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

Homogeneous data belong to the quotient tuple

Compact coupled Ward observables › Homogeneous data belong to the quotient tuple · lines 2279–2305

Every C1 solution decomposes uniquely as

φ = S_{t_{0}} (h) Φ + H_{t_{0}} (u, v) .

Under a residual gauge,

φ + X_{ξ} Φ - S_{t_{0}} (h + K ξ) Φ = φ - S_{t_{0}} (h) Φ + H_{t_{0}} (X_{ξ} Φ) .

The homogeneous scalar data change by (eq:scalar-gauge-data). They therefore belong in the full quotient datum.

The conclusion is existence:

compact coupled Ward pair \neq ⟹ zero on shell .

No complete compact-current classification is claimed.

Displayed formulas in this passage

unlabeled · line 2282
$φ = S_{t_{0}} (h) Φ + H_{t_{0}} (u, v) .$
unlabeled · line 2286
$φ + X_{ξ} Φ - S_{t_{0}} (h + K ξ) Φ = φ - S_{t_{0}} (h) Φ + H_{t_{0}} (X_{ξ} Φ) .$
unlabeled · line 2298
$compact coupled Ward pair \neq ⟹ zero on shell .$