Manuscript III@2 · III.tex · section · 82dad3e5

The four results

Introduction: the equation space precedes descent › The four results · lines 252–295

The first negative result fixes a free scalar and varies only its metric representative. A compact gauge change of a Poisson compensator produces

- κ \int f ξ^{μ} \partial_{μ} ϕ .

We choose $f, ϕ, ξ$ once so that this integral is nonzero. No function of the unchanged label $([h], ϕ)$ can then agree with the symbol on both representatives. The same representatives yield a quantitative regulated obstruction.

The second negative result allows the scalar to move but retains its free equation. The required condition is

P (X_{ξ} ϕ) = 0 for every compact-data solution of P ϕ = 0.

A solution-jet argument proves that this holds exactly for Minkowski Killing fields. In particular, no nonzero compactly supported gauge vector satisfies it.

The first positive result changes the equation space:

P ϕ_{0} = 0, P ϕ_{1} = - L_{h} ϕ_{0} .

The coefficient transformation is an action on these equations. For a named past-trivial gauge group, a retarded harmonic compensator defines a unique orbit representative. The complete first-order scalar then descends.

The second positive result fixes a free background $Φ$ and uses the linear triangular system

G^{(1)} (h) = 0, P φ = - L_{h} Φ.

Compact constrained harmonic data modulo residual wave gauges form a Hausdorff Cauchy quotient. A compact coupled Ward pair has a strictly positive value on an explicit solution.

Displayed formulas in this passage

unlabeled · line 257
$- κ \int f ξ^{μ} \partial_{μ} ϕ .$
unlabeled · line 268
$P (X_{ξ} ϕ) = 0 for every compact-data solution of P ϕ = 0.$
unlabeled · line 278
$P ϕ_{0} = 0, P ϕ_{1} = - L_{h} ϕ_{0} .$
unlabeled · line 288
$G^{(1)} (h) = 0, P φ = - L_{h} Φ.$

Nearby in this section

figure: Four descent problems