Manuscript V@1 · V.tex · section · e7ddf37d

Option A and Option B are different inverses

Option A and Option B are different inverses · lines 4057–4128

All entries in the following table refer to the trace-reversed metric. Let

ρ = Q_{00} (t_{0}), J_{i} = Q_{0 i} (t_{0}),

and use the same Newton inverse in both columns.

Datum	Option A, used in Section sec:contrast	Option B, comparison only
$j ˉ_{00} (t_{0})$	$- 2 Δ^{- 1} ρ$	$- 2 Δ^{- 1} ρ$
$j ˉ_{0 i} (t_{0})$	$0$	$- 2 Δ^{- 1} J_{i}$
$j ˉ_{ij} (t_{0})$	$0$	$0$
$\partial_{t} j ˉ_{00} (t_{0})$	$0$	$- 2 Δ^{- 1} \partial_{i} J_{i}$
$\partial_{t} j ˉ_{0 i} (t_{0})$	$0$	$0$
$\partial_{t} j ˉ_{ij} (t_{0})$	$\partial_{i} V_{j} + \partial_{j} V_{i} - δ_{ij} \partial_{ℓ} V_{ℓ}$ , $V_{i} = 2 Δ^{- 1} J_{i}$	$0$
Wave-source part	$2 \int_{t_{0}}^{t} S_{w} (t - s) Q (s) d s$	The same

Figure fig:Option-A-composition displays the different initial-data prescriptions and identifies the option used in the cubic theorem.

The two sourced solutions have the same wave forcing, so their difference is homogeneous. In general their Cauchy data differ, and therefore their inverses differ. Homogeneity of the difference does not identify their scalar responses.

Even a separately proved pure-gauge relation would require the fixed-time boundary-data calculation in (eq:boundarygauge). With a scalar response constrained to zero Cauchy data, a gauge change can alter the selected functional.

The contrast theorem uses Option A alone. Neither option is used to define the external metric of Section sec:external or the independent oscillator-valued metric of Section sec:mixed. Replacing those metrics by $R^{A} T (ϕ)$ changes the object and its field degree.

Displayed formulas in this passage

unlabeled · line 4062
$ρ = Q_{00} (t_{0}), J_{i} = Q_{0 i} (t_{0}),$