Manuscript X-E-ST@1 · X-E-ST.tex · section · 44614221

A nonempty scalar-source diagnostic

Diagnostics and named omissions › A nonempty scalar-source diagnostic · lines 2061–2101

X-E-ST@1#eq:sourcecheck

Let $m = 0$ , $u = t$ , and let $g = g_{0} (t)$ on a causal cylinder containing all relevant retained vertices. Use spatial cutoffs outside that cylinder. Define

G_{1} (t) = \int_{0}^{t} g_{0} (s) d s, G_{2} (t) = \int_{0}^{t} G_{1} (s) d s, G_{3} (t) = \int_{0}^{t} G_{2} (s) d s .

Since $P_{tr} T_{0} [t] = n \otimes n$ , direct zero-data integration gives

k_{1} [t] = - 2 G_{2} n \otimes n, C^{0} [k_{1} [t]] = - G_{1}, w^{0} [k_{1} [t]] = - G_{3} .

Also

- E_{m}^{R} L_{1} [k_{1} [t]] t = - \int_{0}^{t} g_{0} (s) G_{2} (s) d s .

Therefore

τ [t] = \int f (x) (\int_{0}^{t_{x}} (1 - g_{0} (s)) G_{2} (s) d s) d^{4} x .

This is nonnegative for $f \geq 0$ , and is nonzero for suitable smooth switching and observation choices. It checks the $- 2$ response normalization, the coordinate sign, and inclusion of the scalar-response part.

This same free scalar can obstruct the invariant coupled jet by Theorem thm:bianchi. A defined gauge-fixed tree and an empty invariant-jet fibre can therefore coexist without contradiction.

Displayed formulas in this passage

X-E-ST@1#eq:sourcecheck · line 2085
$τ [t] = \int f (x) (\int_{0}^{t_{x}} (1 - g_{0} (s)) G_{2} (s) d s) d^{4} x .$
unlabeled · line 2067
$G_{1} (t) = \int_{0}^{t} g_{0} (s) d s, G_{2} (t) = \int_{0}^{t} G_{1} (s) d s, G_{3} (t) = \int_{0}^{t} G_{2} (s) d s .$
unlabeled · line 2074
$k_{1} [t] = - 2 G_{2} n \otimes n, C^{0} [k_{1} [t]] = - G_{1}, w^{0} [k_{1} [t]] = - G_{3} .$
unlabeled · line 2080
$- E_{m}^{R} L_{1} [k_{1} [t]] t = - \int_{0}^{t} g_{0} (s) G_{2} (s) d s .$