The Laguerre Structure of Root Moments and Exceptional Integrated Correlators

Gaussian root moments determine perturbative integrated correlators and occur as the leading affine-level coefficients of contracted adjoint traces. We prove that, for each root orbit of every simple Lie algebra, their generating function is a Gaussian factor times a positive linear combination of odd Laguerre polynomials. The weights are Fischer norms obtained from the highest-root factorization of the Weyl discriminant. This decomposition establishes kernel inversion symmetry and, in the simply laced case, expresses the lattice completion as a polynomial in the modular Laplacian applied to the SU(2) correlator. A degree bound and Chinese remainder certificates determine the complete E₇ and E₈ moment polynomials, yielding exact kernels and adjoint Wilson loops. The same decomposition reduces the sign-corrected one-instanton root formula to one-dimensional integrals. We prove their equality with the lattice prediction for every odd level, and hence for all simply laced algebras and for G₂ and F₄. For SU(N), a direct comparison with the ADHM formula establishes the physical one-instanton equality at every rank. The exceptional interpretation remains conditional on the root formula.