Thresholds on the Exceptional Line: Colour Factors, Universal Formulas and Supergravity Counts

We compute permutation characters, colour factors and supergravity candidate counts using exceptional invariant theory. At ten tensor factors, the fixed spaces for D₄ and F₄ fail to contain the E₇ symmetric-group module, while the E₆ fixed space contains it with a complement of dimension 7805. For each of the six members other than A₁ and A₂, connected trees have a common permutation module through ten legs. For A₂, we determine the ten-leg tree character, of dimension 37142, and the irreducible decomposition of the 1302-dimensional quotient of degree-nine Lie identities by lower-degree consequences. Classical groups determine connected vacuum colour factors through twelve loops, with an obstruction at fourteen; the thirteen-loop case remains open. For restricted quantum-dimension factors, sign cancellations raise the sharp uniqueness threshold from total line weight three to five. We also construct an unrestricted symmetric factor equal to one on all twelve Vogel lines. Quartic trace identities give exceptional-group specialisations of the four-loop lightlike cusp and five-loop beta function, including vector fermions for SO(8). In the seven-loop scalar-polynomial sector of N = 8 supergravity, generating functions count scalar seeds for counterterms and candidate soft operators. The latter strictly exceed the former at every even field number at least eight, and their ratio is 7 − 420/n + O(n⁻²). Applying standard Euler homotopy and Koszul exactness to the weighted invariant ring gives a canonical complement to invariant gradients, generated by 21 explicit forms. Using established supersymmetric descent, we express local soft obstructions as one linear condition at eight fields and two at ten.