Manuscript Em01-C@3 · Em01-C-v3.tex · proposition · f0953702

Proposition: Exact local plateau reduction

Component conventions and the retained internal expression › Exact plateau localization · lines 749–803 · proof included or linked

Em01-C@3#prop:plateau, Em01-C@3#eq:plateau-identity

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On the retained patch,

K_{ret} = K_{ret}^{pl}, K_{8} = K_{8} .

The first equality holds as an equality of internal distributions; it therefore persists in every positive-scale regularized contraction on tests in the patch.

Source-ramp derivative terms are exactly absent from this local expression. No smoothing gain follows merely from their absence.

Proof. The weighted source is

J_{D}^{μν} = σ_{ε} J_{3}^{μν} + 2 C_{2}^{μν α β} \partial_{α} \partial_{β} σ_{ε} - 2 M_{2}^{μν ρ} \partial_{ρ} σ_{ε} .

The source incidence enforces $v = a = z$ . Near its retained support, $v^{0} > ε$ , so the source and all its coefficient jets agree with the unweighted source there. The finite reduction (eq:response-reduction) consequently gives identical response distributions with $Q$ and $Q_{3}$ .

The explicit factors $σ_{ε} (a), σ_{ε} (z)$ are one on open neighborhoods of their retained variables; all their derivatives vanish there. This proves the first equality.

For (eq:K8-unreduced), retarded support enforces

v = a = z, s \in J^{+} (a) \cap J^{-} (b) .

Hence $s^{0} \geq a^{0} > ε$ . Replacing $σ_{ε} (s)$ by one, including inside the displayed derivative, leaves the distribution unchanged. Differentiation does not enlarge support.

The intermediate integration is support-proper on compact endpoint sets, since $s$ lies in compact causal diamonds. Causal convolution then gives

\int R (b - s) \partial_{s^{α}} R (s - v) d^{4} s = \partial_{b^{α}} B (b - v),

which is exactly the reduced expression (eq:K8).

Displayed formulas in this passage

Em01-C@3#eq:plateau-identity · line 752
$K_{ret} = K_{ret}^{pl}, K_{8} = K_{8} .$
unlabeled · line 768
$J_{D}^{μν} = σ_{ε} J_{3}^{μν} + 2 C_{2}^{μν α β} \partial_{α} \partial_{β} σ_{ε} - 2 M_{2}^{μν ρ} \partial_{ρ} σ_{ε} .$
unlabeled · line 787
$v = a = z, s \in J^{+} (a) \cap J^{-} (b) .$
unlabeled · line 797
$\int R (b - s) \partial_{s^{α}} R (s - v) d^{4} s = \partial_{b^{α}} B (b - v),$

Proposition: Exact local plateau reduction

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