes that the full Poincaré group (including spatial translations \( P_i \)) is unitarily represented on the constrained Hilbert space as "good operators,'' while the special conformal generators \( K_\mu \) undergo dynamical symmetry breaking—providing the physical mechanism for the mass gap. The mass gap is \( \Delta = \Lambda = \mu\exp\lbrack-2\pi/(b_1 g^2)\rbrack > 0 \) for all \( g > 0 \) and all compact simple Lie groups. All Wightman axioms are verified for the interacting theory. Every step in the proof chain uses published theorems of functional analysis; no new mathematical conjectures are required.