Manuscript X-G-ST@1 · X-G-ST.tex · proposition · 3c97567f

Proposition: Evaluated reciprocity defect

Stage 3: variational reciprocity and the positive witness › A compact background inside the onset · lines 1438–1495 · proof included or linked

X-G-ST@1#prop:Dsigned, X-G-ST@1#eq:positive

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For (eq:witnessdata),

D_{w} (u, v) = \frac{5 M _{χ}}{8} \int_{0}^{ε} σ_{ε}^{''} (t) ψ (t) d t = \frac{5 M _{χ}}{8} \int_{0}^{ε} ζ (t) (σ_{ε}^{''} (t))^{2} d t > 0.

Proof. The typed commutator (eq:commutator), evaluated on a free input, gives $Q_{w} u = - P_{gr} L_{ξ_{on}}^{cov} u$ , and the same formula for $v$ .

For these tensors the nonzero Lie components are

(L_{ξ_{on}}^{cov} u)_{00} (L_{ξ_{on}}^{cov} v)_{00} = 2 ψ^{'} χ_{0}, = (ψ + 2 t ψ^{'}) χ_{0}, (L_{ξ_{on}}^{cov} u)_{0 i} (L_{ξ_{on}}^{cov} v)_{0 i} = ψ \partial_{i} χ_{0}, = t ψ \partial_{i} χ_{0} .

There are no spatial–spatial components. For any tensor $a$ of this form,

(P_{gr} a)^{00} = - \frac{1}{8} □ a_{00} .

Only the $00$ output is paired with $u, v$ . Therefore

D_{w} (u, v) = \frac{1}{8} \int σ_{ε} [□ ((ψ + 2 t ψ^{'}) χ_{0}) - t □ (2 ψ^{'} χ_{0})] d^{4} x .

The spatial Laplacian terms integrate to zero, since $χ_{0}$ is compactly supported. The remaining temporal coefficient is

(ψ + 2 t ψ^{'})^{''} - t (2 ψ^{'})^{''} = 5 ψ^{''} .

Thus $D_{w} = (5 M_{χ} /8) \int σ_{ε} ψ^{''} d t$ . Two integrations by parts have no boundary terms because $ψ$ is compactly supported inside the onset interval. They give (eq:positive). Strict positivity follows from the choices of $ζ, χ_{0}$ .

Displayed formulas in this passage

X-G-ST@1#eq:positive · line 1441
$D_{w} (u, v) = \frac{5 M _{χ}}{8} \int_{0}^{ε} σ_{ε}^{''} (t) ψ (t) d t = \frac{5 M _{χ}}{8} \int_{0}^{ε} ζ (t) (σ_{ε}^{''} (t))^{2} d t > 0.$
unlabeled · line 1459
$(L_{ξ_{on}}^{cov} u)_{00} (L_{ξ_{on}}^{cov} v)_{00} = 2 ψ^{'} χ_{0}, = (ψ + 2 t ψ^{'}) χ_{0}, (L_{ξ_{on}}^{cov} u)_{0 i} (L_{ξ_{on}}^{cov} v)_{0 i} = ψ \partial_{i} χ_{0}, = t ψ \partial_{i} χ_{0} .$
unlabeled · line 1471
$(P_{gr} a)^{00} = - \frac{1}{8} □ a_{00} .$
unlabeled · line 1476
$D_{w} (u, v) = \frac{1}{8} \int σ_{ε} [□ ((ψ + 2 t ψ^{'}) χ_{0}) - t □ (2 ψ^{'} χ_{0})] d^{4} x .$
unlabeled · line 1485
$(ψ + 2 t ψ^{'})^{''} - t (2 ψ^{'})^{''} = 5 ψ^{''} .$

Proposition: Evaluated reciprocity defect

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