Manuscript X-A-ST@1 · X-A-ST.tex · section · d9ccbde1

Second-order operator audit

Relational expansion and covariance: Route I › Second-order operator audit · lines 1215–1242

X-A-ST@1#eq:operatoraudit

For either the coordinate or scalar operator write $P + κV + κ^{2} W$ , and let primes denote the transformed coefficients. Put

X = X_{ξ}, T_{2} = X_{ζ} + \frac{1}{2} X_{ξ}^{2}, D = D_{ξ}, Δ_{2} = D_{ζ} + \frac{1}{2} D_{ξ}^{2} .

The proven covariance expands to

V^{'} W^{'} + V^{'} X + P T_{2} = V + D P - P X, = W + D V + Δ_{2} P .

These are identities on arbitrary smooth test scalars, not only on solutions. For the coordinates, $P = □$ and $P x^{μ} = 0$ ; inserting (eq:operatoraudit) verifies directly that

v_{cand} = v + X w + T_{2} x

solves the transformed second-order equation with zero data. This checks the quadratic metric law without deleting the terms proportional to $P$ .

Displayed formulas in this passage

X-A-ST@1#eq:operatoraudit · line 1225
$V^{'} W^{'} + V^{'} X + P T_{2} = V + D P - P X, = W + D V + Δ_{2} P .$
unlabeled · line 1220
$X = X_{ξ}, T_{2} = X_{ζ} + \frac{1}{2} X_{ξ}^{2}, D = D_{ξ}, Δ_{2} = D_{ζ} + \frac{1}{2} D_{ξ}^{2} .$
unlabeled · line 1235
$v_{cand} = v + X w + T_{2} x$