Manuscript XII-F@1 · XII-F.tex · theorem · 54cb3169

Theorem: Nonzero auxiliary pairing under causal plateau

F3: cancellation on a subclass and a nonzero shell witness › Evaluation of the Einstein row · lines 1033–1071 · proof included or linked

XII-F@1#thm:obstruction, XII-F@1#eq:main-value, XII-F@1#eq:pairing-value

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For the compact admissible tuple (eq:main-parameter)–(eq:main-fields), (CP) holds and

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

Equivalently,

\int (D L_{ξ} β_{f, g} [ϕ])_{ν} b^{ν} d^{4} y = 4 \int f (t, x) \int_{0}^{t} (t - s)^{2} a (s) d s d^{4} x > 0.

Thus unqualified invariance of $Φ^{(2)}_{D}$ on the complete gauge-fixed free shell is false.

Proof. The switch satisfies (UCP), so the companion is zero. The shell membership is Lemma lem:main-shell. By the copied root identity, $E_{ξ} = A_{1} [f, g] (e_{ξ}; ϕ)$ . Lemma lem:main-evaluation computes this value. For every observation time, $t > t_{b}$ and $a$ is a nonzero nonnegative bump, so the inner integral is strictly positive. The nonzero nonnegative $f$ makes the displayed value strictly negative. Lemma lem:shell then gives (eq:pairing-value).

Displayed formulas in this passage

XII-F@1#eq:main-value · line 1037
$Δ_{ξ} Φ^{(2)}_{D} = E_{ξ} = B_{ξ} = - 8 \int f (t, x) \int_{0}^{t} (t - s)^{2} a (s) d s d^{4} x < 0.$
XII-F@1#eq:pairing-value · line 1047
$\int (D L_{ξ} β_{f, g} [ϕ])_{ν} b^{ν} d^{4} y = 4 \int f (t, x) \int_{0}^{t} (t - s)^{2} a (s) d s d^{4} x > 0.$

Theorem: Nonzero auxiliary pairing under causal plateau

Displayed formulas in this passage

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