Manuscript X-J-ST@1 · X-J-ST.tex · proposition · 8f0d1a36

Proposition: Exact recovery and its necessary qualification

The retained classical interface and its recovery › Free-limit statements with the tadpole retained · lines 626–660 · proof included or linked

X-J-ST@1#prop:recovery

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The classical structure in X-J-ST is exactly (eq:SI); hence it agrees with X-I-ST at every retained order and with X-H-ST through order two.

At $κ = 0$ its field differential is $s_{0}$ . Its complete BV Hamiltonian has

[κ^{0}] I = I_{free} + ⟨ r, A h ⟩ .

It is the free BV Hamiltonian in the centered sector $h = k = 0$ , and more generally whenever $A h = 0$ . An arbitrary off-shell background tadpole is not removed by taking $κ = 0$ .

Setting $λ = 0$ removes only the additional quartic interaction; it does not remove all reference Einstein–scalar interactions.

Proof. There is no action modification after (eq:SI). At order zero, the gravitational quadratic coefficient minus its background-only contribution is

\frac{1}{2} ⟨ h + r, A (h + r) ⟩ - \frac{1}{2} ⟨ h, A h ⟩ = \frac{1}{2} ⟨ r, A r ⟩ + ⟨ r, A h ⟩ .

The other zero-order terms are those displayed in $I_{free}$ . The field-limit and coupling statements follow directly from (eq:smetric)–(eq:SI).

Displayed formulas in this passage

unlabeled · line 634
$[κ^{0}] I = I_{free} + ⟨ r, A h ⟩ .$
unlabeled · line 651
$\frac{1}{2} ⟨ h + r, A (h + r) ⟩ - \frac{1}{2} ⟨ h, A h ⟩ = \frac{1}{2} ⟨ r, A r ⟩ + ⟨ r, A h ⟩ .$

Proposition: Exact recovery and its necessary qualification

Displayed formulas in this passage

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