Manuscript XIII-D@1 · XIII-D.tex · input · bab69024

Input statement: Auxiliary remainder

Copied prerequisite sheet › XII-F: the retained pairing and witness · lines 383–405

XIII-D@1#eq:B-input, XIII-D@1#eq:witness-input

On $C_{sol}$ , under the copied causal-plateau condition,

Δ_{ξ} Φ^{(2)}_{D} = B_{ξ} [b, ϕ] = - 2 \int (D L_{ξ} β_{f, g} [ϕ]) \cdot b d^{4} x .

The companion vanishes under (CP). Local vanishing holds if $D^{*} b = 0$ near $supp ξ$ .

The compact free-shell witness satisfying (CP) has

Δ_{ξ} Φ^{(2)}_{D} = - 8 \int f (t, x) \int_{0}^{t} (t - s)^{2} a (s) d s d^{4} x < 0.

Its carrier class is nonzero. A cancelling co-transformation was not supplied. The scalar-sourced slot has zero metric variation and is included once XIIF.

Displayed formulas in this passage

XIII-D@1#eq:B-input · line 385
$Δ_{ξ} Φ^{(2)}_{D} = B_{ξ} [b, ϕ] = - 2 \int (D L_{ξ} β_{f, g} [ϕ]) \cdot b d^{4} x .$
XIII-D@1#eq:witness-input · line 396
$Δ_{ξ} Φ^{(2)}_{D} = - 8 \int f (t, x) \int_{0}^{t} (t - s)^{2} a (s) d s d^{4} x < 0.$

Nearby in this section

section: XII-F: the retained pairing and witness