Manuscript XI-A-ST@1 · XI-A-ST.tex · section · c3f0929d

The quadratic normalization

The signed action and its Euler derivatives › The quadratic normalization · lines 1308–1398

XI-A-ST@1#eq:puregauge, XI-A-ST@1#sec:different, XI-A-ST@1#eq:wave, XI-A-ST@1#eq:constraintbreaking, XI-A-ST@1#sec:coverage, XI-A-ST@1#prop:coverage

Back to manuscript

For the massless input, the existing XI-A conformal check is

G_{lin} [ψ η]_{μν} = - \partial_{μ} \partial_{ν} ψ + η_{μν} □ ψ .

Selecting $γ = ψ η$ does not change (eq:bump), which depends only on $g, ϕ_{m}, f_{b}$ . It therefore does not repair the sourced equation.

The other retained diagnostic is

G_{lin} [K ξ] = 0, D (K ξ) = □ ξ .

It checks the linearization, not existence of a sourced jet.

The optional compact-scalar-data version of the witness uses a spatial cutoff equal to one on a ball containing the initial causal shadows of $supp g$ . Finite propagation makes its scalar solution agree with (eq:input) near $supp g$ , preserving the detector CheungXIA,BGP. This does not construct a gravitational propagator.

the gauge-fixed wave equation is different

For the source-response subclass, the quadratic gauge-fixed equation would be

- □ γ ˉ_{1}^{μν} = \frac{1}{2} g T_{0}^{μν} [ϕ_{0}] .

Taking its divergence on the free scalar shell yields

□ D^{ν} (γ_{1}) = - \frac{1}{2} (\partial_{μ} g) T_{0}^{μν} [ϕ_{0}] .

For $ϕ_{m}$ , the right side at $ν = 0$ is $- \partial_{t} g /4$ , so $D γ_{1}$ cannot vanish identically.

Componentwise smooth Cauchy theory can solve (eq:wave) without a conserved source. That fact does not solve (eq:sourcejet). This is XI-A's distinction, later used in XI-B CheungXIA,CheungXIB.

only{ Equation (eq:quadraticoperator) specifies the signed quadratic operator. This paper supplies no gravitational retarded or advanced inverse, no two-point function, no ghost or auxiliary multiplet, no antifield, no tree, and no state.

XI-B is a downstream comparison, not a premise used to construct those objects here.}

The source obstruction is not a statement that Einstein theory fails. It is an incompatibility between the specified compact switch, the named free input, and the invariant equation space.

coverage and the limits of a report catalogue

The proposed catalogue records finitely many evaluations. It cannot replace an identity on an unrestricted smooth or distributional space. The following local statements give the coverage and invisibility discipline compared with Oai282-A CheungOai282.

Let $Z$ be a proved finite-dimensional residual space, and let

P : Z ⟶ R^{N}, P (r) = (ℓ_{1} (r), \dots, ℓ_{N} (r))

be a specified linear probe map. The conditions $ℓ_{j} (r) = 0$ for every $j$ force $r = 0$ for every $r \in Z$ if and only if $P$ is injective.

Displayed formulas in this passage

XI-A-ST@1#eq:puregauge · line 1320
$G_{lin} [K ξ] = 0, D (K ξ) = □ ξ .$
XI-A-ST@1#eq:wave · line 1341
$- □ γ ˉ_{1}^{μν} = \frac{1}{2} g T_{0}^{μν} [ϕ_{0}] .$
XI-A-ST@1#eq:constraintbreaking · line 1347
$□ D^{ν} (γ_{1}) = - \frac{1}{2} (\partial_{μ} g) T_{0}^{μν} [ϕ_{0}] .$
unlabeled · line 1311
$G_{lin} [ψ η]_{μν} = - \partial_{μ} \partial_{ν} ψ + η_{μν} □ ψ .$
unlabeled · line 1391
$P : Z ⟶ R^{N}, P (r) = (ℓ_{1} (r), \dots, ℓ_{N} (r))$

The quadratic normalization

Displayed formulas in this passage

Nearby in this section