Proposition: Signature ledger

The two signature conventions are related on the chart by

but are not silently substituted for one another. All density expansions, Euler signs, and Cauchy inverses below use $\eta$ and

The host's unit-lapse stationary proper time is $T$; the developed comparison variable is $t=c(T-T_{\Sigma })$. The formal metric jet introduced below is not a command that evolves $G$.

Proof. The displayed chart metric gives the first identity. The remaining statements specify which objects enter the analytical equations and which enter physical port placement. Reversal of the overall signature leaves their common null directions unchanged, but does not justify changing an Euler or Green-kernel sign.