Manuscript Oai069-A@1 · Oai069-A.tex · proposition · a4913aab

Proposition: Finite Ward-defect identity

Phase 3: an independent free-scalar consistency audit › A finite Ward-contact consistency certificate · lines 1559–1585 · proof included or linked

Oai069-A@1#prop:ward, Oai069-A@1#eq:ward

For a conformal Killing field $X$ and a compact joint test $F$ ,

[\partial^{μ} (X^{ν} t_{μν}) - C_{X}] (F) = ν \sum E_{ν} (X^{ν} F) + E_{tr} (σ_{X} F) + K_{X} (F) .

Thus equality of the named insertion and contact data on both declared sides gives equality of the named finite Ward residuals.

Proof. The improvement has zero divergence and trace $- □ L$ . Contracting the conformal Killing equation with the symmetric insertion gives

\partial^{μ} (X^{ν} t_{μν}) = X^{ν} \partial^{μ} T_{μν} + σ_{X} (T^{μ}_{μ} - □ L) .

Substitution and evaluation on $F$ prove (eq:ward).

Displayed formulas in this passage

Oai069-A@1#eq:ward · line 1562
$[\partial^{μ} (X^{ν} t_{μν}) - C_{X}] (F) = ν \sum E_{ν} (X^{ν} F) + E_{tr} (σ_{X} F) + K_{X} (F) .$
unlabeled · line 1579
$\partial^{μ} (X^{ν} t_{μν}) = X^{ν} \partial^{μ} T_{μν} + σ_{X} (T^{μ}_{μ} - □ L) .$