Manuscript X-G@1 · X-G.tex · theorem · 01566e14
Theorem: No exact source-preserving variational upgrade
Source switching and an exact-upgrade obstruction › A pure-gauge background in the ramp · lines 1258–1281 · proof included or linked
X-G@1#thm:obstruction
There is no upgrade satisfying Definition def:upgrade on X-F's unrestricted background class.
The contradiction remains if off-shell representatives are changed by terms vanishing on all free configurations. It cannot be removed by the algebraic auxiliary field, ghost loops, or counterterms beginning strictly at quantum order.
Proof. Linearize the claimed reduced action response at zero fluctuation, zero scalar, and zero ghosts, in the background (eq:rampgauge). Its mixed metric Hessian must satisfy Lemma lem:reciprocity. Proposition prop:Dsigned contradicts that condition.
Terms vanishing on all free configurations do not change the witness. Algebraic auxiliary elimination yields another action, whose second variation is still symmetric. Ghost loops and positive- counterterms do not change the classical coefficient used in the contradiction.