Manuscript XII-B@1 · XII-B.tex · proposition · 229bc869

Proposition: Nonzero layer term

An explicit Einstein–Hilbert layer discrepancy › The cubic density on the pure-trace sector · lines 683–710 · proof included or linked

XII-B@1#prop:tracelayer, XII-B@1#eq:tracelayer

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Density weighting gives

EL_{ψ} (σ_{ε} ℓ_{3} [ψ η]) = σ_{ε} η_{μν} J_{3}^{μν} [ψ η] - 6 (\partial_{μ} σ_{ε}) ψ \partial^{μ} ψ .

The second term is not identically zero on the unrestricted off-shell configuration class.

Proof. Apply (eq:firstorderlayer) to (eq:tracedensity). To detect the discrepancy, choose a spacetime box with $0 < t < ε$ and a smooth $ψ$ equal to $t$ near that box. Let $h \geq 0$ be a nonzero compact variation supported inside it. The discrepancy pairing is

- 6 \int σ_{ε}^{'} (t) t h (t, x) d^{4} x < 0.

The strict sign follows from $t > 0$ and $σ_{ε}^{'} > 0$ there. All variations are compact and supported away from the layer endpoints, so no boundary convention enters this pairing.

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XII-B@1#eq:tracelayer · line 686
$EL_{ψ} (σ_{ε} ℓ_{3} [ψ η]) = σ_{ε} η_{μν} J_{3}^{μν} [ψ η] - 6 (\partial_{μ} σ_{ε}) ψ \partial^{μ} ψ .$
unlabeled · line 704
$- 6 \int σ_{ε}^{'} (t) t h (t, x) d^{4} x < 0.$

Proposition: Nonzero layer term

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