Infratier Closure Physics - The Eightfold Cost in Color Closure

The proposed SU(3) confinement coordinate subtracts one eighth of a curvature residue. Counting eight generators gives the mean cost per generator once their total has been chosen. To obtain the same cost in every normalized direction requires an additional argument. We prove that full adjoint symmetry enforces this equality for a quadratic response at a symmetric background. The proof leaves the total cost and its relation to a physical depth free. We give a basis-independent test of equality, an invariant nonlinear counterexample, and a numerical concordance for the two descent manuscripts.