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Fredholm ingredients stated in Lemma 3.1

PDF p. 13: Proof of Lemma 3.1

Statement

Assume the Axiom of Choice. In Lemma 3.1 the article uses three distinct ingredients: the scalar zero-Dirichlet Laplacian has index zero for $- 1 < β < 0$ ; multiplication by $V \in C_{- 4}^{2, α}$ is compact from the full $C_{β}^{4, α}$ space to $C_{β - 2}^{2, α}$ ; and a nonnegative potential eliminates its decaying zero-boundary kernel on the article's connected exterior.

Precisely, use the weighted three-dimensional exterior conventions, $0 < α < 1$ , a smooth asymptotically flat metric with $h - δ \in C_{- 1}^{3, α}$ , and $- 1 < β < 0$ . Put $X_{β} = {u \in C_{β}^{4, α} : u ∣_{S} = 0}$ and $Y_{β} = C_{β - 2}^{2, α}$ . The map $Δ_{h} : X_{β} \to Y_{β}$ is Fredholm of index zero. The compact multiplication assertion holds on the full input space without a boundary condition and therefore also on $X_{β}$ . Consequently $Δ_{h} - V : X_{β} \to Y_{β}$ is Fredholm of index zero without a sign assumption. If $V \geq 0$ and the exterior is connected, its kernel is zero. On the more permissive disconnected exterior interface, its only possible kernel consists of constants on closed compact components where $V \equiv 0$ . Thus index zero and kernel elimination are separate assertions, and the Dirichlet condition belongs to the Laplacian's domain rather than the compactness hypothesis.

Statement references

The Axiom of Choice

Weighted Hölder norms on a three-dimensional exterior

Weighted exterior elliptic domains and admissible weights

Fredholm and Banach continuation interfaces

Exterior Dirichlet parametrix and compact decaying potentials

Compact potential multiplication without a Dirichlet restriction

Fredholm index is stable under compact perturbations

Decaying Dirichlet kernel and the exact closed-component exception

Facts & Assumptions

Given: The hypotheses of the Statement.

Goal: Establish its stated analytic ingredients and qualifications.

[F1]

The Axiom of Choice includes the analytic suppliers’ inherited assumptions. The Axiom of Choice

[F2]

The weighted Hölder and exterior interfaces specify $X_{β}$ , $Y_{β}$ , their end chart, inner boundary and metric-decay conventions. Weighted Hölder norms on a three-dimensional exterior Weighted exterior elliptic domains and admissible weights

[F3]

Fredholm index is kernel dimension minus cokernel dimension for a bounded operator with closed range and finite kernel and cokernel. Fredholm and Banach continuation interfaces

[F4]

The independently authored exterior Dirichlet construction gives index zero for $Δ_{h} : X_{β} \to Y_{β}$ , including permissible compact extra components. Exterior Dirichlet parametrix and compact decaying potentials

[F5]

Potential multiplication is compact from the full weighted input space, and remains compact on its zero-Dirichlet subspace. Compact potential multiplication without a Dirichlet restriction

[F6]

A compact perturbation of a Fredholm operator between Banach spaces is Fredholm with the same index. Fredholm index is stable under compact perturbations

[F7]

For nonnegative $C_{- 4}^{2, α}$ potentials the decaying kernel vanishes on the end component and all boundary components; the only exceptions are constants on closed compact components with zero potential. Decaying Dirichlet kernel and the exact closed-component exception

Proof

**Given:**The hypotheses of the Statement.

Goal: Establish its stated analytic ingredients and qualifications.

[F1] The Axiom of Choice includes the analytic suppliers’ inherited assumptions. The Axiom of Choice

[F2] The weighted Hölder and exterior interfaces specify $X_{β}$ , $Y_{β}$ , their end chart, inner boundary and metric-decay conventions. Weighted Hölder norms on a three-dimensional exterior Weighted exterior elliptic domains and admissible weights

[F3] Fredholm index is kernel dimension minus cokernel dimension for a bounded operator with closed range and finite kernel and cokernel. Fredholm and Banach continuation interfaces

[F4] The independently authored exterior Dirichlet construction gives index zero for $Δ_{h} : X_{β} \to Y_{β}$ , including permissible compact extra components. Exterior Dirichlet parametrix and compact decaying potentials

[F5] Potential multiplication is compact from the full weighted input space, and remains compact on its zero-Dirichlet subspace. Compact potential multiplication without a Dirichlet restriction

[F6] A compact perturbation of a Fredholm operator between Banach spaces is Fredholm with the same index. Fredholm index is stable under compact perturbations

[F7] For nonnegative $C_{- 4}^{2, α}$ potentials the decaying kernel vanishes on the end component and all boundary components; the only exceptions are constants on closed compact components with zero potential. Decaying Dirichlet kernel and the exact closed-component exception

1.1F2F4F5

The authored scalar exterior construction applies exactly at the stated orders, metric falloff and weight interval to the zero-trace domain. It supplies Fredholmness and index zero without replacing the scalar operator by a vector Laplacian. The full-space multiplication lemma gives precisely the compactness stated in the article, including inputs with arbitrary inner-boundary trace; its restriction to $X_{β}$ is compact as well.

2.1F1F3F6F7step 1.1∎

Apply compact-perturbation stability to the Dirichlet Laplacian and the compact map $- M_{V}$ between its Banach domain and target. This gives index zero for $Δ_{h} - V$ regardless of the sign of $V$ . When $V \geq 0$ , use the continuous-potential kernel classification. The article’s exterior section is connected, so it has no separate closed compact component and its kernel is zero; in the larger supplier scope the stated closed-component exception is retained. These are independent ingredients for the article’s Fredholm argument, not a claim that index zero alone implies injectivity.

Source notes

The frozen article PDF p.13 and its extracted text were read for attribution. The actual An–Huang source, arXiv:2206.00079v2, Lemma 3.3, printed pp.16–17, concerns a vector operator Γ depending on a positive scalar field and an asymptotically flat pair. Its stated vector theorem is not silently used here as the scalar Dirichlet theorem; the already authored scalar supplier supplies that result. The two new interface extensions are proved locally: no boundary trace is used in compact multiplication, and positive-part comparison only requires a continuous nonnegative potential. No original-article defect or independent proof review is asserted.

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Sources

The uniqueness of smooth stationary black holes · PDF p.13, Lemma 3.1 and its proof; exterior geometry in Definition 1.1 and §3