The Image Lands on the Wrong Category: The Frame's Discreteness Derivation in Yang–Mills Dress, and the Mass-Gap Attempt's Death by Its Own Kill Condition

The mass-gap question has stood open in this program, stated honestly as a Millennium-grade problem: build a Yang–Mills image of this program's discreteness derivation. An earlier survey in this program, of which of its results might transplant to other subjects, specified the attempt and its failure condition in advance: the price of the attempt is a Yang–Mills image of the derivation, and if the proof essentially uses torsion and areal saturation, with no gauge-theory image, the attempt fails at its first working attempt, because it then matches the target in shape only, with no shared mechanism. This paper is the first working attempt. This program's discreteness — derived in an earlier study conditional on Axiom H, its corollary being that the vacuum charge density vanishes identically and every excitation carries one identical quantum — is a discreteness of a quantum number: excitations are sorted into identical, indivisible units of a carried charge. Transported faithfully into Yang–Mills theory, that statement images onto the discreteness Yang–Mills already owns: its charge sectors (topological charge, center flux) are discrete, and its excitations carry quantized labels. The image exists. What it does not reach is the mass gap, because the Millennium statement is not about labels. It is a statement about a spectrum: that pure Yang–Mills theory on R³ exists, and that its Hamiltonian has a positive gap Δ > 0 above the vacuum. Label discreteness and spectral gap are logically independent, and this paper exhibits the independence in both directions with textbook counterexamples: quantum electrodynamics carries quantized charge and no gap; a free massive field carries a gap and no charge quantum of any kind. A derivation whose conclusion is a label discreteness therefore cannot, by any imaging, become a derivation of a spectral gap: the conclusion is in the wrong category before any dynamics is discussed. The failure condition is then evaluated on the merits: the derivation is, by its own filing, conditional on areal saturation (Axiom H) inside a torsion frame — it essentially uses exactly the two ingredients the condition names, and the image it admits is a shape match, not a mechanism. Verdict: the mass-gap attempt fails at its first working attempt, exactly as the condition stated in advance said it would. The mass gap stands untouched — and the paper closes by naming, exactly, the kind of result in this program that would not fail this way, so the wall's height is measured rather than gestured at.