Manuscript X-A-conv@1 · X-A-conv.tex · section · c76fffa3

The three signs that enter the witness

Sign and regulator audit › The three signs that enter the witness · lines 2351–2379

With signature $+ - - -$ ,

□_{η} = \partial_{t}^{2} - Δ, E_{0} s = + \int_{0}^{t} \frac{sin (( t - u ) ∣ D ∣ )}{∣ D ∣} s (u) d u .

The scalar action has a positive kinetic density, so its Euler derivative is $- D_{g} [q] ϕ$ . The first scalar coefficient satisfies

□_{η} ρ_{1, g} = - V_{1, g} [h] ρ_{0}, ρ_{1, g} = - E_{0} V_{1, g} [h] ρ_{0} .

The inverse coordinate map begins with $- w$ , so the first bare relational insertion is

- \int f X_{w} ϕ .

Consequently the scalar pullback and coordinate variation combine as

+ \int f X_{ξ} ϕ - \int f X_{S_{ε} ξ} ϕ,

which has the positive witness (eq:positiveenergy).

Displayed formulas in this passage

unlabeled · line 2354
$□_{η} = \partial_{t}^{2} - Δ, E_{0} s = + \int_{0}^{t} \frac{sin (( t - u ) ∣ D ∣ )}{∣ D ∣} s (u) d u .$
unlabeled · line 2364
$□_{η} ρ_{1, g} = - V_{1, g} [h] ρ_{0}, ρ_{1, g} = - E_{0} V_{1, g} [h] ρ_{0} .$
unlabeled · line 2371
$- \int f X_{w} ϕ .$
unlabeled · line 2375
$+ \int f X_{ξ} ϕ - \int f X_{S_{ε} ξ} ϕ,$