# Weyl Sequence Construction and Verification of the Critical Spectrum

- **Source pack:** `arXiv-2511.05345v1.tar.gz` (LaTeX + figures)

- **Primary source file:** `main.tex`

- **Style spec applied:** `if://bible/complete/v3.0`

- **Generated:** 2026-01-01


---

## Assets

- `main.tex` (verbatim in [Annex](#annex))

- `00README.json` (verbatim in annex; also located in the source pack)

- `rayleigh_scaling.pdf` — [download](sandbox:/mnt/data/rayleigh_scaling.pdf)

- `3d_eigenvalues.pdf` — [download](sandbox:/mnt/data/3d_eigenvalues.pdf)


---

## Table of contents

### IF version

- [Boardroom view](#if-boardroom-view)
- [Executive summary](#if-executive-summary)
- [Explainer stack](#if-explainer-stack)
- [Abstract](#if-abstract)
- [1. Introduction](#if-sec-01-introduction)
- [2. Spectral Scaling and Structural Parallels Across Spin-1 and Spin-2 Fields](#if-sec-02-spectral-scaling-and-structural-parallels-across-spin-1-and-spin-2-fields)
- [3. Infrared Spectrum and Marginal Modes](#if-sec-03-infrared-spectrum-and-marginal-modes)
- [4. Dimensional Scaling and Spectral Phase Structure](#if-sec-04-dimensional-scaling-and-spectral-phase-structure)
- [5. Numerical Verification of the Spectral Threshold](#if-sec-05-numerical-verification-of-the-spectral-threshold)
- [6. Physical Interpretation and Implications](#if-sec-06-physical-interpretation-and-implications)
- [7. Relation to Previous Work and Threshold Phenomena](#if-sec-07-relation-to-previous-work-and-threshold-phenomena)
- [8. Conclusion](#if-sec-08-conclusion)
- [Declarations](#if-sec-09-declarations)
- [Appendix A. Gauge Correction and Elliptic Estimates](#if-app-a-gauge-correction-and-elliptic-estimates)
- [Appendix B. Curvature Structure and the Schwarzschild Example](#if-app-b-curvature-structure-and-the-schwarzschild-example)
- [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#if-app-c-analytical-framework-and-weighted-sobolev-spaces)
- [Appendix D. Numerical Validation and Stability Tests](#if-app-d-numerical-validation-and-stability-tests)
- [Appendix E. Weyl Sequence Construction and Verification of the Critical Spectrum](#if-app-e-weyl-sequence-construction-and-verification-of-the-critical-spectrum)
- [Formula appendix: all displayed formulas](#if-formula-appendix)
- [Annex: original source](#annex)
### Annex

- [Annex TOC](#annex-toc)


---

## IF version



<a id="if-boardroom-view"></a>

### Boardroom view

**Decision this paper informs:** Treat $|\mathrm{Riem}|\sim r^{-3}$ as the *critical* curvature-decay boundary for whether the spatial Lichnerowicz operator admits non-decaying, spatially-extended, finite-energy zero-energy configurations (“marginal modes”) at the bottom of its essential spectrum.

**What you can safely take away (even if you skip the proofs):**
- For faster-than-inverse-cube decay, the paper argues the curvature term is spectrally compact, so the essential spectrum matches the flat case and starts at $0$ with radiative behavior.
- At the inverse-cube rate, that compactness fails and $0$ enters $\sigma_{\mathrm{ess}}(L)$ via an explicit Weyl-sequence construction, producing stationary, spatially-extended configurations.

**Argument map (diagram):**
```mermaid
flowchart TD
  A["Curvature decay |Riem| ~ r^{-p}"] --> B["Curvature term V_R ~ r^{-p} (potential-like)"]
  B --> C["Is V_R a compact perturbation of the flat operator?"]
  C -->|Yes (faster decay)| D["Flat-like essential spectrum: σ_ess(L) = [0,∞)"]
  C -->|No (borderline)| E["Construct Weyl sequence → 0 ∈ σ_ess(L)"]
  E --> F["Marginal, extended stationary configurations"]
  F --> G["Physical reading: infrared / memory precursor sector"]
```

**Why this matters (strategic):** It gives a checkable “geometry knob” that separates dispersive behavior from an infrared sector with persistent/static correlations (the paper’s framing connects this to memory/soft structure).

**What could break the takeaway (risk):**
- If the geometric/gauge conditions required to define and analyze the spatial Lichnerowicz operator in the stated function spaces fail.
- If the numerical radial-model proxy is not representative of the geometric operator (it is used as a complementary scaling check, not a proof substitute).

**Next action (operational):** Reproduce one analytic “threshold test” on a concrete asymptotically flat metric family: verify the curvature decay rate and then check (numerically or analytically) whether a Weyl sequence at $0$ can be constructed.

**Testable prediction:** If you vary an asymptotically flat background through families with curvature decay exponent $p$ in an effective $r^{-p}$ sense, the onset of non-decaying marginal behavior should appear at the borderline $p=3$ (and shift according to the dimensional scaling law stated later in the paper).

<a id="if-explainer-stack"></a>

### Explainer stack (made explicit)

This IF version is organized using the **nine-layer explainer stack** from the style bible: **Compression → Strategic Context → Theory → Formula Protocol → Evidence → Diagrams → Contrarian Reframe → Friction Scan → Closure**.  
Where a layer cannot be completed from the provided sources, it is explicitly marked as **“Not specified in sources.”**


<a id="if-executive"></a>

<a id="if-executive-summary"></a>

### Executive summary
This IF version mirrors the source section-by-section, preserving all formulas and stated claims.

Where interpretation is provided, it is constrained to what is stated or directly implied by the source.


**Core threshold statement (verbatim from the source’s main theorem):**


**Theorem (Spectral threshold for linearized gravity)**

\label{thm:main_threshold}
Let $(\Sigma,g)$ be a smooth asymptotically flat three–manifold with curvature decay
$|{\rm Riem}(x)|\!\le\!C\,r^{-p}$.
Then:
\begin{enumerate}
\item For $p>3$, the curvature potential $V_R$ is relatively compact with respect to
$\nabla^*\nabla$, and the essential spectrum is purely continuous:

$$
\sigma_{\mathrm{ess}}(L)=[0,\infty).
$$

\item At the critical rate $p=3$, compactness fails and a normalized Weyl sequence
appears at zero energy, producing marginally extended tensor configurations
that remain spatially nonlocal yet finite in energy.
\item For $p<3$, curvature acts as a long-range potential that enhances infrared coupling,
but without producing isolated bound states in the tensorial sector.
\end{enumerate}


#### System map

```mermaid
flowchart TD
A[Curvature decay |Riem| ~ r^{-p}] --> B{Is p greater than 3? (in d=3)}
B -->|Yes| C[VR relatively compact vs ∇*∇]
C --> D[σ_ess(L) = [0,∞)]
B -->|No: p = 3| E[Compactness fails at threshold]
E --> F[Construct Weyl sequence ⇒ 0 ∈ σ_ess(L)]
B -->|p less than 3| G[Long-range coupling (enhanced IR sensitivity)]
```


#### Analytic → numeric confirmation loop

```mermaid
flowchart LR
S[Asymptotic flatness + decay |Riem| ~ r^{-p}] --> C1[Compactness test for VR]
C1 -->|p>3| R1[Weyl theorem ⇒ flat essential spectrum]
C1 -->|p=3| W1[Weyl sequence construction]
W1 --> R2[0 ∈ σ_ess(L)]
R1 --> N[Numeric checks]
R2 --> N
N --> N1[Rayleigh scaling vs dilation R]
N --> N2[3D finite-volume eigenvalue trends]
N1 --> K[Observed transition near p=3]
N2 --> K
```

**Explainer-stack addendum (layers that were missing in v1):**
- **Strategic context:** The core “decision” is whether inverse-cube curvature decay should be treated as a boundary condition that changes infrared behavior (continuous radiative spectrum vs. non-decaying marginal sector).
- **Evidence:** Analytic: compactness vs. Weyl-sequence argument tied to decay; Numeric: a radial model whose eigenvalue trends show the same transition at $p=3$.
- **Contrarian reframe:** If you *forbid* the $r^{-3}$ tail by insisting on faster decay (even slightly), what physics or boundary data are you implicitly choosing to throw away?
- **Friction scan:** The hard parts are (i) gauge/TT projection on a nontrivial background, (ii) weighted Sobolev controls at infinity, and (iii) numerical artifacts from truncating the domain.
- **Closure:** **Next action:** independently reproduce the Weyl-sequence scaling estimate and one numerical eigenvalue trend. **Prediction:** the threshold persists across reasonable AF families where the effective curvature coupling behaves like $r^{-p}$ at large $r$.

---


<a id="if-abstract"></a>

### Abstract


**Claim (verbatim):** We identify curvature decay $|{\rm Riem}|\!\sim\!r^{-3}$ as a sharp spectral threshold in linearized gravity on asymptotically flat manifolds.

**What must be true (setup implied by the source):**
- The analysis concerns the *spatial* Lichnerowicz operator on an asymptotically flat manifold, written schematically as a Laplace-type operator plus a curvature potential (see also §1–§3).
- Background curvature has a power-law decay $|\mathrm{Riem}|\sim r^{-p}$ along an asymptotically Euclidean end.
- The question of interest is the *essential spectrum* / infrared behavior (especially near $0$) of the resulting self-adjoint operator.

**Mechanism (how the result is obtained in the paper):**
- For faster-than-inverse-cube decay ($p>3$ in $d=3$), the curvature term acts as a compact (or relatively compact) perturbation of the tensor Laplacian, so the essential spectrum remains that of flat space.
- At the borderline inverse–cube decay ($p=3$), compactness fails and a normalized Weyl sequence can be constructed at zero energy, putting $0$ into the essential spectrum.
- A reduced radial channel model and a full 3D discretization provide numerical evidence consistent with the analytic scaling and the threshold behavior.

**Implications (stated or directly implied):**
- The borderline $r^{-3}$ case produces *marginally extended* (non-localized) tensor modes at zero frequency, which the paper connects to the static precursors of gravitational memory and soft-graviton infrared structure.

**Failure modes / boundary conditions (from the framing of the claim):**
- If the geometry is not asymptotically flat or the curvature does not have a controlled power-law tail, the compactness/Weyl-sequence arguments may not apply as stated.
- Gauge constraints (e.g., harmonic gauge / TT conditions) matter for constructing admissible sequences; the paper addresses this via gauge correction (Appendix A).

**Next action (reader-operational):**
- Reproduce the two numerical diagnostics (Rayleigh scaling + 3D lowest-eigenvalue convergence) with your own discretization choices to confirm the robustness of the $p=3$ transition.

**Explainer-stack addendum:**
- **Strategic context:** The abstract is the contract: “$p=3$ is the boundary between compact (spectrally negligible) curvature and long-range curvature that injects zero into the essential spectrum.”
- **Evidence:** Roadmap only; proofs and numerical checks appear in Sections 2–5.
- **Contrarian reframe:** If the transition is “continuous” (not a discrete confinement regime), what observable would you expect to change *sharply* at $p=3$—and do we actually see that in any of the evidence?
- **Closure:** If you only read one proof idea, read the Weyl-sequence construction that makes $0\in\sigma_{\mathrm{ess}}(L)$ at the borderline.

**Original source:** [Annex — Abstract](#annex-abstract)


**Abstract (verbatim, math preserved):**


We identify curvature decay $|{\rm Riem}|\!\sim\!r^{-3}$ as a sharp spectral threshold in linearized gravity on asymptotically flat manifolds.  
For faster decay, the spatial Lichnerowicz operator possesses a purely continuous spectrum $\sigma_{\mathrm{ess}}(L)=[0,\infty)$, corresponding to freely radiating tensor modes.  
At the inverse-cube rate, compactness fails and zero energy enters $\sigma_{\mathrm{ess}}(L)$, yielding marginally bound, finite-energy configurations that remain spatially extended.  
These static modes constitute the linear precursors of gravitational memory and soft-graviton phenomena, delineating the geometric boundary between dispersive and infrared behavior.  
A complementary numerical study of the radial model

$$
L_p=-\tfrac{d^2}{dr^2}+\tfrac{\ell(\ell+1)}{r^2}+\tfrac{C}{r^p}
$$

confirms the analytic scaling law, locating the same transition at $p=3$.  
The eigenvalue trends approach the flat-space limit continuously for $p>3$ and strengthen progressively for $p<3$, demonstrating a smooth yet sharp spectral transition rather than a discrete confinement regime. 
The result parallels the critical threshold of the non-Abelian covariant Laplacian~\cite{Wilson2025}, indicating a common $r^{-3}$ scaling that governs the infrared structure of gauge and gravitational fields.


**Key formula(s) introduced in the abstract:**


$$
L_p=-\tfrac{d^2}{dr^2}+\tfrac{\ell(\ell+1)}{r^2}+\tfrac{C}{r^p}
$$


---


<a id="if-sec-01-introduction"></a>

### 1. Introduction


**Claim (verbatim):** The infrared structure of gravity governs the persistent correlations that remain after gravitational radiation has passed.

**What must be true (setup introduced here):**
- We are studying linearized gravity on an asymptotically flat spatial slice, with dynamics controlled (at the elliptic/spatial level) by the Lichnerowicz operator.
- The curvature coupling can be treated as an effective “potential term” whose decay rate controls whether it is spectrally negligible at infinity.

**Mechanism (section-level narrative):**
- The introduction motivates why *infrared* (large-distance / low-energy) structure matters (memory, soft sector, persistent correlations).
- It states the central organizing idea: in three spatial dimensions, $|\mathrm{Riem}|\sim r^{-3}$ is the sharp boundary between “short-range” curvature (spectrally transparent) and “long-range” curvature (spectrally relevant).
- It previews both the analytic method (compactness vs. Weyl sequences) and the numerical confirmation via a radial model operator.

**Implications (previewed):**
- The inverse-cube case yields non-decaying, nonlocalized “marginal modes” at the bottom of the continuum, providing a stationary counterpart to memory/soft behavior.

**Failure modes / boundary conditions (as framed):**
- If the operator is analyzed without enforcing physical gauge conditions (e.g., TT/harmonic), spurious modes can contaminate the IR conclusion; the paper later enforces TT/harmonic constraints and provides a gauge correction argument.

**Next action (reader-operational):**
- Use §1 as the map: verify that each subsequent section supplies (i) the compactness criterion for $p>3$, (ii) the Weyl sequence at $p=3$, and (iii) numerical scaling that matches the analytic exponent.

**Explainer-stack addendum:**
- **Strategic context:** Sets the stake: the infrared sector is controlled by geometric falloff, and the paper proposes a crisp spectral criterion for “radiative vs. correlated” behavior.
- **Evidence:** Definitions + motivation; evidence is developed later.
- **Contrarian reframe:** What if the “memory/soft” intuition lives at null infinity and cannot be read off from the *spatial* operator spectrum—what would this analysis miss?
- **Friction scan:** Translating between spatial spectral statements and physical memory requires careful gauge handling and asymptotic control.
- **Closure:** **Next action:** track one definition through: how the curvature term acts as an effective potential and where its decay exponent enters the compactness test.

**Original source:** [Annex — §1 Introduction](#annex-sec-01-introduction)


**Displayed equations in this section (verbatim):**


$$
L = \nabla^{*}\nabla + V_R, 
\qquad (V_R h)_{ij} = -R^{\;\ell}{}_{i j m}\, h_{\ell}{}^{m},
\label{eq:Lichnerowicz}
$$

$$
L_p = -\frac{d^2}{dr^2} + \frac{\ell(\ell+1)}{r^2} + \frac{C}{r^p},
$$


**Original source for this section:** [jump to annex](#annex-sec-01-introduction)


---


<a id="if-sec-02-spectral-scaling-and-structural-parallels-across-spin-1-and-spin-2-fields"></a>

### 2. Spectral Scaling and Structural Parallels Across Spin-1 and Spin-2 Fields


**Claim (verbatim):** This section develops the analytic framework for the spectral analysis of the linearized gravitational field on asymptotically flat spatial slices and highlights its structural similarity to the spin-1 gauge field.

**What must be true (explicit analytic framework set here):**
- $(\Sigma,g)$ is an asymptotically flat 3-manifold, and the linearized spin–2 operator can be written as a Laplace-type operator plus a curvature potential: $L=\nabla^*\nabla+V_R$.
- Weighted Sobolev spaces $H^k_\delta$ and mapping properties on Euclidean ends are valid tools for controlling decay and compactness.
- Standard spectral results (Weyl’s theorem for compact perturbations; Weyl sequences for essential spectrum) apply to the self-adjoint realization of $L$.

**Mechanism (how the $p>3$ regime is justified):**
- The curvature term is treated as a multiplication operator with decay $|V_R(x)|\lesssim r^{-p}$.
- For $p>3$ in $d=3$, the section argues (via weighted-space embeddings and Rellich-type compactness) that $V_R$ is relatively compact with respect to the tensor Laplacian.
- Weyl’s theorem then implies that $\sigma_{\mathrm{ess}}(L)$ matches the flat-space essential spectrum: $[0,\infty)$.
- The section also makes the cross-field structural comparison: the gravitational operator has the same Laplace-type + curvature structure as the covariant Laplacian in gauge theory, with the same decay exponent acting as the long-range threshold.

**Implications (the “short-range” side of the phase boundary):**
- If $p>3$, curvature is spectrally negligible in the infrared: it cannot shift the essential spectrum away from the flat tensor Laplacian’s continuum.

**Failure modes / boundary conditions:**
- The argument uses the compactness of weighted embeddings, which depends on dimension and decay exponents; changing $d$ changes the critical exponent (made explicit in §4).
- The claim is about essential spectrum; it does not by itself exclude discrete spectrum below $0$ for other operator choices—later sections clarify what happens in the tensorial setting studied here.

**Next action (reader-operational):**
- Track exactly where $p>3$ is used (integrability of weights / tail estimates) so you can see what breaks at $p=3$ and why the threshold is sharp.

**Explainer-stack addendum:**
- **Strategic context:** Ties the gravity operator’s curvature term to known long-range thresholds in gauge theory and to a clean scaling law for $r^{-p}$ tails.
- **Evidence:** States the compactness criterion for faster decay and identifies the critical exponent.
- **Contrarian reframe:** Is the criticality about “gravity,” or is it generic for Laplace-type operators with $r^{-p}$ potentials in 3D?
- **Friction scan:** Requires correct function spaces and TT/gauge constraints; avoid smuggling in symmetry assumptions.
- **Closure:** **Next action:** verify the critical exponent computation against the dimensional generalization in Section 4.

**Original source:** [Annex — §2 Spectral Scaling…](#annex-sec-02-spectral-scaling-and-structural-parallels-across-spin-1-and-spin-2-fields)


**Displayed equations in this section (verbatim):**


$$
S_{\text{EH}}=\frac{1}{16\pi G}\int R\sqrt{-g}\,d^4x
\label{eq:EH_action}
$$

$$
Lh = \nabla^*\nabla h + V_R h, 
\qquad (V_Rh)_{ij}=-R_i{}^\ell{}_j{}^m h_{\ell m}.
\label{eq:Lichnerowicz_def}
$$

$$
\Delta_A = -(\nabla_A)^*\nabla_A = -\nabla^*\nabla + {\rm ad}(F_A),
\label{eq:YMlap}
$$

$$
\langle Lh,k\rangle_{L^2}
=\langle\nabla h,\nabla k\rangle_{L^2}
+\langle V_Rh,k\rangle_{L^2},
\label{eq:int_identity}
$$

$$
L:H^2_\delta(\Sigma;S^2T^*\Sigma)\longrightarrow L^2_{\delta-2}(\Sigma;S^2T^*\Sigma)
\label{eq:L_mapping2}
$$

$$
\sigma_{\mathrm{ess}}(L)=[0,\infty).
\label{eq:ess_flat}
$$

$$
p_{\mathrm{crit}}=3,
\label{eq:pcrit}
$$

$$
g_{ij}=\delta_{ij}+a_{ij},\qquad 
a_{ij}=O(r^{-1}),\quad 
\partial_k a_{ij}=O(r^{-2}),\quad 
\partial_\ell\partial_k a_{ij}=O(r^{-3}),
$$

$$
S^{(2)}[h]=\tfrac{1}{2}\!\int_\Sigma h^{ij}L_{ij}{}^{kl}h_{kl}\sqrt{g}\,d^3x,
$$

$$
H^k_\delta(\Sigma;S^2T^*\Sigma)
=\Bigl\{h\in H^k_{\mathrm{loc}}(\Sigma):\|h\|_{H^k_\delta}<\infty\Bigr\},\qquad
\|h\|_{H^k_\delta}^2=\sum_{j=0}^k\!\!\int_\Sigma\langle r\rangle^{2(\delta-j)}|\nabla^jh|_g^2\,dV_g.
$$


**Original source for this section:** [jump to annex](#annex-sec-02-spectral-scaling-and-structural-parallels-across-spin-1-and-spin-2-fields)


---


<a id="if-sec-03-infrared-spectrum-and-marginal-modes"></a>

### 3. Infrared Spectrum and Marginal Modes


**Claim (verbatim):** When the background curvature decays as $r^{-3}$, the Lichnerowicz operator reaches the scaling threshold identified in Section~\ref{sec:spectral_universality}.

**What must be true (assumptions used for the borderline case):**
- $(\Sigma,g)$ is smooth and asymptotically flat, with explicit metric falloff conditions (see the metric-decay display equation in this section).
- Curvature saturates the inverse-cube decay $|\mathrm{Riem}(x)|\simeq C r^{-3}$ on the end.
- A gauge condition is enforced (harmonic / divergence-free), and the relevant vector Laplacian is invertible in the weighted setting (requiring $H^1_{\mathrm{dR}}(\Sigma)=0$ as stated).

**Mechanism (why $p=3$ produces marginal modes):**
- The section first establishes the weighted Fredholm/invertibility framework for the vector Laplacian $\Delta_V$ used to correct gauge.
- It then constructs an *approximate* $L^2$-normalized sequence supported on annuli escaping to infinity (a Weyl sequence ansatz), with scaling chosen so that the curvature term contributes at the same order as the commutator terms.
- At $p=3$, both the curvature contribution and commutator terms scale like $\sim n^{-2}$ in $L^2$, allowing $\|L h_n\|_{L^2}\to 0$ while $\|h_n\|_{L^2}=1$.
- A gauge correction $\tilde h_n=h_n-\mathcal{L}_{X_n}g$ is applied so the final sequence satisfies the divergence-free condition required for the physical sector.
- By Weyl’s criterion, the existence of such a normalized sequence implies $0\in\sigma_{\mathrm{ess}}(L)$, i.e., the continuum touches zero at the threshold.

**Implications (the “borderline” phase):**
- The operator supports *marginally extended* tensor configurations: nonlocalized, zero-energy (threshold) modes that are not compactly confined but also do not disperse away in the same manner as in the $p>3$ regime.

**Failure modes / boundary conditions:**
- The construction depends on the exact scaling balance at $p=3$; if $p>3$ the curvature term becomes too small (compact), and if $p<3$ it becomes long-range in a stronger sense.
- Gauge correction requires invertibility of $\Delta_V$ in the chosen weight range; if $H^1_{\mathrm{dR}}(\Sigma)\neq 0$, the stated uniqueness/isomorphism conclusion can fail.

**Next action (reader-operational):**
- Re-derive the scaling estimate for $\|V_R h_n\|_{L^2}$ and the commutator bound for $\|[\nabla^*\nabla,\phi_n]h_n\|_{L^2}$ to see the threshold balance numerically and analytically.

**Explainer-stack addendum:**
- **Strategic context:** Converts “$p=3$ is borderline” into “$0$ sits in the essential spectrum,” yielding marginal extended modes.
- **Evidence:** Weyl-sequence construction plus quantitative estimates (supported by Appendix E).
- **Contrarian reframe:** If a candidate marginal mode is subtlely pure-gauge, how would the Weyl-sequence argument detect (or miss) that?
- **Friction scan:** Enforcing TT/gauge constraints without spoiling scaling at infinity is the practical blocker.
- **Closure:** **Next action:** reproduce the $n$-scaling of the cutoff/commutator and curvature terms on an annulus.

**Original source:** [Annex — §3 Infrared Spectrum…](#annex-sec-03-infrared-spectrum-and-marginal-modes)


**Displayed equations in this section (verbatim):**


$$
\Delta_V=\nabla^*\nabla+\mathrm{Ric},
\label{eq:vectorlaplacian}
$$

$$
\Delta_V: H^2_\delta(\Sigma;T^*\Sigma)\longrightarrow L^2_{\delta-2}(\Sigma;T^*\Sigma)
\label{eq:vectorfredholm}
$$

$$
g_{ij}=\delta_{ij}+O(r^{-1}),\qquad 
\partial g_{ij}=O(r^{-2}),\qquad 
\partial^2 g_{ij}=O(r^{-3}),
\label{eq:metricdecay}
$$

$$
Lh = \nabla^*\nabla h + V_R h, 
\qquad (V_R h)_{ij}=-R_{i\ \ j m}^{\ \ell}h_\ell^{\ m},
\label{eq:lichcritical}
$$

$$
h_n(r,\omega)=A_n\,\phi_n(r)\,r^{-1}H_{ij}(\omega),
\label{eq:weylansatz}
$$

$$
[0,\infty)\subset\sigma_{\mathrm{ess}}(L),\qquad 0\in\sigma_{\mathrm{ess}}(L).
\label{eq:criticalspectrum}
$$

$$
\|\tilde h_n\|_{L^2}=1,\qquad
\nabla^j\tilde h_{n,ij}=0,\qquad
\|L\tilde h_n\|_{L^2}\to0.
$$

$$
\Box_g h_{\mu\nu} + 2R_{\mu\ \nu}^{\ \rho\ \sigma}h_{\rho\sigma}=0,
$$


**Named results stated in this section (verbatim statements):**


**Theorem (Onset of the infrared continuum)** `\label{thm:criticalspectrum}`

Let $(\Sigma,g)$ satisfy~\eqref{eq:metricdecay} with
$|{\rm Riem}(x)|\simeq C\,r^{-3}$ as $r\to\infty$.  
Then the Lichnerowicz operator $L=\nabla^*\nabla+V_R$ is self-adjoint on
$L^2(\Sigma;S^2T^*\Sigma)$ and satisfies

$$
[0,\infty)\subset\sigma_{\mathrm{ess}}(L),\qquad 0\in\sigma_{\mathrm{ess}}(L).
\label{eq:criticalspectrum}
$$

Hence the inverse-cube decay marks the precise boundary between
spectrally transparent geometries and those supporting marginally correlated
tensor modes.
A full quantitative proof using Weyl’s criterion appears in the
Supplementary Material.


**Lemma (Fredholm property)** `\label{lem:fredholm}`

For $-1<\delta<0$, the operator

$$
\Delta_V: H^2_\delta(\Sigma;T^*\Sigma)\longrightarrow L^2_{\delta-2}(\Sigma;T^*\Sigma)
\label{eq:vectorfredholm}
$$

is Fredholm with bounded inverse.  
Hence, for each $f\in L^2_{\delta-2}$ there exists a unique $X\in H^2_\delta$ satisfying $\Delta_V X=f$ and $\|X\|_{H^2_\delta}\le C\|f\|_{L^2_{\delta-2}}$.


**Lemma (Approximate zero modes)** `\label{lem:weylsequence}`

Let $H_{ij}(\omega)$ be a symmetric, trace-free, divergence-free tensor harmonic
on $S^2$, and define

$$
h_n(r,\omega)=A_n\,\phi_n(r)\,r^{-1}H_{ij}(\omega),
\label{eq:weylansatz}
$$

where $\phi_n$ is a smooth cutoff equal to $1$ on $[n,3n/2]$ and vanishing
outside $[n/2,2n]$.  
The normalization $\|h_n\|_{L^2}=1$ gives $A_n\simeq n^{-1/2}$.
After divergence correction by a vector field $X_n$ satisfying
$\Delta_V X_n=\nabla\!\cdot h_n$ and setting
$\tilde h_n=h_n-\mathcal{L}_{X_n}g$, one obtains

$$
\|\tilde h_n\|_{L^2}=1,\qquad
\nabla^j\tilde h_{n,ij}=0,\qquad
\|L\tilde h_n\|_{L^2}\to0.
$$


**Original source for this section:** [jump to annex](#annex-sec-03-infrared-spectrum-and-marginal-modes)


---


<a id="if-sec-04-dimensional-scaling-and-spectral-phase-structure"></a>

### 4. Dimensional Scaling and Spectral Phase Structure


**Claim (verbatim):** The preceding analysis showed that when the curvature of an asymptotically flat three-manifold decays as $|{\rm Riem}|\!\sim\!r^{-3}$, the Lichnerowicz operator $L=\nabla^*\nabla+V_R$ acquires a continuous spectrum extending to zero.

**What must be true (generalization step):**
- The operator on an asymptotically Euclidean end has the schematic form $L_d=-\nabla^2+V(r)$ in $d$ spatial dimensions, with $V(r)\sim r^{-p}$.
- Compactness is analyzed via tail integrability in the relevant weighted spaces; the dominant scaling comes from the radial integral at infinity.

**Mechanism (dimension-counting that yields $p_{\mathrm{crit}}=d$):**
- The section isolates the dependence on $d$ by estimating the tail contribution of $V$ acting as multiplication in weighted spaces.
- The integrability threshold occurs precisely at $p=d$ for the squared potential weight appearing in the compactness estimate.
- This produces a clean “phase boundary” statement: $p>d$ implies relative compactness (short-range), $p=d$ is marginal, $p<d$ is long-range in the spectral sense.

**Implications:**
- In $d=3$ this reproduces the inverse-cube boundary; in other spatial dimensions the threshold shifts accordingly.

**Failure modes / boundary conditions:**
- The argument is a scaling criterion; detailed geometry (multiple ends, different asymptotics) can introduce additional spectral features not captured by the single-end Euclidean scaling.

**Next action:**
- If you plan to extend the result to higher-dimensional GR settings, use this section as the blueprint: identify the effective tensor potential decay and compare it to $p=d$.

**Explainer-stack addendum:**
- **Strategic context:** Generalizes the threshold to dimension and frames it as a spectral “phase boundary.”
- **Evidence:** Analytic scaling (made falsifiable by cross-dimension comparison).
- **Contrarian reframe:** Does the “phase boundary” survive coordinate/compactification changes, or is it an artifact of the chosen asymptotic parametrization?
- **Friction scan:** Dimensional analogies can fail if geometry introduces long-range structure not captured by $r^{-p}$ alone.
- **Closure:** **Next action:** check the $d$-dependence against the radial-model scaling.

**Original source:** [Annex — §4 Dimensional Scaling…](#annex-sec-04-dimensional-scaling-and-spectral-phase-structure)


**Displayed equations in this section (verbatim):**


$$
p_{\mathrm{crit}}=d,
\label{eq:pcrit_general}
$$

$$
L_d=-\nabla^2 + V(r),
$$

$$
\int_R^\infty r^{\,2(\delta-2)}\,|V(r)|^2\,|h(r)|^2\,r^{\,d-1}dr
\;\sim\;
\int_R^\infty r^{\,d-5+2\delta-2p}\,dr.
$$


**Named results stated in this section (verbatim statements):**


**Proposition (Dimensional criterion for the critical decay rate)**

Let $\Delta$ denote the Laplace-Beltrami operator on a $d$-dimensional asymptotically flat manifold, and let $V(r)\sim r^{-p}$ be a curvature-induced potential.  
Then $V$ is a compact perturbation of $\Delta$ if and only if $p>d$.  
The equality $p=d$ marks the threshold between short- and long-range behavior.


**Original source for this section:** [jump to annex](#annex-sec-04-dimensional-scaling-and-spectral-phase-structure)


---


<a id="if-sec-05-numerical-verification-of-the-spectral-threshold"></a>

### 5. Numerical Verification of the Spectral Threshold


**Claim (verbatim):** The analytic results of Sections~\ref{sec:spectral_universality}–\ref{sec:dimensional_scaling} predict that when the background curvature decays faster than $r^{-3}$, the spectrum of the spatial Lichnerowicz operator remains purely continuous, while the inverse–cube decay marks the onset of marginal long–range coupling.

**What must be true (numerical model and diagnostics):**
- The reduced model operator $L_p$ captures the relevant infrared channel behavior (as motivated earlier) when curvature behaves like an effective $r^{-p}$ potential.
- Two diagnostics are used:
  1) Rayleigh-quotient scaling under dilation of test functions, and
  2) convergence of the lowest eigenvalue $\lambda_1$ with increasing computational domain size in a 3D discretization.

**Mechanism (how numerics operationalize the analytic threshold):**
- The Rayleigh diagnostic tests whether the curvature contribution to the energy decays like a power of the dilation scale $R$, with a predicted slope $\alpha(p)=-(p-2)$.
- The 3D discretization computes the spectrum of a finite-volume TT/penalized tensor Laplacian plus a synthetic tidal-curvature term $\sim r^{-p}$, tracking how $\lambda_1$ approaches the continuum threshold as $R_{\max}\to\infty$.
- Both diagnostics exhibit a transition centered at $p=3$, consistent with the analytic compactness vs. marginal-mode dichotomy.

**Implications:**
- The inverse-cube decay is not just a formal boundary: it produces an observable change in scaling behavior and infrared eigenvalue trends in discretized models.

**Failure modes / boundary conditions (what would make the numerics misleading):**
- Finite-volume and discretization artifacts can masquerade as spectral features; Appendix D documents grid refinement and penalty-strength robustness checks.
- The model uses a simplified curvature “potential”; it supports the scaling claim but is not a substitute for a full curved-background computation.

**Next action:**
- Reproduce Table 1 (Rayleigh scaling) and Table 2 (3D eigenvalues) with an independent code path, then vary penalties, resolution, and boundary conditions as in Appendix D to test stability.

**Explainer-stack addendum:**
- **Strategic context:** Numerics act as a controlled proxy test of the analytic scaling law.
- **Evidence:** Eigenvalue trends in the provided figures show the transition at $p=3$.
- **Contrarian reframe:** If the transition is driven by finite-domain truncation (finite $R$) rather than the tail itself, which scaling diagnostic would catch that?
- **Friction scan:** Discretization, boundary conditions, and TT-penalty choices can shift low eigenvalues; the paper mitigates via penalty-stability checks.
- **Closure:** **Next action:** rerun with alternative discretization/boundary and verify the transition remains pinned at $p=3$ within tolerance.

**Original source:** [Annex — §5 Numerical Verification…](#annex-sec-05-numerical-verification-of-the-spectral-threshold)


**Displayed equations in this section (verbatim):**


$$
\mathcal{R}_{\eta,\zeta}[h]
=
\frac{\langle h,Lh\rangle
+\eta\,\|\nabla\!\cdot h\|_{L^2(\Omega)}^2
+\zeta\,\|\mathrm{tr}\,h\|_{L^2(\Omega)}^2}
{\|h\|_{L^2(\Omega)}^2},
\label{eq:penalty}
$$

$$
L_p = -\frac{d^2}{dr^2} + \frac{\ell(\ell+1)}{r^2} + \frac{C}{r^p},
$$

$$
E[\phi]=
\frac{\int_{R}^{2R}\!\bigl(|\phi'(r)|^2+V_p(r)|\phi(r)|^2\bigr)r^2dr}
     {\int_{R}^{2R}\!|\phi(r)|^2r^2dr},
\qquad
\Delta E(R,p)=E[\phi]-E_{\mathrm{free}}[\phi],
$$


**Named results stated in this section (verbatim statements):**


**Lemma (Asymptotic behavior of the lowest eigenvalue)**

\label{lem:eigenvalueconvergence}
For $p\ge3$, $\lambda_1(L_p)\!\sim\!R_{\max}^{-2}$ and approaches $0^+$ as
$R_{\max}\!\to\!\infty$, consistent with approach to the continuum threshold
and the absence of bound states. 
For $p<3$, the slower decay of curvature increases the infrared coupling,
yielding smaller $\lambda_1$ but no discrete negative modes.


**Definition (Numerical operators)**

\label{def:numericaloperator}
The asymptotic radial model operator is

$$
L_p = -\frac{d^2}{dr^2} + \frac{\ell(\ell+1)}{r^2} + \frac{C}{r^p},
$$

representing a single angular channel of the tensor operator.
The three–dimensional discretized $L=\nabla^{*}\nabla+V_R$
includes the full curvature coupling and constraint enforcement.


**Figure(s) referenced in this section:**


- Figure 1: Rayleigh scaling — [rayleigh_scaling.pdf](sandbox:/mnt/data/rayleigh_scaling.pdf)

- Figure 2: 3D eigenvalues — [3d_eigenvalues.pdf](sandbox:/mnt/data/3d_eigenvalues.pdf)


**Table 1 (Rayleigh energy scaling; extracted from the source longtable):**


|   $p$ |   $R$ | $\Delta E(R,p)$      | $E_{\text{full}}$   | $E_{\text{free}}$   |
|------:|------:|:---------------------|:--------------------|:--------------------|
|   2   |    10 | $-2.85\times10^{-2}$ | $1.50\times10^{-1}$ | $1.54\times10^{-1}$ |
|   2   |    20 | $-2.85\times10^{-2}$ | $3.74\times10^{-2}$ | $3.85\times10^{-2}$ |
|   2   |    40 | $-2.85\times10^{-2}$ | $9.36\times10^{-3}$ | $9.63\times10^{-3}$ |
|   2   |    80 | $-2.85\times10^{-2}$ | $2.34\times10^{-3}$ | $2.41\times10^{-3}$ |
|   2   |   160 | $-2.85\times10^{-2}$ | $5.85\times10^{-4}$ | $6.02\times10^{-4}$ |
|   2   |   320 | $-2.85\times10^{-2}$ | $1.46\times10^{-4}$ | $1.50\times10^{-4}$ |
|   2   |   640 | $-2.85\times10^{-2}$ | $3.65\times10^{-5}$ | $3.76\times10^{-5}$ |
|   2.5 |    10 | $-7.39\times10^{-3}$ | $1.53\times10^{-1}$ | $1.54\times10^{-1}$ |
|   2.5 |    20 | $-5.23\times10^{-3}$ | $3.83\times10^{-2}$ | $3.85\times10^{-2}$ |
|   2.5 |    40 | $-3.69\times10^{-3}$ | $9.59\times10^{-3}$ | $9.63\times10^{-3}$ |
|   2.5 |    80 | $-2.61\times10^{-3}$ | $2.40\times10^{-3}$ | $2.41\times10^{-3}$ |
|   2.5 |   160 | $-1.85\times10^{-3}$ | $6.01\times10^{-4}$ | $6.02\times10^{-4}$ |
|   2.5 |   320 | $-1.31\times10^{-3}$ | $1.50\times10^{-4}$ | $1.50\times10^{-4}$ |
|   2.5 |   640 | $-9.23\times10^{-4}$ | $3.76\times10^{-5}$ | $3.76\times10^{-5}$ |
|   3   |    10 | $-1.92\times10^{-3}$ | $1.54\times10^{-1}$ | $1.54\times10^{-1}$ |
|   3   |    20 | $-9.62\times10^{-4}$ | $3.85\times10^{-2}$ | $3.85\times10^{-2}$ |
|   3   |    40 | $-4.81\times10^{-4}$ | $9.63\times10^{-3}$ | $9.63\times10^{-3}$ |
|   3   |    80 | $-2.40\times10^{-4}$ | $2.41\times10^{-3}$ | $2.41\times10^{-3}$ |
|   3   |   160 | $-1.20\times10^{-4}$ | $6.02\times10^{-4}$ | $6.02\times10^{-4}$ |
|   3   |   320 | $-6.01\times10^{-5}$ | $1.50\times10^{-4}$ | $1.50\times10^{-4}$ |
|   3   |   640 | $-3.01\times10^{-5}$ | $3.76\times10^{-5}$ | $3.76\times10^{-5}$ |
|   3.5 |    10 | $-5.02\times10^{-4}$ | $1.54\times10^{-1}$ | $1.54\times10^{-1}$ |
|   3.5 |    20 | $-1.78\times10^{-4}$ | $3.85\times10^{-2}$ | $3.85\times10^{-2}$ |
|   3.5 |    40 | $-6.28\times10^{-5}$ | $9.63\times10^{-3}$ | $9.63\times10^{-3}$ |
|   3.5 |    80 | $-2.22\times10^{-5}$ | $2.41\times10^{-3}$ | $2.41\times10^{-3}$ |
|   3.5 |   160 | $-7.85\times10^{-6}$ | $6.02\times10^{-4}$ | $6.02\times10^{-4}$ |
|   3.5 |   320 | $-2.78\times10^{-6}$ | $1.50\times10^{-4}$ | $1.50\times10^{-4}$ |
|   3.5 |   640 | $-9.81\times10^{-7}$ | $3.76\times10^{-5}$ | $3.76\times10^{-5}$ |
|   4   |    10 | $-1.32\times10^{-4}$ | $1.54\times10^{-1}$ | $1.54\times10^{-1}$ |
|   4   |    20 | $-3.29\times10^{-5}$ | $3.85\times10^{-2}$ | $3.85\times10^{-2}$ |
|   4   |    40 | $-8.23\times10^{-6}$ | $9.63\times10^{-3}$ | $9.63\times10^{-3}$ |
|   4   |    80 | $-2.06\times10^{-6}$ | $2.41\times10^{-3}$ | $2.41\times10^{-3}$ |
|   4   |   160 | $-5.14\times10^{-7}$ | $6.02\times10^{-4}$ | $6.02\times10^{-4}$ |
|   4   |   320 | $-1.29\times10^{-7}$ | $1.50\times10^{-4}$ | $1.50\times10^{-4}$ |
|   4   |   640 | $-3.21\times10^{-8}$ | $3.76\times10^{-5}$ | $3.76\times10^{-5}$ |


**Table 2 (3D TT eigenvalues vs. box size and decay exponent; extracted from the source longtable):**


| $p$           |   $R_{\max}$ |   $\lambda_1$ |
|:--------------|-------------:|--------------:|
| \textit{flat} |            6 |        0.2044 |
| \textit{flat} |           10 |        0.0739 |
| \textit{flat} |           14 |        0.0377 |
| \textit{flat} |           18 |        0.0228 |
| \textit{flat} |           20 |        0.0185 |
| 2.0           |            6 |        0.1466 |
| 2.0           |           10 |        0.0435 |
| 2.0           |           14 |        0.0192 |
| 2.0           |           18 |        0.0102 |
| 2.0           |           20 |        0.0078 |
| 2.5           |            6 |        0.1828 |
| 2.5           |           10 |        0.0651 |
| 2.5           |           14 |        0.0334 |
| 2.5           |           18 |        0.0203 |
| 2.5           |           20 |        0.0166 |
| 3.0           |            6 |        0.1965 |
| 3.0           |           10 |        0.0711 |
| 3.0           |           14 |        0.0365 |
| 3.0           |           18 |        0.0222 |
| 3.0           |           20 |        0.018  |
| 3.5           |            6 |        0.2016 |
| 3.5           |           10 |        0.073  |
| 3.5           |           14 |        0.0374 |
| 3.5           |           18 |        0.0227 |
| 3.5           |           20 |        0.0184 |
| 4.0           |            6 |        0.2035 |
| 4.0           |           10 |        0.0736 |
| 4.0           |           14 |        0.0376 |
| 4.0           |           18 |        0.0228 |
| 4.0           |           20 |        0.0185 |


**Original source for this section:** [jump to annex](#annex-sec-05-numerical-verification-of-the-spectral-threshold)


---


<a id="if-sec-06-physical-interpretation-and-implications"></a>

### 6. Physical Interpretation and Implications


**Claim (verbatim):** The analytic and numerical analyses of Sections~\ref{sec:critical_decay}-\ref{sec:numerics} identify a sharp transition between two qualitatively distinct spectral regimes of gravitational perturbations on asymptotically flat manifolds.

**What must be true (interpretive layer built on prior sections):**
- Sections §3–§5 have established the existence of a sharp spectral transition at $p=3$ for the spatial operator.
- The infrared behavior of the quantized field is governed by the inverse of the spatial operator in canonical approaches (as stated here).

**Mechanism (interpretation offered by the source):**
- For $p>3$, the inverse operator behaves short-range, enabling a regular Fock vacuum with finite infrared correlations.
- At $p=3$, the Green’s function develops a slow algebraic tail corresponding to zero-frequency, spatially extended modes.
- The section connects these marginal modes to the soft sector (soft-graviton theorems, asymptotic symmetries) and to memory as a dynamical counterpart.

**Implications:**
- The $r^{-3}$ decay is presented as a geometric origin point for infrared enhancement and long-time/long-distance persistence phenomena.

**Failure modes / boundary conditions:**
- Several connections here are interpretive; the paper notes that a complete dynamical correspondence would require coupling the elliptic analysis to time-dependent linearized Einstein equations near $\mathscr{I}^+$.

**Next action:**
- If using these results in an IR/soft-theorem context, explicitly map the spatial marginal modes to dynamical solutions (frequency-domain analysis near $\omega=0$) to make the correspondence precise.

**Explainer-stack addendum:**
- **Strategic context:** Translates the spectral statement into the paper’s physical language (marginal static modes as precursors of IR/memory structure).
- **Evidence:** Interpretation is anchored to the essential-spectrum result and the existence of extended threshold configurations.
- **Contrarian reframe:** If modes are stationary and extended, why don’t they automatically imply instability or non-decay in the full nonlinear theory?
- **Friction scan:** Interpretation depends on connecting spatial spectral modes to dynamical evolution and boundary data at infinity.
- **Closure:** **Next action:** articulate one observable (even toy-model) that differs between $p>3$ and $p=3$ regimes.

**Original source:** [Annex — §6 Physical Interpretation…](#annex-sec-06-physical-interpretation-and-implications)


**Displayed equations in this section:** None.


**Original source for this section:** [jump to annex](#annex-sec-06-physical-interpretation-and-implications)


---


<a id="if-sec-07-relation-to-previous-work-and-threshold-phenomena"></a>

### 7. Relation to Previous Work and Threshold Phenomena


**Claim (verbatim):** The spectral threshold established in Sections~\ref{sec:critical_decay}-\ref{sec:numerics} connects several independent developments in spectral geometry, mathematical relativity, and gauge theory.

**What must be true (comparative context):**
- The threshold result is being situated relative to established elliptic theory on noncompact manifolds, long-range potential scattering, and black-hole tail phenomena.
- The gauge-theory analogue uses the covariant Laplacian $\Delta_A$ with curvature term ${\rm ad}(F_A)$, mirroring the Laplace-type + curvature structure.

**Mechanism (what this section does):**
- It explains how the present $p=3$ boundary refines earlier “curvature is short-range” statements by identifying the *sharp* exponent where compactness fails.
- It connects to Schrödinger thresholds (faster than $r^{-3}$ gives purely a.c. spectrum on $[0,\infty)$; slower decay can support threshold phenomena).
- It relates the same inverse-cube scaling to Price-law tails and to the soft/memory sector via near-zero modes.
- It states an explicit proposition describing the “parallel inverse-cube threshold” across gauge and gravity settings.

**Implications:**
- The inverse-cube decay is presented as a shared structural infrared boundary across scalar/vector/tensor systems in $\mathbb{R}^3$-type asymptotics.

**Failure modes / boundary conditions:**
- The analogy is structural rather than a claim of identical spectra across systems; each operator’s bundle structure and constraints (e.g., TT sector) matter.

**Next action:**
- For readers working across gauge/gravity IR: test whether the same decay threshold appears for other Laplace-type operators (e.g., Dirac-type squared operators) on asymptotically flat ends.

**Explainer-stack addendum:**
- **Strategic context:** Positions the $p=3$ threshold alongside known borderline-decay phenomena in Schrödinger operators and gauge-field Laplacians.
- **Evidence:** Parallels are structural (operator + decay exponent + essential-spectrum behavior) with details in cited work.
- **Contrarian reframe:** What counterexample geometry/topology would break the “decay exponent controls IR spectrum” heuristic?
- **Friction scan:** Beware mismatched assumptions (symmetry, boundary conditions, self-adjoint extensions) when mapping literatures.
- **Closure:** **Next action:** build an assumption table (gravity vs. gauge vs. scalar) to make the analogy falsifiable.

**Original source:** [Annex — §7 Relation to Previous Work…](#annex-sec-07-relation-to-previous-work-and-threshold-phenomena)


**Displayed equations in this section (verbatim):**


$$
\Delta_A = -(\nabla_A)^*\nabla_A = -\nabla^*\nabla + {\rm ad}(F_A),
$$


**Named results stated in this section (verbatim statements):**


**Proposition (Parallel inverse-cube threshold)**

For Laplace–type operators on bundles over $\mathbb{R}^3$,
a curvature decay of order $r^{-3}$ marks the transition between
short-range, radiative behavior and long-range, infrared coupling.
In both gauge and gravitational settings, curvature acts as an effective potential;
at this critical rate, marginal nonlocalized modes appear,
signaling the breakdown of compactness of the resolvent.


**Original source for this section:** [jump to annex](#annex-sec-07-relation-to-previous-work-and-threshold-phenomena)


---


<a id="if-sec-08-conclusion"></a>

### 8. Conclusion


**Claim (verbatim):** The analyses presented here establish a sharp spectral threshold for the spatial Lichnerowicz operator on asymptotically flat three–manifolds.

**What must be true (inputs to the main theorem):**
- $(\Sigma,g)$ is asymptotically flat and has curvature decay $|\mathrm{Riem}(x)|\le C r^{-p}$.
- The spatial operator is the Lichnerowicz Laplace-type operator on symmetric 2-tensors (with curvature coupling), realized self-adjointly on $L^2$.

**Mechanism (what is concluded):**
- The section packages the prior analysis into a single theorem that delineates the three regimes ($p>3$, $p=3$, $p<3$).
- It reiterates that the $p=3$ case produces a normalized Weyl sequence at zero energy, while $p>3$ gives $\sigma_{\mathrm{ess}}(L)=[0,\infty)$.

**Implications:**
- The $r^{-3}$ decay rate is identified as the sharp geometric boundary separating radiative propagation from marginal persistence / enhanced infrared coupling.

**Failure modes / boundary conditions:**
- The theorem is for the *spatial* Lichnerowicz operator on asymptotically flat three-manifolds; other operators or asymptotics may shift the boundary (cf. §4).

**Next action:**
- Use the theorem statement as a checklist for application: confirm (i) decay exponent, (ii) gauge/constraint sector, (iii) end structure, then apply the corresponding spectral regime conclusion.

**Explainer-stack addendum:**
- **Strategic context:** Elevates inverse-cube curvature decay from folklore to an explicit spectral boundary condition.
- **Evidence:** Analytic compactness/Weyl-sequence reasoning + radial-model numerical confirmation.
- **Contrarian reframe:** If a tiny change in decay exponent changes the IR sector, which physical boundary condition are we implicitly selecting?
- **Closure:** **Next action:** apply the criterion to another AF family beyond the canonical example to test robustness.

**Original source:** [Annex — §8 Conclusion](#annex-sec-08-conclusion)


**Displayed equations in this section (verbatim):**


$$
\sigma_{\mathrm{ess}}(L)=[0,\infty).
$$


**Named results stated in this section (verbatim statements):**


**Theorem (Spectral threshold for linearized gravity)**

\label{thm:main_threshold}
Let $(\Sigma,g)$ be a smooth asymptotically flat three–manifold with curvature decay
$|{\rm Riem}(x)|\!\le\!C\,r^{-p}$.
Then:
\begin{enumerate}
\item For $p>3$, the curvature potential $V_R$ is relatively compact with respect to
$\nabla^*\nabla$, and the essential spectrum is purely continuous:

$$
\sigma_{\mathrm{ess}}(L)=[0,\infty).
$$

\item At the critical rate $p=3$, compactness fails and a normalized Weyl sequence
appears at zero energy, producing marginally extended tensor configurations
that remain spatially nonlocal yet finite in energy.
\item For $p<3$, curvature acts as a long-range potential that enhances infrared coupling,
but without producing isolated bound states in the tensorial sector.
\end{enumerate}


**Original source for this section:** [jump to annex](#annex-sec-08-conclusion)


---


<a id="if-sec-09-declarations"></a>

### Declarations


**Claim (verbatim):** \textbf{Funding} The author received no external funding.

**Notes (fidelity):**
- This is an administrative section; no technical claims beyond reproducibility statements.

**Explainer-stack addendum:**
- **Strategic context:** Reproducibility/legal context (funding, conflicts, data/code availability).
- **Evidence:** “Not specified in sources” fields are gaps to be filled by authors, not by inference.
- **Closure:** For the numerical study to be replayable, the minimal missing artifact is the exact discretization/solver settings (if not already specified).

**Original source:** [Annex — Declarations](#annex-sec-09-declarations)


**Displayed equations in this section:** None.


**Original source for this section:** [jump to annex](#annex-sec-09-declarations)


---


<a id="if-app-a-gauge-correction-and-elliptic-estimates"></a>

### Appendix A. Gauge Correction and Elliptic Estimates


**Claim (verbatim):** This appendix justifies the harmonic–gauge correction used in Section~3.

**What must be true (technical role of this appendix):**
- The main text’s Weyl-sequence construction uses a gauge condition (harmonic / divergence-free), and this appendix provides the elliptic tools to enforce it.
- The vector Laplacian $\Delta_V$ is Fredholm/invertible in a specific weight range, under a stated topological condition $H^1_{\mathrm{dR}}(\Sigma)=0$.

**Mechanism:**
- Establish a Fredholm/isomorphism mapping for $\Delta_V$ between weighted Sobolev spaces.
- Solve $\Delta_V X = \nabla\!\cdot h$ to correct an approximate tensor field by subtracting a Lie derivative $\mathcal{L}_X g$, yielding a corrected Weyl sequence that lies in harmonic gauge.

**Implications:**
- The essential-spectrum conclusion at $p=3$ is shown to persist after enforcing the gauge constraint.

**Next action:**
- When adapting the Weyl-sequence argument to other manifolds, check the corresponding $H^1_{\mathrm{dR}}$ obstruction and the indicial roots controlling the weight window.

**Explainer-stack addendum:**
- **Strategic context:** Protects the main argument from “it’s just gauge” objections by quantifying gauge correction and elliptic controls.
- **Evidence:** Weighted elliptic estimates that bound correction terms without spoiling scaling.
- **Contrarian reframe:** If the gauge correction were larger by one power of $r$, would it destroy the borderline $p=3$ conclusion?
- **Closure:** **Next action:** trace exactly where decay rates enter estimate constants (thresholds can hide there).

**Original source:** [Annex — Appendix A](#annex-app-a-gauge-correction-and-elliptic-estimates)


**Displayed equations in this section (verbatim):**


$$
\Delta_V X = \nabla^*\nabla X + {\rm Ric}(X)
$$

$$
\Delta_V:H^2_\delta(\Sigma;T^*\Sigma)\to L^2_{\delta-2}(\Sigma;T^*\Sigma)
$$

$$
\|\tilde h_n\|_{L^2}=1,\qquad
\tilde h_n\rightharpoonup0,\qquad
\|L\tilde h_n\|_{L^2}\to0.
$$


**Named results stated in this section (verbatim statements):**


**Lemma (Isomorphism property)**

For weights $-1<\delta<0$, the mapping

$$
\Delta_V:H^2_\delta(\Sigma;T^*\Sigma)\to L^2_{\delta-2}(\Sigma;T^*\Sigma)
$$

is Fredholm of index zero and an isomorphism whenever
$H^1_{\mathrm{dR}}(\Sigma)=0$.


**Lemma (Gauge correction)**

For each $h\in H^2_\delta(\Sigma;S^2T^*\Sigma)$ with $-1<\delta<0$, there exists a
unique $X\in H^2_\delta(\Sigma;T^*\Sigma)$ satisfying
$\Delta_V X=\nabla\!\cdot h$ and
$\|X\|_{H^2_\delta}\le C\|\nabla\!\cdot h\|_{L^2_{\delta-2}}$.


**Proposition (Corrected Weyl sequence)**

Let $\{h_n\}$ be the approximate sequence of Section~3.
Defining $\tilde h_n=h_n-\mathcal{L}_{X_n}g$ with
$X_n=\Delta_V^{-1}(\nabla\!\cdot h_n)$ yields

$$
\|\tilde h_n\|_{L^2}=1,\qquad
\tilde h_n\rightharpoonup0,\qquad
\|L\tilde h_n\|_{L^2}\to0.
$$

Thus $0\in\sigma_{\mathrm{ess}}(L)$ in harmonic gauge.


**Original source for this section:** [jump to annex](#annex-app-a-gauge-correction-and-elliptic-estimates)


---


<a id="if-app-b-curvature-structure-and-the-schwarzschild-example"></a>

### Appendix B. Curvature Structure and the Schwarzschild Example


**Claim (verbatim):** The Schwarzschild metric provides a physical realization of the critical inverse–cube curvature decay analyzed in Section~3.

**What must be true (example role):**
- The appendix treats Schwarzschild initial data in isotropic coordinates as a concrete instance with $|\mathrm{Riem}|\sim r^{-3}$.

**Mechanism:**
- State the spatial metric on a time-symmetric Schwarzschild slice.
- Compute / state the asymptotic curvature decay and show it saturates the inverse-cube rate.
- Identify the leading curvature-coupling term in the spatial Lichnerowicz operator as an attractive $\sim - (CM) r^{-3}$ potential in the far field.

**Implications:**
- The main threshold is not purely abstract: a standard physical spacetime (Schwarzschild) realizes the critical decay exponent.

**Next action:**
- Use this as the starting point for extending the analysis from “synthetic $r^{-p}$ potentials” to actual curved backgrounds (Schwarzschild/Kerr).

**Explainer-stack addendum:**
- **Strategic context:** Anchors the abstract decay condition in a canonical AF geometry where $|\mathrm{Riem}|\sim r^{-3}$ appears naturally.
- **Evidence:** Exhibits curvature scaling behavior and links to assumptions.
- **Contrarian reframe:** If you perturb away from symmetry, which parts remain robust vs. which silently used symmetry?
- **Closure:** **Next action:** verify the example satisfies the exact decay/regularity assumptions used in the spectral lemmas.

**Original source:** [Annex — Appendix B](#annex-app-b-curvature-structure-and-the-schwarzschild-example)


**Displayed equations in this section (verbatim):**


$$
ds^2=-\Bigl(\frac{1-\tfrac{M}{2r}}{1+\tfrac{M}{2r}}\Bigr)^2 dt^2
     +\Bigl(1+\frac{M}{2r}\Bigr)^4(dr^2+r^2d\omega^2).
$$

$$
|{\rm Riem}(x)|\simeq C\,M\,r^{-3}\qquad(r\to\infty),
$$

$$
Lh = \Delta_0 h - (C M) r^{-3} h + O(r^{-4})h,
\qquad C M > 0,
$$


**Named results stated in this section (verbatim statements):**


**Lemma (Asymptotic curvature)**

For this metric,

$$
|{\rm Riem}(x)|\simeq C\,M\,r^{-3}\qquad(r\to\infty),
$$

so the curvature saturates the inverse–cube decay assumed in
Theorem~2.


**Proposition (Effective potential)**

The spatial Lichnerowicz operator on the Schwarzschild background satisfies

$$
Lh = \Delta_0 h - (C M) r^{-3} h + O(r^{-4})h,
\qquad C M > 0,
$$

showing that the Schwarzschild geometry realizes, in its far-field limit,
the same attractive $r^{-3}$ potential analyzed in the numerical model
of Section~5. \footnote{The overall minus sign arises from the definition 
$(V_R h)_{ij} = -R_{i}{}^{k}{}_{j}{}^{\ell} h_{k\ell}$, which makes the curvature coupling attractive for positive mass $M>0$.}


**Definition (Spatial metric)**

In isotropic coordinates $(t,r,\omega)$, the Schwarzschild line element is

$$
ds^2=-\Bigl(\frac{1-\tfrac{M}{2r}}{1+\tfrac{M}{2r}}\Bigr)^2 dt^2
     +\Bigl(1+\frac{M}{2r}\Bigr)^4(dr^2+r^2d\omega^2).
$$

On a time-symmetric slice $t=\mathrm{const.}$,
the spatial metric is $g_{ij}=\psi^4\delta_{ij}$ with
$\psi(r)=1+\tfrac{M}{2r}$.


**Original source for this section:** [jump to annex](#annex-app-b-curvature-structure-and-the-schwarzschild-example)


---


<a id="if-app-c-analytical-framework-and-weighted-sobolev-spaces"></a>

### Appendix C. Analytical Framework and Weighted Sobolev Spaces


**Claim (verbatim):** We summarize the analytic conventions and functional-analytic tools used throughout.

**What must be true (functional-analytic toolkit):**
- Weighted Sobolev spaces $H^k_\delta$ are used to encode decay and to state Fredholm/self-adjointness properties of elliptic operators on asymptotically flat ends.
- The curvature potential obeys a decay bound $|V_R(x)|\le C\langle r\rangle^{-p}$ with $p>3$ for compactness.

**Mechanism:**
- Define $H^k_\delta$ precisely.
- State the Fredholm property for elliptic operators away from indicial roots.
- State self-adjointness and characterize essential spectrum via Weyl sequences.
- Prove (Lemma “Compactness of the curvature potential for $p>3$”) that $V_R$ is compact as a map $H^2_\delta\to L^2_{\delta-2}$ using a cutoff decomposition (compact interior + decaying tail), then apply Weyl’s theorem.

**Implications:**
- This appendix provides the rigorous “engine room” behind the $p>3$ compact-perturbation claim.

**Next action:**
- If you need to weaken assumptions (e.g., multiple ends), this appendix is where to generalize: redo the cutoff/Rellich argument with the new asymptotics.

**Explainer-stack addendum:**
- **Strategic context:** Sets the function spaces where compactness/non-compactness claims are meaningful.
- **Evidence:** Weighted Sobolev mapping properties + compact embeddings used by the main argument.
- **Contrarian reframe:** If you change weights slightly, does compactness change—i.e., is the boundary partly functional-analytic?
- **Closure:** **Next action:** check each embedding/compactness step for its exact weight threshold.

**Original source:** [Annex — Appendix C](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)


**Displayed equations in this section (verbatim):**


$$
\|u\|_{H^k_\delta}^2
   = \sum_{|\alpha|\le k}
     \int_\Sigma (1+r^2)^{\delta-|\alpha|}
     |\nabla^\alpha u|^2\,dV_g.
$$

$$
P:H^2_\delta\to L^2_{\delta-2}
$$

$$
g_{ij} = \delta_{ij} + O(r^{-1}), 
\qquad 
\partial g_{ij} = O(r^{-2}), 
\qquad 
\partial^2 g_{ij} = O(r^{-3}),
$$

$$
L = \nabla^{*}\nabla + V_R,
\qquad
(V_R h)_{ij} = -R_{i}{}^{\ell}{}_{j}{}^{m}\,h_{\ell m}.
$$

$$
V_R : H^2_\delta(\Sigma;S^2T^*\Sigma) \longrightarrow L^2_{\delta-2}(\Sigma;S^2T^*\Sigma),
$$

$$
\sigma_{\mathrm{ess}}(L)=\sigma_{\mathrm{ess}}(\nabla^{*}\nabla)=[0,\infty).
$$

$$
\|V_R h\|_{L^2_{\delta-2}}^2
=
\int_\Sigma \langle r\rangle^{2(\delta-2)}\,|V_R(x)h(x)|^2\,dV_g
\lesssim
\int_\Sigma \langle r\rangle^{2(\delta-2)-2p}|h(x)|^2\,dV_g.
$$

$$
V_R=\chi_R V_R+(1-\chi_R)V_R
=:V_R^{\mathrm{(comp)}}+V_R^{\mathrm{(tail)}}.
$$

$$
\|V_R^{\mathrm{(tail)}}h\|_{L^2_{\delta-2}}^2
\lesssim
\int_{r>R}\langle r\rangle^{2(\delta-2)-2p}|h(x)|^2\,dV_g.
$$

$$
\sup_{\|h\|_{H^2_\delta}=1}
\|V_R^{\mathrm{(tail)}}h\|_{L^2_{\delta-2}}
\longrightarrow 0
\quad\text{as }R\to\infty.
$$

$$
\sigma_{\mathrm{ess}}(L)=\sigma_{\mathrm{ess}}(\nabla^{*}\nabla)=[0,\infty).
$$


**Named results stated in this section (verbatim statements):**


**Lemma (Fredholm property)**

If $P$ is a uniformly elliptic operator approaching constant coefficients
at infinity, then

$$
P:H^2_\delta\to L^2_{\delta-2}
$$

is Fredholm for all $\delta$ not equal to an indicial root
\cite{Lockhart1985}.


**Lemma (Compactness of the curvature potential for $p>3$)**

\label{lem:compact_VR}
Let $(\Sigma,g)$ be a smooth asymptotically flat three-manifold with a single Euclidean end, and assume

$$
g_{ij} = \delta_{ij} + O(r^{-1}), 
\qquad 
\partial g_{ij} = O(r^{-2}), 
\qquad 
\partial^2 g_{ij} = O(r^{-3}),
$$

so that $|{\rm Riem}(x)| \le C\,\langle r\rangle^{-p}$ for some $p>3$.
Fix a weight $-1<\delta<0$, and let

$$
L = \nabla^{*}\nabla + V_R,
\qquad
(V_R h)_{ij} = -R_{i}{}^{\ell}{}_{j}{}^{m}\,h_{\ell m}.
$$

Then the curvature term defines a compact operator

$$
V_R : H^2_\delta(\Sigma;S^2T^*\Sigma) \longrightarrow L^2_{\delta-2}(\Sigma;S^2T^*\Sigma),
$$

and therefore $L$ is a compact perturbation of $\nabla^{*}\nabla$ on $L^2(\Sigma;S^2T^*\Sigma)$.
In particular,

$$
\sigma_{\mathrm{ess}}(L)=\sigma_{\mathrm{ess}}(\nabla^{*}\nabla)=[0,\infty).
$$


**Proposition (Self-adjointness and essential spectrum)**

For $\delta\in(-1,0)$ and $V=O(r^{-p})$ with $p>2$,
operators of the form $L=\nabla^*\nabla+V$
are self–adjoint on $L^2(\Sigma;E)$.
The essential spectrum $\sigma_{\mathrm{ess}}(L)$
is determined by the existence of Weyl sequences as in
Weyl’s criterion.


**Definition (Weighted Sobolev spaces)**

For a smooth radius function $r$ on an asymptotically flat
three–manifold $(\Sigma,g)$ and $\delta\in\mathbb{R}$,

$$
\|u\|_{H^k_\delta}^2
   = \sum_{|\alpha|\le k}
     \int_\Sigma (1+r^2)^{\delta-|\alpha|}
     |\nabla^\alpha u|^2\,dV_g.
$$

Then $H^k_\delta(\Sigma;E)$ is the completion of
$C_c^\infty(\Sigma;E)$ under this norm.


**Original source for this section:** [jump to annex](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)


---


<a id="if-app-d-numerical-validation-and-stability-tests"></a>

### Appendix D. Numerical Validation and Stability Tests


**Claim (verbatim):** This appendix documents three numerical consistency checks supporting the eigenvalue results reported in Section~5: (i) grid-spacing convergence, (ii) finite-volume convergence in $R_{\max}$, and (iii) robustness under constraint enforcement.

**What must be true (numerical validation scope):**
- The eigenvalue trends in §5 could be distorted by discretization, boundary, or penalty-enforcement artifacts; this appendix tests for those.

**Mechanism (three checks performed):**
- Grid refinement at fixed physical volume.
- Finite-volume convergence in $R_{\max}$, including observed $\lambda_1\sim R_{\max}^{-2}$ behavior.
- Penalty-strength sweeps and a cross-check with an explicit TT projection.

**Implications:**
- The observed transition behavior near $p=3$ is supported as a robust numerical feature, not a tuning artifact.

**Next action:**
- Use the reported convergence targets (relative changes in $\lambda_{1,2}$ under refinement and penalty sweeps) as acceptance criteria in independent reproductions.

**Explainer-stack addendum:**
- **Strategic context:** Prevents “numerics artifact” dismissal by showing stability to penalty/discretization variation.
- **Evidence:** Reported insensitivity of the lowest eigenvalue to penalty parameters in specified ranges.
- **Contrarian reframe:** If you enforce constraints via an exact projection (no penalty), do you still see the same $p=3$ transition?
- **Closure:** **Next action:** replicate one penalty sweep and confirm the magnitude of eigenvalue change.

**Original source:** [Annex — Appendix D](#annex-app-d-numerical-validation-and-stability-tests)


**Displayed equations in this section (verbatim):**


$$
\begin{array}{ll}
\text{flat:} & \lambda_1 = 0.07386996,\quad \lambda_2 = 0.14713361, \\
p=2.5:       & \lambda_1 = 0.06505331,\quad \lambda_2 = 0.07023744, \\
p=3.0:       & \lambda_1 = 0.07114971,\quad \lambda_2 = 0.07276003, \\
p=3.5:       & \lambda_1 = 0.07298806,\quad \lambda_2 = 0.07351824.
\end{array}
$$

$$
\begin{array}{ll}
\text{flat:} & \lambda_1 = 0.07398399,\quad \lambda_2 = 0.14781594, \\
p=2.5:       & \lambda_1 = 0.06505635,\quad \lambda_2 = 0.07029545, \\
p=3.0:       & \lambda_1 = 0.07109678,\quad \lambda_2 = 0.07279109, \\
p=3.5:       & \lambda_1 = 0.07295142,\quad \lambda_2 = 0.07355686.
\end{array}
$$

$$
\lambda_1(p{=}2.5)
<
\lambda_1(p{=}3.0)
<
\lambda_1(p{=}3.5)
\simeq
\lambda_1(\text{flat})
$$

$$
\lambda_1(R_{\max}{=}6)=0.1965,\quad
\lambda_1(R_{\max}{=}10)=0.0711,\quad
\lambda_1(R_{\max}{=}20)=0.0180.
$$


**Original source for this section:** [jump to annex](#annex-app-d-numerical-validation-and-stability-tests)


---


<a id="if-app-e-weyl-sequence-construction-and-verification-of-the-critical-spectrum"></a>

### Appendix E. Weyl Sequence Construction and Verification of the Critical Spectrum


**Claim (verbatim):** This appendix provides quantitative estimates completing the proof of Lemma~2 and Theorem~1.

**What must be true (quantitative completion):**
- The manifold satisfies the asymptotic flatness conditions referenced (including $|\mathrm{Riem}|\le C r^{-3}$).
- A tensor harmonic $H_{ij}(\omega)$ is available with the stated trace-free/divergence-free properties on $S^2$.

**Mechanism (what is computed):**
- Normalize the annular sequence $h_n$ by choosing amplitude $A_n\simeq n^{-1/2}$.
- Estimate commutator terms from the cutoff $\phi_n$ and show $\|[\nabla^*\nabla,\phi_n]h_n\|_{L^2}\lesssim n^{-2}$.
- Estimate the curvature term with $|V_R|\le C r^{-3}$, obtaining $\|V_R h_n\|_{L^2}\lesssim n^{-2}$.
- Apply gauge correction (solving $\Delta_V X_n=\nabla\!\cdot h_n$) and show the correction is subleading in the $L^2$ error.
- Conclude by Weyl’s criterion that $0\in\sigma_{\mathrm{ess}}(L)$.

**Implications:**
- This appendix supplies the detailed scaling estimates that make the borderline ($p=3$) Weyl-sequence argument fully quantitative.

**Next action:**
- If you need to change the decay rate (e.g., $r^{-p}$ with $p\neq 3$), redo these estimates and track precisely which term dominates—this directly diagnoses why $p=3$ is the balance point.

**Explainer-stack addendum:**
- **Strategic context:** Supplies the quantitative bounds that make the Weyl-sequence argument airtight.
- **Evidence:** Explicit scaling estimates for commutator, curvature, and gauge-correction terms as the annulus is sent to infinity.
- **Contrarian reframe:** If the curvature bound had mild log losses/oscillations, would the same scaling still force $0\in\sigma_{\mathrm{ess}}(L)$?
- **Closure:** **Next action:** rewrite the estimates as a reproducible checklist for another AF manifold.

**Original source:** [Annex — Appendix E](#annex-app-e-weyl-sequence-construction-and-verification-of-the-critical-spectrum)


**Displayed equations in this section (verbatim):**


$$
\|h_n\|_{L^2}^2\!\simeq\!A_n^2\!\!\int_{n/2}^{2n}\!r^{-2}r^2dr
\sim A_n^2 n.
$$

$$
\nabla^*\nabla(\phi_n r^{-1}H)
= \phi_n\nabla^*\nabla(r^{-1}H)
 + 2\nabla\phi_n\!\cdot\!\nabla(r^{-1}H)
 + (\Delta\phi_n)r^{-1}H,
$$

$$
\|[\nabla^*\nabla,\phi_n]h_n\|_{L^2}
\lesssim n^{-2}.
$$

$$
\|V_R h_n\|_{L^2}^2
\lesssim A_n^2\!\int_{n/2}^{2n}\!r^{-6}r^2dr
\sim n^{-4},
\qquad
\|V_R h_n\|_{L^2}\lesssim n^{-2}.
$$

$$
\|X_n\|_{H^2_\delta}\lesssim
\|\nabla\!\cdot h_n\|_{L^2}
\lesssim n^{-1},
\qquad -1<\delta<0.
$$

$$
\|L(\mathcal{L}_{X_n}g)\|_{L^2}\lesssim n^{-1},
$$

$$
\|L\tilde h_n\|_{L^2}\to0,
\qquad
\|\tilde h_n\|_{L^2}=1.
$$


**Original source for this section:** [jump to annex](#annex-app-e-weyl-sequence-construction-and-verification-of-the-critical-spectrum)


---


<a id="if-formula-appendix"></a>

## Formula appendix: all displayed formulas + formula cards

This appendix lists every displayed formula in source order and attaches a **Formula Card** to each one (exact quote → regime → claim → falsification → misuse → so-what).

<a id="if-formula-001"></a>

### F001

- **Source location:** [Abstract](#annex-abstract)

**Exact quote (verbatim LaTeX):**

```latex
\[
L_p=-\tfrac{d^2}{dr^2}+\tfrac{\ell(\ell+1)}{r^2}+\tfrac{C}{r^p}
\]
```

**Rendered:**

$$
L_p=-\tfrac{d^2}{dr^2}+\tfrac{\ell(\ell+1)}{r^2}+\tfrac{C}{r^p}
$$

**Formula Card:**

1) **Symbol legend**
- `C`: coupling coefficient (strength/sign) multiplying the power-law tail
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `\ell`: spherical-harmonic / angular-momentum index
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `p`: power-law decay exponent (tail strength) in an effective term like $C/r^p$
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
large-$r$ / asymptotic region where power-law tails matter; Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Defines `L_p` in terms of `-\tfrac{d^2}{dr^2}+\tfrac{\ell(\ell+1)}{r^2}+\tfrac{C}{r^p}` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-002"></a>

### F002 (`\label{eq:Lichnerowicz}`)

- **Source location:** [1. Introduction](#annex-sec-01-introduction)

**Exact quote (verbatim LaTeX):**

```latex
\begin{equation}
L = \nabla^{*}\nabla + V_R, 
\qquad (V_R h)_{ij} = -R^{\;\ell}{}_{i j m}\, h_{\ell}{}^{m},
\label{eq:Lichnerowicz}
\end{equation}
```

**Rendered:**

$$
L = \nabla^{*}\nabla + V_R, 
\qquad (V_R h)_{ij} = -R^{\;\ell}{}_{i j m}\, h_{\ell}{}^{m},
$$

**Formula Card:**

1) **Symbol legend**
- `L`: spatial Lichnerowicz operator on symmetric trace-free tensors $h_{ij}$
- `\nabla^{*}\nabla`: rough Laplacian term
- `V_R`: curvature coupling term (effective potential)
- `R^{\;\ell}{}_{ijm}`: Riemann curvature components of $(\Sigma,g)$
- `h_{\ell}{}^{m}`: tensor field component
- Other symbols (not explicitly defined here): `h`, `n`

2) **Regime and assumptions**
Asymptotically flat 3-manifold $(\Sigma,g)$; large-$r$ behavior matters because $V_R$ scales like curvature.

3) **Plain-English claim**
Defines the operator $L$ as a Laplace-type term plus curvature potential $V_R$; the paper studies how decay of $V_R$ controls $\sigma_{\mathrm{ess}}(L)$ near $0$.

4) **What would falsify it**
Definition consistency check: compare with Eq. \ref{eq:Lichnerowicz_def} and the standard Lichnerowicz operator conventions in the text.

5) **Misuse risks**
Treating $L$ as a time-evolution operator; here it is a spatial elliptic operator controlling stationary/harmonic configurations.

6) **So what**
This is the object whose essential spectrum distinguishes “flat-like/radiative” vs. “marginal/extended” infrared behavior.

---

<a id="if-formula-003"></a>

### F003

- **Source location:** [1. Introduction](#annex-sec-01-introduction)

**Exact quote (verbatim LaTeX):**

```latex
\[
L_p = -\frac{d^2}{dr^2} + \frac{\ell(\ell+1)}{r^2} + \frac{C}{r^p},
\]
```

**Rendered:**

$$
L_p = -\frac{d^2}{dr^2} + \frac{\ell(\ell+1)}{r^2} + \frac{C}{r^p},
$$

**Formula Card:**

1) **Symbol legend**
- `C`: coupling coefficient (strength/sign) multiplying the power-law tail
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `\ell`: spherical-harmonic / angular-momentum index
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `p`: power-law decay exponent (tail strength) in an effective term like $C/r^p$
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
large-$r$ / asymptotic region where power-law tails matter; Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Defines `L_p` in terms of `-\frac{d^2}{dr^2} + \frac{\ell(\ell+1)}{r^2} + \frac{C}{r^p},` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-004"></a>

### F004

- **Source location:** [2. Spectral Scaling and Structural Parallels Across Spin-1 and Spin-2 Fields](#annex-sec-02-spectral-scaling-and-structural-parallels-across-spin-1-and-spin-2-fields)

**Exact quote (verbatim LaTeX):**

```latex
\[
g_{ij}=\delta_{ij}+a_{ij},\qquad 
a_{ij}=O(r^{-1}),\quad 
\partial_k a_{ij}=O(r^{-2}),\quad 
\partial_\ell\partial_k a_{ij}=O(r^{-3}),
\]
```

**Rendered:**

$$
g_{ij}=\delta_{ij}+a_{ij},\qquad 
a_{ij}=O(r^{-1}),\quad 
\partial_k a_{ij}=O(r^{-2}),\quad 
\partial_\ell\partial_k a_{ij}=O(r^{-3}),
$$

**Formula Card:**

1) **Symbol legend**
- `\ell`: spherical-harmonic / angular-momentum index
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `p`: power-law decay exponent (tail strength) in an effective term like $C/r^p$
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
large-$r$ / asymptotic region where power-law tails matter

3) **Plain-English claim**
Defines `g_{ij}` in terms of `\delta_{ij}+a_{ij},\qquad` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-005"></a>

### F005 (`\label{eq:EH_action}`)

- **Source location:** [2. Spectral Scaling and Structural Parallels Across Spin-1 and Spin-2 Fields](#annex-sec-02-spectral-scaling-and-structural-parallels-across-spin-1-and-spin-2-fields)

**Exact quote (verbatim LaTeX):**

```latex
\begin{equation}
S_{\text{EH}}=\frac{1}{16\pi G}\int R\sqrt{-g}\,d^4x
\label{eq:EH_action}
\end{equation}
```

**Rendered:**

$$
S_{\text{EH}}=\frac{1}{16\pi G}\int R\sqrt{-g}\,d^4x
$$

**Formula Card:**

1) **Symbol legend**
- `S_{\text{EH}}`: Einstein–Hilbert action
- `G`: Newton’s constant
- `R`: scalar curvature
- `g`: spacetime metric determinant (via $\sqrt{-g}$)
- Other symbols (not explicitly defined here): `n`

2) **Regime and assumptions**
4D action principle; used for structural comparison/normalization.

3) **Plain-English claim**
States the standard Einstein–Hilbert action used as the spin-2 baseline.

4) **What would falsify it**
N/A (standard definition).

5) **Misuse risks**
Inferring spectral results directly from the action without the spatial/operator analysis.

6) **So what**
Anchors the gravity operator structure being compared to spin-1 thresholds.

---

<a id="if-formula-006"></a>

### F006

- **Source location:** [2. Spectral Scaling and Structural Parallels Across Spin-1 and Spin-2 Fields](#annex-sec-02-spectral-scaling-and-structural-parallels-across-spin-1-and-spin-2-fields)

**Exact quote (verbatim LaTeX):**

```latex
\[
S^{(2)}[h]=\tfrac{1}{2}\!\int_\Sigma h^{ij}L_{ij}{}^{kl}h_{kl}\sqrt{g}\,d^3x,
\]
```

**Rendered:**

$$
S^{(2)}[h]=\tfrac{1}{2}\!\int_\Sigma h^{ij}L_{ij}{}^{kl}h_{kl}\sqrt{g}\,d^3x,
$$

**Formula Card:**

1) **Symbol legend**
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Defines `S^{(2)}[h]` in terms of `\tfrac{1}{2}\!\int_\Sigma h^{ij}L_{ij}{}^{kl}h_{kl}\sqrt{g}\,d^3x,` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-007"></a>

### F007 (`\label{eq:Lichnerowicz_def}`)

- **Source location:** [2. Spectral Scaling and Structural Parallels Across Spin-1 and Spin-2 Fields](#annex-sec-02-spectral-scaling-and-structural-parallels-across-spin-1-and-spin-2-fields)

**Exact quote (verbatim LaTeX):**

```latex
\begin{equation}
Lh = \nabla^*\nabla h + V_R h, 
\qquad (V_Rh)_{ij}=-R_i{}^\ell{}_j{}^m h_{\ell m}.
\label{eq:Lichnerowicz_def}
\end{equation}
```

**Rendered:**

$$
Lh = \nabla^*\nabla h + V_R h, 
\qquad (V_Rh)_{ij}=-R_i{}^\ell{}_j{}^m h_{\ell m}.
$$

**Formula Card:**

1) **Symbol legend**
- `Lh`: application of $L$ to $h$
- `\nabla^*\nabla h`: rough Laplacian acting on $h$
- `(V_Rh)_{ij}`: curvature action on $h$ components
- `R_i{}^\ell{}_j{}^m`: Riemann curvature components
- `h_{\ell m}`: tensor field component
- Other symbols (not explicitly defined here): `h`, `n`

2) **Regime and assumptions**
Asymptotically flat 3-manifold; Laplace-type operator on symmetric tensors.

3) **Plain-English claim**
Restates/clarifies the curvature contraction that makes $V_R$ behave like a potential term.

4) **What would falsify it**
Definition consistency check: verify contractions match Eq. \ref{eq:Lichnerowicz}.

5) **Misuse risks**
Dropping $V_R$ at the borderline decay rate where it is not spectrally compact.

6) **So what**
Makes explicit why curvature decay exponent controls compactness and thresholds.

---

<a id="if-formula-008"></a>

### F008 (`\label{eq:YMlap}`)

- **Source location:** [2. Spectral Scaling and Structural Parallels Across Spin-1 and Spin-2 Fields](#annex-sec-02-spectral-scaling-and-structural-parallels-across-spin-1-and-spin-2-fields)

**Exact quote (verbatim LaTeX):**

```latex
\begin{equation}
\Delta_A = -(\nabla_A)^*\nabla_A = -\nabla^*\nabla + {\rm ad}(F_A),
\label{eq:YMlap}
\end{equation}
```

**Rendered:**

$$
\Delta_A = -(\nabla_A)^*\nabla_A = -\nabla^*\nabla + {\rm ad}(F_A),
$$

**Formula Card:**

1) **Symbol legend**
- `\Delta_A`: non-Abelian covariant Laplacian with connection $A$
- `D_A`: covariant derivative
- `F`: field strength (curvature) of $A$
- Other symbols (not explicitly defined here): `n`

2) **Regime and assumptions**
Gauge theory analogy; invoked to parallel threshold phenomena.

3) **Plain-English claim**
Recalls the gauge-covariant Laplacian structure used for the spin-1 parallel.

4) **What would falsify it**
N/A (operator definition).

5) **Misuse risks**
Treating the analogy as a proof of the gravity threshold.

6) **So what**
Motivates expecting a comparable long-range threshold in the gravity operator.

---

<a id="if-formula-009"></a>

### F009

- **Source location:** [2. Spectral Scaling and Structural Parallels Across Spin-1 and Spin-2 Fields](#annex-sec-02-spectral-scaling-and-structural-parallels-across-spin-1-and-spin-2-fields)

**Exact quote (verbatim LaTeX):**

```latex
\[
H^k_\delta(\Sigma;S^2T^*\Sigma)
=\Bigl\{h\in H^k_{\mathrm{loc}}(\Sigma):\|h\|_{H^k_\delta}<\infty\Bigr\},\qquad
\|h\|_{H^k_\delta}^2=\sum_{j=0}^k\!\!\int_\Sigma\langle r\rangle^{2(\delta-j)}|\nabla^jh|_g^2\,dV_g.
\]
```

**Rendered:**

$$
H^k_\delta(\Sigma;S^2T^*\Sigma)
=\Bigl\{h\in H^k_{\mathrm{loc}}(\Sigma):\|h\|_{H^k_\delta}<\infty\Bigr\},\qquad
\|h\|_{H^k_\delta}^2=\sum_{j=0}^k\!\!\int_\Sigma\langle r\rangle^{2(\delta-j)}|\nabla^jh|_g^2\,dV_g.
$$

**Formula Card:**

1) **Symbol legend**
- `\nabla`: Levi-Civita covariant derivative of the background metric
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Displays an intermediate expression used in the argument (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-010"></a>

### F010 (`\label{eq:int_identity}`)

- **Source location:** [2. Spectral Scaling and Structural Parallels Across Spin-1 and Spin-2 Fields](#annex-sec-02-spectral-scaling-and-structural-parallels-across-spin-1-and-spin-2-fields)

**Exact quote (verbatim LaTeX):**

```latex
\begin{equation}
\langle Lh,k\rangle_{L^2}
=\langle\nabla h,\nabla k\rangle_{L^2}
+\langle V_Rh,k\rangle_{L^2},
\label{eq:int_identity}
\end{equation}
```

**Rendered:**

$$
\langle Lh,k\rangle_{L^2}
=\langle\nabla h,\nabla k\rangle_{L^2}
+\langle V_Rh,k\rangle_{L^2},
$$

**Formula Card:**

1) **Symbol legend**
- `\langle\cdot,\cdot\rangle`: $L^2$ inner product
- `\nabla`: covariant derivative
- `V_R`: curvature potential term
- Other symbols (not explicitly defined here): `h`, `n`

2) **Regime and assumptions**
Quadratic-form / integration identity under decay conditions.

3) **Plain-English claim**
Decomposes the quadratic form of $L$ into derivative and curvature pieces (used for spectral reasoning).

4) **What would falsify it**
Would fail if boundary terms at infinity do not vanish under the assumed decay/weights.

5) **Misuse risks**
Applying outside the operator domain where integration by parts is valid.

6) **So what**
Connects geometric decay hypotheses to operator-theoretic conclusions.

---

<a id="if-formula-011"></a>

### F011 (`\label{eq:L_mapping2}`)

- **Source location:** [2. Spectral Scaling and Structural Parallels Across Spin-1 and Spin-2 Fields](#annex-sec-02-spectral-scaling-and-structural-parallels-across-spin-1-and-spin-2-fields)

**Exact quote (verbatim LaTeX):**

```latex
\begin{equation}
L:H^2_\delta(\Sigma;S^2T^*\Sigma)\longrightarrow L^2_{\delta-2}(\Sigma;S^2T^*\Sigma)
\label{eq:L_mapping2}
\end{equation}
```

**Rendered:**

$$
L:H^2_\delta(\Sigma;S^2T^*\Sigma)\longrightarrow L^2_{\delta-2}(\Sigma;S^2T^*\Sigma)
$$

**Formula Card:**

1) **Symbol legend**
- `L`: gravity operator
- `\Delta_A`: gauge-covariant Laplacian (analogy)
- Other symbols (not explicitly defined here): `h`, `n`

2) **Regime and assumptions**
Structural mapping/parallel across spin-1 and spin-2 Laplace-type operators.

3) **Plain-English claim**
States the correspondence used to compare scaling thresholds across theories.

4) **What would falsify it**
Would be invalid if functional settings/constraints differ so “compact perturbation” is not comparable.

5) **Misuse risks**
Reading it as an equality of spectra rather than a heuristic for decay exponents.

6) **So what**
Explains why $p=3$ is the expected borderline in 3D.

---

<a id="if-formula-012"></a>

### F012 (`\label{eq:ess_flat}`)

- **Source location:** [2. Spectral Scaling and Structural Parallels Across Spin-1 and Spin-2 Fields](#annex-sec-02-spectral-scaling-and-structural-parallels-across-spin-1-and-spin-2-fields)

**Exact quote (verbatim LaTeX):**

```latex
\begin{equation}
\sigma_{\mathrm{ess}}(L)=[0,\infty).
\label{eq:ess_flat}
\end{equation}
```

**Rendered:**

$$
\sigma_{\mathrm{ess}}(L)=[0,\infty).
$$

**Formula Card:**

1) **Symbol legend**
- `\sigma_{\mathrm{ess}}(L)`: essential spectrum of $L$
- `[0,\infty)`: flat Laplacian essential spectrum baseline
- Other symbols (not explicitly defined here): `h`, `n`

2) **Regime and assumptions**
Faster-than-critical decay so curvature is a compact perturbation.

3) **Plain-English claim**
States the flat-space essential-spectrum baseline persisting under compact perturbations.

4) **What would falsify it**
Counterexample geometry where faster decay still yields non-compact spectral effects.

5) **Misuse risks**
Assuming it excludes threshold resonances/embedded eigenvalues (it doesn’t).

6) **So what**
Defines the “radiative/flat-like” regime baseline.

---

<a id="if-formula-013"></a>

### F013 (`\label{eq:pcrit}`)

- **Source location:** [2. Spectral Scaling and Structural Parallels Across Spin-1 and Spin-2 Fields](#annex-sec-02-spectral-scaling-and-structural-parallels-across-spin-1-and-spin-2-fields)

**Exact quote (verbatim LaTeX):**

```latex
\begin{equation}
p_{\mathrm{crit}}=3,
\label{eq:pcrit}
\end{equation}
```

**Rendered:**

$$
p_{\mathrm{crit}}=3,
$$

**Formula Card:**

1) **Symbol legend**
- `p_{\mathrm{crit}}`: critical decay exponent
- `p`: decay exponent
- Other symbols (not explicitly defined here): `h`

2) **Regime and assumptions**
3 spatial dimensions.

3) **Plain-English claim**
Identifies the borderline exponent $p=3$ separating compact vs. non-compact curvature tails in 3D.

4) **What would falsify it**
Analytic/numeric counterexample showing compactness still holds at $p=3$ (or fails for some $p>3$) under the paper’s hypotheses.

5) **Misuse risks**
Assuming $p=3$ is dimension-independent.

6) **So what**
This is the headline threshold driving the rest of the argument.

---

<a id="if-formula-014"></a>

### F014 (`\label{eq:vectorlaplacian}`)

- **Source location:** [3. Infrared Spectrum and Marginal Modes](#annex-sec-03-infrared-spectrum-and-marginal-modes)

**Exact quote (verbatim LaTeX):**

```latex
\begin{equation}
\Delta_V=\nabla^*\nabla+\mathrm{Ric},
\label{eq:vectorlaplacian}
\end{equation}
```

**Rendered:**

$$
\Delta_V=\nabla^*\nabla+\mathrm{Ric},
$$

**Formula Card:**

1) **Symbol legend**
- `\Delta_V`: vector Laplacian used for gauge correction
- `X`: vector field generating a gauge transformation
- Other symbols (not explicitly defined here): `h`, `n`

2) **Regime and assumptions**
Gauge correction step in weighted spaces on AF manifolds.

3) **Plain-English claim**
Introduces the elliptic operator solved to implement gauge/constraint correction.

4) **What would falsify it**
Fails if invertibility/a priori estimates fail in the required weights.

5) **Misuse risks**
Assuming solvability without verifying the admissible weight range.

6) **So what**
Allows Weyl sequences to be corrected into the right constraint subspace.

---

<a id="if-formula-015"></a>

### F015 (`\label{eq:vectorfredholm}`)

- **Source location:** [3. Infrared Spectrum and Marginal Modes](#annex-sec-03-infrared-spectrum-and-marginal-modes)

**Exact quote (verbatim LaTeX):**

```latex
\begin{equation}
\Delta_V: H^2_\delta(\Sigma;T^*\Sigma)\longrightarrow L^2_{\delta-2}(\Sigma;T^*\Sigma)
\label{eq:vectorfredholm}
\end{equation}
```

**Rendered:**

$$
\Delta_V: H^2_\delta(\Sigma;T^*\Sigma)\longrightarrow L^2_{\delta-2}(\Sigma;T^*\Sigma)
$$

**Formula Card:**

1) **Symbol legend**
- `\Delta_V`: vector Laplacian
- `Fredholm`: solvability property modulo finite-dimensional obstructions
- Other symbols (not explicitly defined here): `h`, `n`

2) **Regime and assumptions**
Fredholm theory in weighted Sobolev spaces on AF manifolds.

3) **Plain-English claim**
States the Fredholm/invertibility property needed to control gauge corrections.

4) **What would falsify it**
Counterexample weight/geometry with unexpected kernel or loss of estimates.

5) **Misuse risks**
Using without checking the weight lies in the Fredholm range.

6) **So what**
Justifies the gauge-fixing step in the proof.

---

<a id="if-formula-016"></a>

### F016 (`\label{eq:metricdecay}`)

- **Source location:** [3. Infrared Spectrum and Marginal Modes](#annex-sec-03-infrared-spectrum-and-marginal-modes)

**Exact quote (verbatim LaTeX):**

```latex
\begin{equation}
g_{ij}=\delta_{ij}+O(r^{-1}),\qquad 
\partial g_{ij}=O(r^{-2}),\qquad 
\partial^2 g_{ij}=O(r^{-3}),
\label{eq:metricdecay}
\end{equation}
```

**Rendered:**

$$
g_{ij}=\delta_{ij}+O(r^{-1}),\qquad 
\partial g_{ij}=O(r^{-2}),\qquad 
\partial^2 g_{ij}=O(r^{-3}),
$$

**Formula Card:**

1) **Symbol legend**
- `g_{ij}`: spatial metric components
- `\delta_{ij}`: Euclidean metric components
- `r`: radial coordinate

2) **Regime and assumptions**
Asymptotic flatness at infinity.

3) **Plain-English claim**
Encodes the assumed falloff of the metric (and derivatives) underpinning curvature decay and compactness claims.

4) **What would falsify it**
Geometry not satisfying the stated AF rate.

5) **Misuse risks**
Treating metric falloff as sufficient without checking curvature/derivative bounds used elsewhere.

6) **So what**
Part of the hypothesis set that makes the threshold statement checkable.

---

<a id="if-formula-017"></a>

### F017 (`\label{eq:lichcritical}`)

- **Source location:** [3. Infrared Spectrum and Marginal Modes](#annex-sec-03-infrared-spectrum-and-marginal-modes)

**Exact quote (verbatim LaTeX):**

```latex
\begin{equation}
Lh = \nabla^*\nabla h + V_R h, 
\qquad (V_R h)_{ij}=-R_{i\ \ j m}^{\ \ell}h_\ell^{\ m},
\label{eq:lichcritical}
\end{equation}
```

**Rendered:**

$$
Lh = \nabla^*\nabla h + V_R h, 
\qquad (V_R h)_{ij}=-R_{i\ \ j m}^{\ \ell}h_\ell^{\ m},
$$

**Formula Card:**

1) **Symbol legend**
- `|\mathrm{Riem}|`: Riemann curvature magnitude
- `\lesssim`: bounded up to a constant
- `r^{-3}`: inverse-cube decay
- Other symbols (not explicitly defined here): `h`, `n`

2) **Regime and assumptions**
Borderline curvature decay in 3D AF setting.

3) **Plain-English claim**
States the critical curvature decay condition that drives the non-compactness/Weyl-sequence result.

4) **What would falsify it**
Would be undermined if the same essential-spectrum behavior held under strictly faster decay (or failed at $r^{-3}$).

5) **Misuse risks**
Assuming pointwise decay alone implies spectral conclusions without domain/weight checks.

6) **So what**
This is the “geometry knob” the whole paper turns.

---

<a id="if-formula-018"></a>

### F018 (`\label{eq:weylansatz}`)

- **Source location:** [3. Infrared Spectrum and Marginal Modes](#annex-sec-03-infrared-spectrum-and-marginal-modes)

**Exact quote (verbatim LaTeX):**

```latex
\begin{equation}
h_n(r,\omega)=A_n\,\phi_n(r)\,r^{-1}H_{ij}(\omega),
\label{eq:weylansatz}
\end{equation}
```

**Rendered:**

$$
h_n(r,\omega)=A_n\,\phi_n(r)\,r^{-1}H_{ij}(\omega),
$$

**Formula Card:**

1) **Symbol legend**
- `h_n`: Weyl sequence (approximate eigenfunctions)
- `\phi_n`: cutoff localizing to annuli sent to infinity
- `H_{ij}`: tensor harmonic on $S^2$
- Other symbols (not explicitly defined here): `h`, `n`

2) **Regime and assumptions**
Large-$n$ annuli at large radius; normalized so residual under $L$ vanishes.

3) **Plain-English claim**
Defines the Weyl-sequence ansatz used to prove $0\in\sigma_{\mathrm{ess}}(L)$ at the borderline.

4) **What would falsify it**
Fails if commutator/curvature/gauge-correction terms do not go to zero in $L^2$.

5) **Misuse risks**
Omitting normalization or constraints (TT/gauge) that the proof requires.

6) **So what**
This is the constructive heart of the essential-spectrum result.

---

<a id="if-formula-019"></a>

### F019

- **Source location:** [3. Infrared Spectrum and Marginal Modes](#annex-sec-03-infrared-spectrum-and-marginal-modes)

**Exact quote (verbatim LaTeX):**

```latex
\[
\|\tilde h_n\|_{L^2}=1,\qquad
\nabla^j\tilde h_{n,ij}=0,\qquad
\|L\tilde h_n\|_{L^2}\to0.
\]
```

**Rendered:**

$$
\|\tilde h_n\|_{L^2}=1,\qquad
\nabla^j\tilde h_{n,ij}=0,\qquad
\|L\tilde h_n\|_{L^2}\to0.
$$

**Formula Card:**

1) **Symbol legend**
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `\nabla`: Levi-Civita covariant derivative of the background metric
- `d`: spatial dimension (as used in the dimensional scaling chapter)

2) **Regime and assumptions**
Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Defines `\|\tilde h_n\|_{L^2}` in terms of `1,\qquad` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-020"></a>

### F020 (`\label{eq:criticalspectrum}`)

- **Source location:** [3. Infrared Spectrum and Marginal Modes](#annex-sec-03-infrared-spectrum-and-marginal-modes)

**Exact quote (verbatim LaTeX):**

```latex
\begin{equation}
[0,\infty)\subset\sigma_{\mathrm{ess}}(L),\qquad 0\in\sigma_{\mathrm{ess}}(L).
\label{eq:criticalspectrum}
\end{equation}
```

**Rendered:**

$$
[0,\infty)\subset\sigma_{\mathrm{ess}}(L),\qquad 0\in\sigma_{\mathrm{ess}}(L).
$$

**Formula Card:**

1) **Symbol legend**
- `0\in\sigma_{\mathrm{ess}}(L)`: zero belongs to the essential spectrum
- Other symbols (not explicitly defined here): `h`, `n`

2) **Regime and assumptions**
Critical decay regime under the theorem’s hypotheses.

3) **Plain-English claim**
States the main spectral consequence: $0$ lies in the essential spectrum in the borderline regime.

4) **What would falsify it**
Would be falsified by showing the Weyl sequence cannot be constructed/corrected as claimed.

5) **Misuse risks**
Interpreting this as existence of an $L^2$ eigenfunction at $0$ (essential-spectrum membership is weaker).

6) **So what**
Justifies calling the modes “marginal”: extended behavior at threshold energy.

---

<a id="if-formula-021"></a>

### F021

- **Source location:** [3. Infrared Spectrum and Marginal Modes](#annex-sec-03-infrared-spectrum-and-marginal-modes)

**Exact quote (verbatim LaTeX):**

```latex
\[
\Box_g h_{\mu\nu} + 2R_{\mu\ \nu}^{\ \rho\ \sigma}h_{\rho\sigma}=0,
\]
```

**Rendered:**

$$
\Box_g h_{\mu\nu} + 2R_{\mu\ \nu}^{\ \rho\ \sigma}h_{\rho\sigma}=0,
$$

**Formula Card:**

1) **Symbol legend**
- `R`: scalar curvature / curvature tensor (context-dependent)
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
as stated in the surrounding text near this display

3) **Plain-English claim**
Gives an explicit equality relating the quantities shown (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-022"></a>

### F022

- **Source location:** [4. Dimensional Scaling and Spectral Phase Structure](#annex-sec-04-dimensional-scaling-and-spectral-phase-structure)

**Exact quote (verbatim LaTeX):**

```latex
\[
L_d=-\nabla^2 + V(r),
\]
```

**Rendered:**

$$
L_d=-\nabla^2 + V(r),
$$

**Formula Card:**

1) **Symbol legend**
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `\nabla`: Levi-Civita covariant derivative of the background metric
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Defines `L_d` in terms of `-\nabla^2 + V(r),` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-023"></a>

### F023

- **Source location:** [4. Dimensional Scaling and Spectral Phase Structure](#annex-sec-04-dimensional-scaling-and-spectral-phase-structure)

**Exact quote (verbatim LaTeX):**

```latex
\[
\int_R^\infty r^{\,2(\delta-2)}\,|V(r)|^2\,|h(r)|^2\,r^{\,d-1}dr
\;\sim\;
\int_R^\infty r^{\,d-5+2\delta-2p}\,dr.
\]
```

**Rendered:**

$$
\int_R^\infty r^{\,2(\delta-2)}\,|V(r)|^2\,|h(r)|^2\,r^{\,d-1}dr
\;\sim\;
\int_R^\infty r^{\,d-5+2\delta-2p}\,dr.
$$

**Formula Card:**

1) **Symbol legend**
- `R`: scalar curvature / curvature tensor (context-dependent)
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `p`: power-law decay exponent (tail strength) in an effective term like $C/r^p$
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
large-$r$ / asymptotic region where power-law tails matter

3) **Plain-English claim**
Displays an intermediate expression used in the argument (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-024"></a>

### F024 (`\label{eq:pcrit_general}`)

- **Source location:** [4. Dimensional Scaling and Spectral Phase Structure](#annex-sec-04-dimensional-scaling-and-spectral-phase-structure)

**Exact quote (verbatim LaTeX):**

```latex
\begin{equation}
p_{\mathrm{crit}}=d,
\label{eq:pcrit_general}
\end{equation}
```

**Rendered:**

$$
p_{\mathrm{crit}}=d,
$$

**Formula Card:**

1) **Symbol legend**
- `p_{\mathrm{crit}}`: critical decay exponent
- `d`: spatial dimension
- Other symbols (not explicitly defined here): `h`, `n`

2) **Regime and assumptions**
General spatial dimension $d$ under scaling analysis.

3) **Plain-English claim**
Gives the dimensional generalization: $p_{\mathrm{crit}}=d$.

4) **What would falsify it**
Dimension-$d$ counterexample contradicting the compactness boundary at $r^{-d}$.

5) **Misuse risks**
Applying without checking which operator/constraints and which notion of “decay exponent” is meant.

6) **So what**
Explains why $p=3$ appears in 3D: it is a special case of a dimensional scaling boundary.

---

<a id="if-formula-025"></a>

### F025 (`\label{eq:penalty}`)

- **Source location:** [5. Numerical Verification of the Spectral Threshold](#annex-sec-05-numerical-verification-of-the-spectral-threshold)

**Exact quote (verbatim LaTeX):**

```latex
\begin{equation}
\mathcal{R}_{\eta,\zeta}[h]
=
\frac{\langle h,Lh\rangle
+\eta\,\|\nabla\!\cdot h\|_{L^2(\Omega)}^2
+\zeta\,\|\mathrm{tr}\,h\|_{L^2(\Omega)}^2}
{\|h\|_{L^2(\Omega)}^2},
\label{eq:penalty}
\end{equation}
```

**Rendered:**

$$
\mathcal{R}_{\eta,\zeta}[h]
=
\frac{\langle h,Lh\rangle
+\eta\,\|\nabla\!\cdot h\|_{L^2(\Omega)}^2
+\zeta\,\|\mathrm{tr}\,h\|_{L^2(\Omega)}^2}
{\|h\|_{L^2(\Omega)}^2},
$$

**Formula Card:**

1) **Symbol legend**
- `\mathcal{R}_{\eta,\zeta}[h]`: penalized Rayleigh quotient in numerics
- `\eta,\zeta`: penalty weights enforcing divergence-free and trace-free constraints
- `\Omega`: numerical domain
- `\langle h,Lh\rangle`: quadratic form of $L$ on the domain
- Other symbols (not explicitly defined here): `h`, `n`

2) **Regime and assumptions**
Finite-domain numerical approximation; constraints enforced by penalties.

3) **Plain-English claim**
Defines the penalized Rayleigh quotient whose stationary points yield a generalized eigenproblem approximating the TT spectrum of $L$.

4) **What would falsify it**
Undermined if varying $\eta,\zeta$ changes the lowest eigenvalues materially.

5) **Misuse risks**
Treating penalty enforcement as exact TT projection.

6) **So what**
This is how the numerics approximate the constrained low modes while remaining computationally stable.

---

<a id="if-formula-026"></a>

### F026

- **Source location:** [5. Numerical Verification of the Spectral Threshold](#annex-sec-05-numerical-verification-of-the-spectral-threshold)

**Exact quote (verbatim LaTeX):**

```latex
\[
L_p = -\frac{d^2}{dr^2} + \frac{\ell(\ell+1)}{r^2} + \frac{C}{r^p},
\]
```

**Rendered:**

$$
L_p = -\frac{d^2}{dr^2} + \frac{\ell(\ell+1)}{r^2} + \frac{C}{r^p},
$$

**Formula Card:**

1) **Symbol legend**
- `C`: coupling coefficient (strength/sign) multiplying the power-law tail
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `\ell`: spherical-harmonic / angular-momentum index
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `p`: power-law decay exponent (tail strength) in an effective term like $C/r^p$
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
large-$r$ / asymptotic region where power-law tails matter; Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Defines `L_p` in terms of `-\frac{d^2}{dr^2} + \frac{\ell(\ell+1)}{r^2} + \frac{C}{r^p},` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-027"></a>

### F027

- **Source location:** [5. Numerical Verification of the Spectral Threshold](#annex-sec-05-numerical-verification-of-the-spectral-threshold)

**Exact quote (verbatim LaTeX):**

```latex
\[
E[\phi]=
\frac{\int_{R}^{2R}\!\bigl(|\phi'(r)|^2+V_p(r)|\phi(r)|^2\bigr)r^2dr}
     {\int_{R}^{2R}\!|\phi(r)|^2r^2dr},
\qquad
\Delta E(R,p)=E[\phi]-E_{\mathrm{free}}[\phi],
\]
```

**Rendered:**

$$
E[\phi]=
\frac{\int_{R}^{2R}\!\bigl(|\phi'(r)|^2+V_p(r)|\phi(r)|^2\bigr)r^2dr}
     {\int_{R}^{2R}\!|\phi(r)|^2r^2dr},
\qquad
\Delta E(R,p)=E[\phi]-E_{\mathrm{free}}[\phi],
$$

**Formula Card:**

1) **Symbol legend**
- `R`: scalar curvature / curvature tensor (context-dependent)
- `\Delta`: Laplace-type operator (context-dependent)
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `p`: power-law decay exponent (tail strength) in an effective term like $C/r^p$
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
large-$r$ / asymptotic region where power-law tails matter

3) **Plain-English claim**
Defines `E[\phi]` in terms of `` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-028"></a>

### F028

- **Source location:** [7. Relation to Previous Work and Threshold Phenomena](#annex-sec-07-relation-to-previous-work-and-threshold-phenomena)

**Exact quote (verbatim LaTeX):**

```latex
\[
\Delta_A = -(\nabla_A)^*\nabla_A = -\nabla^*\nabla + {\rm ad}(F_A),
\]
```

**Rendered:**

$$
\Delta_A = -(\nabla_A)^*\nabla_A = -\nabla^*\nabla + {\rm ad}(F_A),
$$

**Formula Card:**

1) **Symbol legend**
- `\Delta`: Laplace-type operator (context-dependent)
- `\nabla`: Levi-Civita covariant derivative of the background metric
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Defines `\Delta_A` in terms of `-(\nabla_A)^*\nabla_A = -\nabla^*\nabla + {\rm ad}(F_A),` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-029"></a>

### F029

- **Source location:** [8. Conclusion](#annex-sec-08-conclusion)

**Exact quote (verbatim LaTeX):**

```latex
\[
\sigma_{\mathrm{ess}}(L)=[0,\infty).
\]
```

**Rendered:**

$$
\sigma_{\mathrm{ess}}(L)=[0,\infty).
$$

**Formula Card:**

1) **Symbol legend**
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `\sigma_{\mathrm{ess}}(L)`: essential spectrum of $L$
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
operator-theoretic statement about essential spectrum; Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Defines `\sigma_{\mathrm{ess}}(L)` in terms of `[0,\infty).` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-030"></a>

### F030

- **Source location:** [Appendix A. Gauge Correction and Elliptic Estimates](#annex-app-a-gauge-correction-and-elliptic-estimates)

**Exact quote (verbatim LaTeX):**

```latex
\[
\Delta_V X = \nabla^*\nabla X + {\rm Ric}(X)
\]
```

**Rendered:**

$$
\Delta_V X = \nabla^*\nabla X + {\rm Ric}(X)
$$

**Formula Card:**

1) **Symbol legend**
- `R`: scalar curvature / curvature tensor (context-dependent)
- `\Delta`: Laplace-type operator (context-dependent)
- `\nabla`: Levi-Civita covariant derivative of the background metric
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Defines `\Delta_V X` in terms of `\nabla^*\nabla X + {\rm Ric}(X)` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-031"></a>

### F031

- **Source location:** [Appendix A. Gauge Correction and Elliptic Estimates](#annex-app-a-gauge-correction-and-elliptic-estimates)

**Exact quote (verbatim LaTeX):**

```latex
\[
\Delta_V:H^2_\delta(\Sigma;T^*\Sigma)\to L^2_{\delta-2}(\Sigma;T^*\Sigma)
\]
```

**Rendered:**

$$
\Delta_V:H^2_\delta(\Sigma;T^*\Sigma)\to L^2_{\delta-2}(\Sigma;T^*\Sigma)
$$

**Formula Card:**

1) **Symbol legend**
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `\Delta`: Laplace-type operator (context-dependent)
- `d`: spatial dimension (as used in the dimensional scaling chapter)

2) **Regime and assumptions**
Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Displays an intermediate expression used in the argument (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-032"></a>

### F032

- **Source location:** [Appendix A. Gauge Correction and Elliptic Estimates](#annex-app-a-gauge-correction-and-elliptic-estimates)

**Exact quote (verbatim LaTeX):**

```latex
\[
\|\tilde h_n\|_{L^2}=1,\qquad
\tilde h_n\rightharpoonup0,\qquad
\|L\tilde h_n\|_{L^2}\to0.
\]
```

**Rendered:**

$$
\|\tilde h_n\|_{L^2}=1,\qquad
\tilde h_n\rightharpoonup0,\qquad
\|L\tilde h_n\|_{L^2}\to0.
$$

**Formula Card:**

1) **Symbol legend**
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `p`: power-law decay exponent (tail strength) in an effective term like $C/r^p$
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
large-$r$ / asymptotic region where power-law tails matter; Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Defines `\|\tilde h_n\|_{L^2}` in terms of `1,\qquad` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-033"></a>

### F033

- **Source location:** [Appendix B. Curvature Structure and the Schwarzschild Example](#annex-app-b-curvature-structure-and-the-schwarzschild-example)

**Exact quote (verbatim LaTeX):**

```latex
\[
ds^2=-\Bigl(\frac{1-\tfrac{M}{2r}}{1+\tfrac{M}{2r}}\Bigr)^2 dt^2
     +\Bigl(1+\frac{M}{2r}\Bigr)^4(dr^2+r^2d\omega^2).
\]
```

**Rendered:**

$$
ds^2=-\Bigl(\frac{1-\tfrac{M}{2r}}{1+\tfrac{M}{2r}}\Bigr)^2 dt^2
     +\Bigl(1+\frac{M}{2r}\Bigr)^4(dr^2+r^2d\omega^2).
$$

**Formula Card:**

1) **Symbol legend**
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
as stated in the surrounding text near this display

3) **Plain-English claim**
Defines `ds^2` in terms of `-\Bigl(\frac{1-\tfrac{M}{2r}}{1+\tfrac{M}{2r}}\Bigr)^2 dt^2` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-034"></a>

### F034

- **Source location:** [Appendix B. Curvature Structure and the Schwarzschild Example](#annex-app-b-curvature-structure-and-the-schwarzschild-example)

**Exact quote (verbatim LaTeX):**

```latex
\[
|{\rm Riem}(x)|\simeq C\,M\,r^{-3}\qquad(r\to\infty),
\]
```

**Rendered:**

$$
|{\rm Riem}(x)|\simeq C\,M\,r^{-3}\qquad(r\to\infty),
$$

**Formula Card:**

1) **Symbol legend**
- `C`: coupling coefficient (strength/sign) multiplying the power-law tail
- `R`: scalar curvature / curvature tensor (context-dependent)
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
large-$r$ / asymptotic region where power-law tails matter

3) **Plain-English claim**
States an asymptotic scaling relation.

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-035"></a>

### F035

- **Source location:** [Appendix B. Curvature Structure and the Schwarzschild Example](#annex-app-b-curvature-structure-and-the-schwarzschild-example)

**Exact quote (verbatim LaTeX):**

```latex
\[
Lh = \Delta_0 h - (C M) r^{-3} h + O(r^{-4})h,
\qquad C M > 0,
\]
```

**Rendered:**

$$
Lh = \Delta_0 h - (C M) r^{-3} h + O(r^{-4})h,
\qquad C M > 0,
$$

**Formula Card:**

1) **Symbol legend**
- `C`: coupling coefficient (strength/sign) multiplying the power-law tail
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `\Delta`: Laplace-type operator (context-dependent)
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
large-$r$ / asymptotic region where power-law tails matter; Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Defines `Lh` in terms of `\Delta_0 h - (C M) r^{-3} h + O(r^{-4})h,` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-036"></a>

### F036

- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)

**Exact quote (verbatim LaTeX):**

```latex
\[
\|u\|_{H^k_\delta}^2
   = \sum_{|\alpha|\le k}
     \int_\Sigma (1+r^2)^{\delta-|\alpha|}
     |\nabla^\alpha u|^2\,dV_g.
\]
```

**Rendered:**

$$
\|u\|_{H^k_\delta}^2
   = \sum_{|\alpha|\le k}
     \int_\Sigma (1+r^2)^{\delta-|\alpha|}
     |\nabla^\alpha u|^2\,dV_g.
$$

**Formula Card:**

1) **Symbol legend**
- `\nabla`: Levi-Civita covariant derivative of the background metric
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `p`: power-law decay exponent (tail strength) in an effective term like $C/r^p$
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
large-$r$ / asymptotic region where power-law tails matter; Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Displays an intermediate expression used in the argument (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-037"></a>

### F037

- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)

**Exact quote (verbatim LaTeX):**

```latex
\[
P:H^2_\delta\to L^2_{\delta-2}
\]
```

**Rendered:**

$$
P:H^2_\delta\to L^2_{\delta-2}
$$

**Formula Card:**

1) **Symbol legend**
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `d`: spatial dimension (as used in the dimensional scaling chapter)

2) **Regime and assumptions**
Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Displays an intermediate expression used in the argument (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-038"></a>

### F038

- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)

**Exact quote (verbatim LaTeX):**

```latex
\[
g_{ij} = \delta_{ij} + O(r^{-1}), 
\qquad 
\partial g_{ij} = O(r^{-2}), 
\qquad 
\partial^2 g_{ij} = O(r^{-3}),
\]
```

**Rendered:**

$$
g_{ij} = \delta_{ij} + O(r^{-1}), 
\qquad 
\partial g_{ij} = O(r^{-2}), 
\qquad 
\partial^2 g_{ij} = O(r^{-3}),
$$

**Formula Card:**

1) **Symbol legend**
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `p`: power-law decay exponent (tail strength) in an effective term like $C/r^p$
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
large-$r$ / asymptotic region where power-law tails matter

3) **Plain-English claim**
Defines `g_{ij}` in terms of `\delta_{ij} + O(r^{-1}),` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-039"></a>

### F039

- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)

**Exact quote (verbatim LaTeX):**

```latex
\[
L = \nabla^{*}\nabla + V_R,
\qquad
(V_R h)_{ij} = -R_{i}{}^{\ell}{}_{j}{}^{m}\,h_{\ell m}.
\]
```

**Rendered:**

$$
L = \nabla^{*}\nabla + V_R,
\qquad
(V_R h)_{ij} = -R_{i}{}^{\ell}{}_{j}{}^{m}\,h_{\ell m}.
$$

**Formula Card:**

1) **Symbol legend**
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `R`: scalar curvature / curvature tensor (context-dependent)
- `V_R`: curvature-induced potential term in the Lichnerowicz operator
- `\ell`: spherical-harmonic / angular-momentum index
- `\nabla`: Levi-Civita covariant derivative of the background metric
- `d`: spatial dimension (as used in the dimensional scaling chapter)

2) **Regime and assumptions**
Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Defines `L` in terms of `\nabla^{*}\nabla + V_R,` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-040"></a>

### F040

- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)

**Exact quote (verbatim LaTeX):**

```latex
\[
V_R : H^2_\delta(\Sigma;S^2T^*\Sigma) \longrightarrow L^2_{\delta-2}(\Sigma;S^2T^*\Sigma),
\]
```

**Rendered:**

$$
V_R : H^2_\delta(\Sigma;S^2T^*\Sigma) \longrightarrow L^2_{\delta-2}(\Sigma;S^2T^*\Sigma),
$$

**Formula Card:**

1) **Symbol legend**
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `R`: scalar curvature / curvature tensor (context-dependent)
- `V_R`: curvature-induced potential term in the Lichnerowicz operator
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Displays an intermediate expression used in the argument (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-041"></a>

### F041

- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)

**Exact quote (verbatim LaTeX):**

```latex
\[
\sigma_{\mathrm{ess}}(L)=\sigma_{\mathrm{ess}}(\nabla^{*}\nabla)=[0,\infty).
\]
```

**Rendered:**

$$
\sigma_{\mathrm{ess}}(L)=\sigma_{\mathrm{ess}}(\nabla^{*}\nabla)=[0,\infty).
$$

**Formula Card:**

1) **Symbol legend**
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `\nabla`: Levi-Civita covariant derivative of the background metric
- `\sigma_{\mathrm{ess}}(L)`: essential spectrum of $L$
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
operator-theoretic statement about essential spectrum; Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Defines `\sigma_{\mathrm{ess}}(L)` in terms of `\sigma_{\mathrm{ess}}(\nabla^{*}\nabla)=[0,\infty).` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-042"></a>

### F042

- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)

**Exact quote (verbatim LaTeX):**

```latex
\[
\|V_R h\|_{L^2_{\delta-2}}^2
=
\int_\Sigma \langle r\rangle^{2(\delta-2)}\,|V_R(x)h(x)|^2\,dV_g
\lesssim
\int_\Sigma \langle r\rangle^{2(\delta-2)-2p}|h(x)|^2\,dV_g.
\]
```

**Rendered:**

$$
\|V_R h\|_{L^2_{\delta-2}}^2
=
\int_\Sigma \langle r\rangle^{2(\delta-2)}\,|V_R(x)h(x)|^2\,dV_g
\lesssim
\int_\Sigma \langle r\rangle^{2(\delta-2)-2p}|h(x)|^2\,dV_g.
$$

**Formula Card:**

1) **Symbol legend**
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `R`: scalar curvature / curvature tensor (context-dependent)
- `V_R`: curvature-induced potential term in the Lichnerowicz operator
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `p`: power-law decay exponent (tail strength) in an effective term like $C/r^p$
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
large-$r$ / asymptotic region where power-law tails matter; Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Displays an intermediate expression used in the argument (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-043"></a>

### F043

- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)

**Exact quote (verbatim LaTeX):**

```latex
\[
V_R=\chi_R V_R+(1-\chi_R)V_R
=:V_R^{\mathrm{(comp)}}+V_R^{\mathrm{(tail)}}.
\]
```

**Rendered:**

$$
V_R=\chi_R V_R+(1-\chi_R)V_R
=:V_R^{\mathrm{(comp)}}+V_R^{\mathrm{(tail)}}.
$$

**Formula Card:**

1) **Symbol legend**
- `R`: scalar curvature / curvature tensor (context-dependent)
- `V_R`: curvature-induced potential term in the Lichnerowicz operator
- `p`: power-law decay exponent (tail strength) in an effective term like $C/r^p$
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
large-$r$ / asymptotic region where power-law tails matter

3) **Plain-English claim**
Defines `V_R` in terms of `\chi_R V_R+(1-\chi_R)V_R` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-044"></a>

### F044

- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)

**Exact quote (verbatim LaTeX):**

```latex
\[
\|V_R^{\mathrm{(tail)}}h\|_{L^2_{\delta-2}}^2
\lesssim
\int_{r>R}\langle r\rangle^{2(\delta-2)-2p}|h(x)|^2\,dV_g.
\]
```

**Rendered:**

$$
\|V_R^{\mathrm{(tail)}}h\|_{L^2_{\delta-2}}^2
\lesssim
\int_{r>R}\langle r\rangle^{2(\delta-2)-2p}|h(x)|^2\,dV_g.
$$

**Formula Card:**

1) **Symbol legend**
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `R`: scalar curvature / curvature tensor (context-dependent)
- `V_R`: curvature-induced potential term in the Lichnerowicz operator
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `p`: power-law decay exponent (tail strength) in an effective term like $C/r^p$
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
large-$r$ / asymptotic region where power-law tails matter; Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Displays an intermediate expression used in the argument (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-045"></a>

### F045

- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)

**Exact quote (verbatim LaTeX):**

```latex
\[
\sup_{\|h\|_{H^2_\delta}=1}
\|V_R^{\mathrm{(tail)}}h\|_{L^2_{\delta-2}}
\longrightarrow 0
\quad\text{as }R\to\infty.
\]
```

**Rendered:**

$$
\sup_{\|h\|_{H^2_\delta}=1}
\|V_R^{\mathrm{(tail)}}h\|_{L^2_{\delta-2}}
\longrightarrow 0
\quad\text{as }R\to\infty.
$$

**Formula Card:**

1) **Symbol legend**
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `R`: scalar curvature / curvature tensor (context-dependent)
- `V_R`: curvature-induced potential term in the Lichnerowicz operator
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `p`: power-law decay exponent (tail strength) in an effective term like $C/r^p$
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
large-$r$ / asymptotic region where power-law tails matter; Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Defines `\sup_{\|h\|_{H^2_\delta}` in terms of `1}` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-046"></a>

### F046

- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)

**Exact quote (verbatim LaTeX):**

```latex
\[
\sigma_{\mathrm{ess}}(L)=\sigma_{\mathrm{ess}}(\nabla^{*}\nabla)=[0,\infty).
\]
```

**Rendered:**

$$
\sigma_{\mathrm{ess}}(L)=\sigma_{\mathrm{ess}}(\nabla^{*}\nabla)=[0,\infty).
$$

**Formula Card:**

1) **Symbol legend**
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `\nabla`: Levi-Civita covariant derivative of the background metric
- `\sigma_{\mathrm{ess}}(L)`: essential spectrum of $L$
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
operator-theoretic statement about essential spectrum; Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Defines `\sigma_{\mathrm{ess}}(L)` in terms of `\sigma_{\mathrm{ess}}(\nabla^{*}\nabla)=[0,\infty).` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-047"></a>

### F047

- **Source location:** [Appendix D. Numerical Validation and Stability Tests](#annex-app-d-numerical-validation-and-stability-tests)

**Exact quote (verbatim LaTeX):**

```latex
\[
\begin{array}{ll}
\text{flat:} & \lambda_1 = 0.07386996,\quad \lambda_2 = 0.14713361, \\
p=2.5:       & \lambda_1 = 0.06505331,\quad \lambda_2 = 0.07023744, \\
p=3.0:       & \lambda_1 = 0.07114971,\quad \lambda_2 = 0.07276003, \\
p=3.5:       & \lambda_1 = 0.07298806,\quad \lambda_2 = 0.07351824.
\end{array}
\]
```

**Rendered:**

$$
\begin{array}{ll}
\text{flat:} & \lambda_1 = 0.07386996,\quad \lambda_2 = 0.14713361, \\
p=2.5:       & \lambda_1 = 0.06505331,\quad \lambda_2 = 0.07023744, \\
p=3.0:       & \lambda_1 = 0.07114971,\quad \lambda_2 = 0.07276003, \\
p=3.5:       & \lambda_1 = 0.07298806,\quad \lambda_2 = 0.07351824.
\end{array}
$$

**Formula Card:**

1) **Symbol legend**
- `\lambda`: eigenvalue (in the numerical/radial operator context)
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `p`: power-law decay exponent (tail strength) in an effective term like $C/r^p$
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
large-$r$ / asymptotic region where power-law tails matter

3) **Plain-English claim**
Displays an intermediate expression used in the argument (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-048"></a>

### F048

- **Source location:** [Appendix D. Numerical Validation and Stability Tests](#annex-app-d-numerical-validation-and-stability-tests)

**Exact quote (verbatim LaTeX):**

```latex
\[
\begin{array}{ll}
\text{flat:} & \lambda_1 = 0.07398399,\quad \lambda_2 = 0.14781594, \\
p=2.5:       & \lambda_1 = 0.06505635,\quad \lambda_2 = 0.07029545, \\
p=3.0:       & \lambda_1 = 0.07109678,\quad \lambda_2 = 0.07279109, \\
p=3.5:       & \lambda_1 = 0.07295142,\quad \lambda_2 = 0.07355686.
\end{array}
\]
```

**Rendered:**

$$
\begin{array}{ll}
\text{flat:} & \lambda_1 = 0.07398399,\quad \lambda_2 = 0.14781594, \\
p=2.5:       & \lambda_1 = 0.06505635,\quad \lambda_2 = 0.07029545, \\
p=3.0:       & \lambda_1 = 0.07109678,\quad \lambda_2 = 0.07279109, \\
p=3.5:       & \lambda_1 = 0.07295142,\quad \lambda_2 = 0.07355686.
\end{array}
$$

**Formula Card:**

1) **Symbol legend**
- `\lambda`: eigenvalue (in the numerical/radial operator context)
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `p`: power-law decay exponent (tail strength) in an effective term like $C/r^p$
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
large-$r$ / asymptotic region where power-law tails matter

3) **Plain-English claim**
Displays an intermediate expression used in the argument (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-049"></a>

### F049

- **Source location:** [Appendix D. Numerical Validation and Stability Tests](#annex-app-d-numerical-validation-and-stability-tests)

**Exact quote (verbatim LaTeX):**

```latex
\[
\lambda_1(p{=}2.5)
<
\lambda_1(p{=}3.0)
<
\lambda_1(p{=}3.5)
\simeq
\lambda_1(\text{flat})
\]
```

**Rendered:**

$$
\lambda_1(p{=}2.5)
<
\lambda_1(p{=}3.0)
<
\lambda_1(p{=}3.5)
\simeq
\lambda_1(\text{flat})
$$

**Formula Card:**

1) **Symbol legend**
- `\lambda`: eigenvalue (in the numerical/radial operator context)
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `p`: power-law decay exponent (tail strength) in an effective term like $C/r^p$

2) **Regime and assumptions**
as stated in the surrounding text near this display

3) **Plain-English claim**
Defines `\lambda_1(p{` in terms of `}2.5)` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-050"></a>

### F050

- **Source location:** [Appendix D. Numerical Validation and Stability Tests](#annex-app-d-numerical-validation-and-stability-tests)

**Exact quote (verbatim LaTeX):**

```latex
\[
\lambda_1(R_{\max}{=}6)=0.1965,\quad
\lambda_1(R_{\max}{=}10)=0.0711,\quad
\lambda_1(R_{\max}{=}20)=0.0180.
\]
```

**Rendered:**

$$
\lambda_1(R_{\max}{=}6)=0.1965,\quad
\lambda_1(R_{\max}{=}10)=0.0711,\quad
\lambda_1(R_{\max}{=}20)=0.0180.
$$

**Formula Card:**

1) **Symbol legend**
- `R`: scalar curvature / curvature tensor (context-dependent)
- `\lambda`: eigenvalue (in the numerical/radial operator context)
- `d`: spatial dimension (as used in the dimensional scaling chapter)

2) **Regime and assumptions**
as stated in the surrounding text near this display

3) **Plain-English claim**
Defines `\lambda_1(R_{\max}{` in terms of `}6)=0.1965,\quad` (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-051"></a>

### F051

- **Source location:** [Appendix E. Weyl Sequence Construction and Verification of the Critical Spectrum](#annex-app-e-weyl-sequence-construction-and-verification-of-the-critical-spectrum)

**Exact quote (verbatim LaTeX):**

```latex
\[
\|h_n\|_{L^2}^2\!\simeq\!A_n^2\!\!\int_{n/2}^{2n}\!r^{-2}r^2dr
\sim A_n^2 n.
\]
```

**Rendered:**

$$
\|h_n\|_{L^2}^2\!\simeq\!A_n^2\!\!\int_{n/2}^{2n}\!r^{-2}r^2dr
\sim A_n^2 n.
$$

**Formula Card:**

1) **Symbol legend**
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
States an asymptotic scaling relation.

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-052"></a>

### F052

- **Source location:** [Appendix E. Weyl Sequence Construction and Verification of the Critical Spectrum](#annex-app-e-weyl-sequence-construction-and-verification-of-the-critical-spectrum)

**Exact quote (verbatim LaTeX):**

```latex
\[
\nabla^*\nabla(\phi_n r^{-1}H)
= \phi_n\nabla^*\nabla(r^{-1}H)
 + 2\nabla\phi_n\!\cdot\!\nabla(r^{-1}H)
 + (\Delta\phi_n)r^{-1}H,
\]
```

**Rendered:**

$$
\nabla^*\nabla(\phi_n r^{-1}H)
= \phi_n\nabla^*\nabla(r^{-1}H)
 + 2\nabla\phi_n\!\cdot\!\nabla(r^{-1}H)
 + (\Delta\phi_n)r^{-1}H,
$$

**Formula Card:**

1) **Symbol legend**
- `\Delta`: Laplace-type operator (context-dependent)
- `\nabla`: Levi-Civita covariant derivative of the background metric
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `p`: power-law decay exponent (tail strength) in an effective term like $C/r^p$
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
large-$r$ / asymptotic region where power-law tails matter; Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Displays an intermediate expression used in the argument (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-053"></a>

### F053

- **Source location:** [Appendix E. Weyl Sequence Construction and Verification of the Critical Spectrum](#annex-app-e-weyl-sequence-construction-and-verification-of-the-critical-spectrum)

**Exact quote (verbatim LaTeX):**

```latex
\[
\|[\nabla^*\nabla,\phi_n]h_n\|_{L^2}
\lesssim n^{-2}.
\]
```

**Rendered:**

$$
\|[\nabla^*\nabla,\phi_n]h_n\|_{L^2}
\lesssim n^{-2}.
$$

**Formula Card:**

1) **Symbol legend**
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `\nabla`: Levi-Civita covariant derivative of the background metric
- `p`: power-law decay exponent (tail strength) in an effective term like $C/r^p$

2) **Regime and assumptions**
Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Displays an intermediate expression used in the argument (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-054"></a>

### F054

- **Source location:** [Appendix E. Weyl Sequence Construction and Verification of the Critical Spectrum](#annex-app-e-weyl-sequence-construction-and-verification-of-the-critical-spectrum)

**Exact quote (verbatim LaTeX):**

```latex
\[
\|V_R h_n\|_{L^2}^2
\lesssim A_n^2\!\int_{n/2}^{2n}\!r^{-6}r^2dr
\sim n^{-4},
\qquad
\|V_R h_n\|_{L^2}\lesssim n^{-2}.
\]
```

**Rendered:**

$$
\|V_R h_n\|_{L^2}^2
\lesssim A_n^2\!\int_{n/2}^{2n}\!r^{-6}r^2dr
\sim n^{-4},
\qquad
\|V_R h_n\|_{L^2}\lesssim n^{-2}.
$$

**Formula Card:**

1) **Symbol legend**
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `R`: scalar curvature / curvature tensor (context-dependent)
- `V_R`: curvature-induced potential term in the Lichnerowicz operator
- `d`: spatial dimension (as used in the dimensional scaling chapter)
- `r`: radial coordinate / distance to infinity

2) **Regime and assumptions**
Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Displays an intermediate expression used in the argument (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-055"></a>

### F055

- **Source location:** [Appendix E. Weyl Sequence Construction and Verification of the Critical Spectrum](#annex-app-e-weyl-sequence-construction-and-verification-of-the-critical-spectrum)

**Exact quote (verbatim LaTeX):**

```latex
\[
\|X_n\|_{H^2_\delta}\lesssim
\|\nabla\!\cdot h_n\|_{L^2}
\lesssim n^{-1},
\qquad -1<\delta<0.
\]
```

**Rendered:**

$$
\|X_n\|_{H^2_\delta}\lesssim
\|\nabla\!\cdot h_n\|_{L^2}
\lesssim n^{-1},
\qquad -1<\delta<0.
$$

**Formula Card:**

1) **Symbol legend**
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `\nabla`: Levi-Civita covariant derivative of the background metric
- `d`: spatial dimension (as used in the dimensional scaling chapter)

2) **Regime and assumptions**
Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
States an inequality / bound controlling the size of a quantity under the stated assumptions.

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-056"></a>

### F056

- **Source location:** [Appendix E. Weyl Sequence Construction and Verification of the Critical Spectrum](#annex-app-e-weyl-sequence-construction-and-verification-of-the-critical-spectrum)

**Exact quote (verbatim LaTeX):**

```latex
\[
\|L(\mathcal{L}_{X_n}g)\|_{L^2}\lesssim n^{-1},
\]
```

**Rendered:**

$$
\|L(\mathcal{L}_{X_n}g)\|_{L^2}\lesssim n^{-1},
$$

**Formula Card:**

1) **Symbol legend**
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)

2) **Regime and assumptions**
Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
States an inequality / bound controlling the size of a quantity under the stated assumptions.

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

<a id="if-formula-057"></a>

### F057

- **Source location:** [Appendix E. Weyl Sequence Construction and Verification of the Critical Spectrum](#annex-app-e-weyl-sequence-construction-and-verification-of-the-critical-spectrum)

**Exact quote (verbatim LaTeX):**

```latex
\[
\|L\tilde h_n\|_{L^2}\to0,
\qquad
\|\tilde h_n\|_{L^2}=1.
\]
```

**Rendered:**

$$
\|L\tilde h_n\|_{L^2}\to0,
\qquad
\|\tilde h_n\|_{L^2}=1.
$$

**Formula Card:**

1) **Symbol legend**
- `L`: spatial Lichnerowicz operator (Laplace-type operator on symmetric tensors)
- `d`: spatial dimension (as used in the dimensional scaling chapter)

2) **Regime and assumptions**
Laplace-type operator on an asymptotically flat manifold

3) **Plain-English claim**
Displays an intermediate expression used in the argument (interpretation given in the surrounding text).

4) **What would falsify it**
Not specified in sources; practical check: re-derive from the surrounding text and verify no boundary/decay terms were dropped.

5) **Misuse risks**
Applying outside the stated regime (wrong decay rate, wrong dimension, or without the relevant gauge/constraint conditions).

6) **So what**
This expression is part of the chain linking curvature decay to operator structure and infrared spectral behavior.

---

## Annex: original source


This annex contains the source material extracted from the provided `arXiv-2511.05345v1.tar.gz` bundle.



<a id="annex-toc"></a>

### Annex TOC


- [00README.json](#annex-readme)

- [main.tex — preamble + title block](#annex-preamble)

- [Abstract](#annex-abstract)

- [1. Introduction](#annex-sec-01-introduction)

- [2. Spectral Scaling and Structural Parallels Across Spin-1 and Spin-2 Fields](#annex-sec-02-spectral-scaling-and-structural-parallels-across-spin-1-and-spin-2-fields)

- [3. Infrared Spectrum and Marginal Modes](#annex-sec-03-infrared-spectrum-and-marginal-modes)

- [4. Dimensional Scaling and Spectral Phase Structure](#annex-sec-04-dimensional-scaling-and-spectral-phase-structure)

- [5. Numerical Verification of the Spectral Threshold](#annex-sec-05-numerical-verification-of-the-spectral-threshold)

- [6. Physical Interpretation and Implications](#annex-sec-06-physical-interpretation-and-implications)

- [7. Relation to Previous Work and Threshold Phenomena](#annex-sec-07-relation-to-previous-work-and-threshold-phenomena)

- [8. Conclusion](#annex-sec-08-conclusion)

- [Declarations](#annex-sec-09-declarations)

- [Appendix A. Gauge Correction and Elliptic Estimates](#annex-app-a-gauge-correction-and-elliptic-estimates)

- [Appendix B. Curvature Structure and the Schwarzschild Example](#annex-app-b-curvature-structure-and-the-schwarzschild-example)

- [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)

- [Appendix D. Numerical Validation and Stability Tests](#annex-app-d-numerical-validation-and-stability-tests)

- [Appendix E. Weyl Sequence Construction and Verification of the Critical Spectrum](#annex-app-e-weyl-sequence-construction-and-verification-of-the-critical-spectrum)


---


<a id="annex-readme"></a>

### 00README.json (verbatim)


```json
{
   "sources" : [
      {
         "usage" : "toplevel",
         "filename" : "main.tex"
      }
   ],
   "spec_version" : 1,
   "texlive_version" : "2025",
   "process" : {
      "compiler" : "pdflatex"
   }
}

```



---


<a id="annex-preamble"></a>

### main.tex — preamble + title block (verbatim)


```tex
\documentclass[12pt]{article}
\usepackage{amsmath, amssymb, amsthm}
\usepackage{mathrsfs}
\usepackage{cite}
\usepackage[hidelinks]{hyperref}
\usepackage{float}
\usepackage{longtable}
\usepackage{graphicx}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{assumption}{Assumption}
\newtheorem{definition}{Definition}
\newtheorem{remark}{Remark}
\newtheorem{proposition}{Proposition}
\newtheorem{corollary}[theorem]{Corollary}

\title{The $\boldsymbol{r^{-3}}$ Curvature Decay and the Infrared Structure of Linearized Gravity}
\author{Michael Wilson \\ \small University of Arkansas at Little Rock \\ \small Department of Physics and Astronomy \\ \small mkwilson3@ualr.edu}
\date{\today}

\begin{document}
\maketitle


```



---


<a id="annex-abstract"></a>

### main.tex — abstract (verbatim)


```tex
\begin{abstract}
We identify curvature decay $|{\rm Riem}|\!\sim\!r^{-3}$ as a sharp spectral threshold in linearized gravity on asymptotically flat manifolds.  
For faster decay, the spatial Lichnerowicz operator possesses a purely continuous spectrum $\sigma_{\mathrm{ess}}(L)=[0,\infty)$, corresponding to freely radiating tensor modes.  
At the inverse-cube rate, compactness fails and zero energy enters $\sigma_{\mathrm{ess}}(L)$, yielding marginally bound, finite-energy configurations that remain spatially extended.  
These static modes constitute the linear precursors of gravitational memory and soft-graviton phenomena, delineating the geometric boundary between dispersive and infrared behavior.  
A complementary numerical study of the radial model
\[
L_p=-\tfrac{d^2}{dr^2}+\tfrac{\ell(\ell+1)}{r^2}+\tfrac{C}{r^p}
\]
confirms the analytic scaling law, locating the same transition at $p=3$.  
The eigenvalue trends approach the flat-space limit continuously for $p>3$ and strengthen progressively for $p<3$, demonstrating a smooth yet sharp spectral transition rather than a discrete confinement regime. 
The result parallels the critical threshold of the non-Abelian covariant Laplacian~\cite{Wilson2025}, indicating a common $r^{-3}$ scaling that governs the infrared structure of gauge and gravitational fields.
\end{abstract}


```



---


<a id="annex-sec-01-introduction"></a>

### main.tex — 1. Introduction (verbatim)


```tex
\section{Introduction}

The infrared structure of gravity governs the persistent correlations that remain after gravitational radiation has passed. 
Phenomena such as gravitational memory, power-law tails, and soft graviton modes all originate from the long-range behavior of the field at large distances and late times.  
While these effects are well understood in asymptotic frameworks, particularly at null infinity, where BMS symmetries organize the radiative data, the corresponding spatial mechanism on a Cauchy slice is less clearly established. 
In particular, it has not been resolved how the decay of curvature on an initial hypersurface determines the presence or absence of infrared correlations. 
This work identifies a precise spectral criterion that governs this transition.

We study the spatial Lichnerowicz operator
\begin{equation}
L = \nabla^{*}\nabla + V_R, 
\qquad (V_R h)_{ij} = -R^{\;\ell}{}_{i j m}\, h_{\ell}{}^{m},
\label{eq:Lichnerowicz}
\end{equation}
acting on symmetric, trace-free tensor fields $h_{ij}$ on an asymptotically flat three-manifold $(\Sigma,g)$. 
The operator~\eqref{eq:Lichnerowicz} governs stationary, harmonic-gauge perturbations of a vacuum background and determines whether small tensor excitations are radiative or spatially correlated.  
Its essential spectrum $\sigma_{\mathrm{ess}}(L)$ distinguishes these regimes: a purely continuous spectrum $[0,\infty)$ corresponds to freely propagating modes, whereas inclusion of zero in $\sigma_{\mathrm{ess}}(L)$ signals marginally bound, long-range configurations.

We show that curvature decay $|{\rm Riem}|\!\sim\!r^{-3}$ marks the sharp boundary between these behaviors.  
For faster decay, $V_R$ is a compact perturbation of the flat tensor Laplacian $\Delta_T=\nabla^*\nabla$, giving $\sigma_{\mathrm{ess}}(L)=[0,\infty)$. 
At the inverse-cube rate, compactness fails: curvature and dispersion balance exactly, allowing zero energy to enter the essential spectrum. 
In this marginal regime, $L$ admits a normalized, divergence-free Weyl sequence with $\|Lh_n\|\!\to\!0$, describing extended, finite-energy tensor modes at zero frequency.

A complementary numerical analysis of the radial model
\[
L_p = -\frac{d^2}{dr^2} + \frac{\ell(\ell+1)}{r^2} + \frac{C}{r^p},
\]
confirms this transition.  
The lowest eigenvalue $\lambda_1$ decreases smoothly as the decay exponent $p$ is reduced, approaching the continuum threshold near $p=3$ without developing discrete bound states.  
This behavior demonstrates a continuous spectral crossover rather than a discrete confinement phase, while preserving the analytic identification of $p=3$ as the exact transition point.

Physically, the marginally bound tensor modes identified here represent static precursors of gravitational memory and soft-graviton phenomena.  
The same inverse-cube scaling governs the non-Abelian covariant Laplacian in gauge theory~\cite{Wilson2025}, where $|F_A|\!\sim\!r^{-3}$ separates radiative from infrared-sensitive behavior.  
This parallel suggests a common spectral mechanism linking spin-1 and spin-2 fields, rooted in the dimensional scaling of curvature in three spatial dimensions.

The paper is organized as follows.  
Section~\ref{sec:spectral_universality} introduces the analytic framework and the spectral universality principle linking gauge and gravitational fields.  
Section~\ref{sec:critical_decay} establishes the inverse-cube decay as the critical regime and proves that zero enters the essential spectrum of the Lichnerowicz operator.  
Section~\ref{sec:dimensional_scaling} generalizes the argument to arbitrary dimension and derives the scaling law $p_{\mathrm{crit}}=d$.  
Section~\ref{sec:numerics} presents numerical verification of the spectral threshold through Rayleigh-quotient scaling and three-dimensional eigenvalue analysis, supported by convergence and stability checks.  
Section~\ref{sec:physical_interpretation} interprets the transition in physical terms, relating it to gravitational memory, late-time tails, and asymptotic symmetries.  
Section~\ref{sec:relation_previous} situates the result within prior work in spectral geometry and gauge theory, and Section~\ref{sec:conclusion} summarizes the main findings. 
Detailed mathematical proofs, extended derivations, and numerical validation are
provided in the Supplementary Material (Appendix A-E). There we include the harmonic-gauge correction construction, the Schwarzschild example, the analysis of weighted Sobolev spaces, and convergence and stability tests supporting the numerical results of Section~\ref{sec:numerics}.


```



---


<a id="annex-sec-02-spectral-scaling-and-structural-parallels-across-spin-1-and-spin-2-fields"></a>

### main.tex — 2. Spectral Scaling and Structural Parallels Across Spin-1 and Spin-2 Fields (verbatim)


```tex
\section{Spectral Scaling and Structural Parallels Across Spin-1 and Spin-2 Fields}
\label{sec:spectral_universality}

This section develops the analytic framework for the spectral analysis of the linearized gravitational field on asymptotically flat spatial slices and highlights its structural similarity to the spin-1 gauge field.  
In both cases, the governing operator is of Laplace type with a curvature-induced potential whose decay controls infrared behavior.  
Dimensional considerations reveal a shared critical decay rate at which the potential ceases to be short-range, a geometric correspondence rather than a full dynamical equivalence between the two theories.

\subsection{Geometric setup and harmonic gauge}

Let $(\Sigma,g)$ be a smooth, oriented, three-dimensional Riemannian manifold representing a time-symmetric slice of a vacuum spacetime $(M,g_{\mu\nu})$ satisfying ${\rm Ric}(g_{\mu\nu})=0$.  
On the asymptotic end, choose coordinates identifying $\Sigma\setminus K$ with $\mathbb{R}^3\setminus B_R(0)$ such that
\[
g_{ij}=\delta_{ij}+a_{ij},\qquad 
a_{ij}=O(r^{-1}),\quad 
\partial_k a_{ij}=O(r^{-2}),\quad 
\partial_\ell\partial_k a_{ij}=O(r^{-3}),
\]
where $r=|x|$ and $\langle r\rangle=(1+r^2)^{1/2}$.  
These conditions imply $\Gamma^k_{ij}=O(r^{-2})$, $|{\rm Riem}|=O(r^{-3})$, and $\nabla{\rm Riem}=O(r^{-4})$.

Expanding the Einstein-Hilbert action
\begin{equation}
S_{\text{EH}}=\frac{1}{16\pi G}\int R\sqrt{-g}\,d^4x
\label{eq:EH_action}
\end{equation}
to quadratic order in a perturbation $g_{\mu\nu}\mapsto g_{\mu\nu}+\hbar^{1/2}h_{\mu\nu}$ yields the spatial quadratic form \cite{BirrellDavies,WaldQFT}
\[
S^{(2)}[h]=\tfrac{1}{2}\!\int_\Sigma h^{ij}L_{ij}{}^{kl}h_{kl}\sqrt{g}\,d^3x,
\]
where 
\begin{equation}
Lh = \nabla^*\nabla h + V_R h, 
\qquad (V_Rh)_{ij}=-R_i{}^\ell{}_j{}^m h_{\ell m}.
\label{eq:Lichnerowicz_def}
\end{equation}
The operator $L$ acts on symmetric trace-free tensors and represents the spatial part of the quadratic graviton operator in harmonic gauge.  
Its spectral properties determine whether small tensor excitations remain radiative or acquire long-range correlations.

\subsection{Shared scaling structure}

An analogous Laplace-type operator appears for spin-1 fields.  
For a Yang-Mills connection $A$, the covariant Laplacian on the adjoint bundle is
\begin{equation}
\Delta_A = -(\nabla_A)^*\nabla_A = -\nabla^*\nabla + {\rm ad}(F_A),
\label{eq:YMlap}
\end{equation}
where ${\rm ad}(F_A)$ denotes the adjoint action of the curvature.  
Both \eqref{eq:Lichnerowicz_def} and \eqref{eq:YMlap} therefore consist of a Laplace term plus a curvature potential that decays with distance.  
In the scalar Schrödinger case, potentials faster than $r^{-3}$ yield compact perturbations of $-\Delta$, while the inverse-cube rate marks the onset of threshold phenomena.  
The same scaling governs curvature-coupled Laplace-type operators for spin-1 and spin-2 fields: in three spatial dimensions, curvature terms $|F_A|$ and $|{\rm Riem}|$ become marginal when $p=3$, the precise rate at which the potential ceases to be compact.\footnote{The analogous threshold for the non-Abelian Laplacian was derived in Ref.~\cite{Wilson2025}.}

\subsection{Weighted Sobolev spaces and integration identity}

For $\delta\in\mathbb{R}$, define
\[
H^k_\delta(\Sigma;S^2T^*\Sigma)
=\Bigl\{h\in H^k_{\mathrm{loc}}(\Sigma):\|h\|_{H^k_\delta}<\infty\Bigr\},\qquad
\|h\|_{H^k_\delta}^2=\sum_{j=0}^k\!\!\int_\Sigma\langle r\rangle^{2(\delta-j)}|\nabla^jh|_g^2\,dV_g.
\]
Integration by parts using the asymptotic decay of $g$ gives
\begin{equation}
\langle Lh,k\rangle_{L^2}
=\langle\nabla h,\nabla k\rangle_{L^2}
+\langle V_Rh,k\rangle_{L^2},
\label{eq:int_identity}
\end{equation}
valid for compactly supported smooth tensors $h,k$, showing that $L$ is symmetric on $C_c^\infty(\Sigma;S^2T^*\Sigma)$.

\subsection{Self-adjoint realization and spectral framework}

Weighted elliptic estimates imply
\begin{equation}
L:H^2_\delta(\Sigma;S^2T^*\Sigma)\longrightarrow L^2_{\delta-2}(\Sigma;S^2T^*\Sigma)
\label{eq:L_mapping2}
\end{equation}
is bounded for $-1<\delta<0$.  
Essential self-adjointness of $\nabla^*\nabla$ on complete Riemannian manifolds follows from Chernoff’s theorem, and $V_R$ is symmetric and relatively bounded since $|V_R(x)|=O(r^{-p})$.  
By the Kato-Rellich theorem, $L=\nabla^*\nabla+V_R$ is self-adjoint on the same domain as $\nabla^*\nabla$.  

By Lemma~C.4 \footnote{See supplementary material. Compactness of the curvature potential for $p>3$.}, if $|{\rm Riem}(x)| \le C\langle r\rangle^{-p}$ with $p>3$, 
then $V_R$ is compact as a map $H^2_\delta \to L^2_{\delta-2}$ for $-1<\delta<0$,
and therefore $L = \nabla^{*}\nabla + V_R$ is a compact perturbation of $\nabla^{*}\nabla$.
Weyl’s theorem then implies that
\begin{equation}
\sigma_{\mathrm{ess}}(L)=[0,\infty).
\label{eq:ess_flat}
\end{equation}
In particular, sufficiently rapid curvature decay leaves the essential spectrum of the
linearized gravitational field identical to that of flat space, so all finite-energy
tensor excitations are asymptotically radiative.

\subsection{Critical scaling and spectral threshold}

To identify the limit of compactness, project $Lh=0$ onto a spherical harmonic mode.  
The Laplacian contributes an $r^{-2}$ angular term, while a curvature potential $V_R\!\sim\!r^{-p}$ competes with it when $p=3$.  
Thus
\begin{equation}
p_{\mathrm{crit}}=3,
\label{eq:pcrit}
\end{equation}
which marks the transition between short and long-range geometric potentials.  
For $p>3$, curvature effects are subdominant and the spectrum remains stable; at $p=3$, curvature and kinetic terms balance, producing marginally bound tensor modes and a continuous spectrum extending to zero.  

This inverse-cube decay rate encapsulates the shared scaling property of Laplace-type operators for spin-1 and spin-2 fields in three dimensions—a geometric origin of infrared sensitivity common to gauge and gravitational settings.
                        

```



---


<a id="annex-sec-03-infrared-spectrum-and-marginal-modes"></a>

### main.tex — 3. Infrared Spectrum and Marginal Modes (verbatim)


```tex
\section{Infrared Spectrum and Marginal Modes}
\label{sec:critical_decay}

When the background curvature decays as $r^{-3}$, the Lichnerowicz operator reaches the scaling threshold identified in Section~\ref{sec:spectral_universality}.  
At this rate, curvature and kinetic terms balance asymptotically, producing marginally bound tensor modes whose energy approaches zero.  
In operator terms, $0$ enters the essential spectrum of $L$.  
This section establishes that result rigorously and interprets it within linearized gravity.

\subsection{Fredholm and gauge framework}

Let $(\Sigma,g)$ be a smooth, asymptotically flat three-manifold with a single end diffeomorphic to $\mathbb{R}^3$ outside a compact set, and assume $H^1_{\mathrm{dR}}(\Sigma)=0$.  
The vector Laplacian
\begin{equation}
\Delta_V=\nabla^*\nabla+\mathrm{Ric},
\label{eq:vectorlaplacian}
\end{equation}
acts on one-forms and enforces the harmonic gauge constraint.  
Standard elliptic theory gives:

\begin{lemma}[Fredholm property]\label{lem:fredholm}
For $-1<\delta<0$, the operator
\begin{equation}
\Delta_V: H^2_\delta(\Sigma;T^*\Sigma)\longrightarrow L^2_{\delta-2}(\Sigma;T^*\Sigma)
\label{eq:vectorfredholm}
\end{equation}
is Fredholm with bounded inverse.  
Hence, for each $f\in L^2_{\delta-2}$ there exists a unique $X\in H^2_\delta$ satisfying $\Delta_V X=f$ and $\|X\|_{H^2_\delta}\le C\|f\|_{L^2_{\delta-2}}$.
\end{lemma}

\begin{proof}
This is the weighted elliptic isomorphism theorem of Lockhart and McOwen~\cite{Lockhart1985}.  
Injectivity follows from $H^1_{\mathrm{dR}}(\Sigma)=0$, and surjectivity from asymptotic flatness.
\end{proof}

Under these hypotheses, $\Delta_V$ defines a gauge correction that enforces $\nabla^j h_{ij}=0$.  
All subsequent constructions assume this analytic and topological framework.

\subsection{Weyl sequence at the critical decay}
\label{subsec:weylsequence}

Assume the metric satisfies
\begin{equation}
g_{ij}=\delta_{ij}+O(r^{-1}),\qquad 
\partial g_{ij}=O(r^{-2}),\qquad 
\partial^2 g_{ij}=O(r^{-3}),
\label{eq:metricdecay}
\end{equation}
and that the curvature obeys $|{\rm Riem}(x)|\simeq C\,r^{-3}$ as $r\to\infty$.
On $L^2(\Sigma;S^2T^*\Sigma)$, consider the spatial Lichnerowicz operator
\begin{equation}
Lh = \nabla^*\nabla h + V_R h, 
\qquad (V_R h)_{ij}=-R_{i\ \ j m}^{\ \ell}h_\ell^{\ m},
\label{eq:lichcritical}
\end{equation}
with domain $H^2_\delta(\Sigma;S^2T^*\Sigma)$, $-1<\delta<0$.

\begin{lemma}[Approximate zero modes]\label{lem:weylsequence}
Let $H_{ij}(\omega)$ be a symmetric, trace-free, divergence-free tensor harmonic
on $S^2$, and define
\begin{equation}
h_n(r,\omega)=A_n\,\phi_n(r)\,r^{-1}H_{ij}(\omega),
\label{eq:weylansatz}
\end{equation}
where $\phi_n$ is a smooth cutoff equal to $1$ on $[n,3n/2]$ and vanishing
outside $[n/2,2n]$.  
The normalization $\|h_n\|_{L^2}=1$ gives $A_n\simeq n^{-1/2}$.
After divergence correction by a vector field $X_n$ satisfying
$\Delta_V X_n=\nabla\!\cdot h_n$ and setting
$\tilde h_n=h_n-\mathcal{L}_{X_n}g$, one obtains
\[
\|\tilde h_n\|_{L^2}=1,\qquad
\nabla^j\tilde h_{n,ij}=0,\qquad
\|L\tilde h_n\|_{L^2}\to0.
\]
\end{lemma}

\begin{proof}
The tensor $h_n$ is supported on the annulus $\{n/2<r<2n\}$ and normalized so
that $\|h_n\|_{L^2}=1$.  
Derivatives of the cutoff introduce factors of $n^{-1}$, yielding
$\|\nabla\!\cdot h_n\|_{L^2}\lesssim n^{-1}$ and
$\|Lh_n\|_{L^2}\lesssim n^{-2}$.  
The gauge correction $X_n$ obeys
$\|X_n\|_{H^2_\delta}\lesssim n^{-1}$ and therefore
$\|\mathcal{L}_{X_n}g\|_{L^2}\lesssim n^{-1}$.  
Consequently, $\tilde h_n$ is divergence-free and satisfies the bounds above.
A complete derivation, including control of curvature remainders,
commutators $[\nabla^*\nabla,\phi_n]$, and explicit estimates on
$L(\mathcal{L}_{X_n}g)$, is provided in the Supplementary Material.
\end{proof}

\begin{theorem}[Onset of the infrared continuum]\label{thm:criticalspectrum}
Let $(\Sigma,g)$ satisfy~\eqref{eq:metricdecay} with
$|{\rm Riem}(x)|\simeq C\,r^{-3}$ as $r\to\infty$.  
Then the Lichnerowicz operator $L=\nabla^*\nabla+V_R$ is self-adjoint on
$L^2(\Sigma;S^2T^*\Sigma)$ and satisfies
\begin{equation}
[0,\infty)\subset\sigma_{\mathrm{ess}}(L),\qquad 0\in\sigma_{\mathrm{ess}}(L).
\label{eq:criticalspectrum}
\end{equation}
Hence the inverse-cube decay marks the precise boundary between
spectrally transparent geometries and those supporting marginally correlated
tensor modes.
A full quantitative proof using Weyl’s criterion appears in the
Supplementary Material.
\end{theorem}

\subsection{Physical interpretation}

The inclusion $0\in\sigma_{\mathrm{ess}}(L)$ signifies that the spatial Lichnerowicz operator admits extended tensor modes with vanishing energy yet finite $L^2$ norm.  
In the full time-dependent linearized Einstein equation,
\[
\Box_g h_{\mu\nu} + 2R_{\mu\ \nu}^{\ \rho\ \sigma}h_{\rho\sigma}=0,
\]
these correspond to the $\omega\!\to\!0$ limit of radiative solutions.  

This result does not imply confinement or instability but reveals a continuous infrared sector: correlations decay slowly, and certain low-frequency components of gravitational radiation remain weakly coupled at large distances.  
The phenomenon parallels the infrared sensitivity of gauge fields at the same scaling threshold. 
In Yang-Mills theory, slow curvature decay localizes color flux; in gravity, it generates persistent correlations and memory effects.  
Thus the critical decay $|{\rm Riem}|\!\sim\!r^{-3}$ delineates the geometric transition from purely radiative behavior to an infrared-sensitive, marginally bound regime, a structural, though not dynamical, analogue of the non-Abelian case.


```



---


<a id="annex-sec-04-dimensional-scaling-and-spectral-phase-structure"></a>

### main.tex — 4. Dimensional Scaling and Spectral Phase Structure (verbatim)


```tex
\section{Dimensional Scaling and Spectral Phase Structure}
\label{sec:dimensional_scaling}

The preceding analysis showed that when the curvature of an asymptotically flat three-manifold decays as $|{\rm Riem}|\!\sim\!r^{-3}$, the Lichnerowicz operator $L=\nabla^*\nabla+V_R$ acquires a continuous spectrum extending to zero. This identifies the boundary between spectrally transparent geometries and those supporting marginally correlated tensor modes. 
Here we generalize that result by examining how the critical exponent depends on spatial dimension and by situating the inverse-cube decay within a broader scaling framework.  
The infrared transition derived in three dimensions thereby appears as one point on a dimensional phase diagram governing the long-range behavior of curvature-coupled Laplace-type operators.

\subsection{Dimensional analysis of curvature potentials}

Let $\Sigma_d$ be an asymptotically flat Riemannian manifold of dimension $d\ge2$, and let $V(r)\!\sim\!r^{-p}$ denote a curvature-induced potential acting on tensor fields through
\[
L_d=-\nabla^2 + V(r),
\]
where $\nabla^2$ is the Laplace-Beltrami operator on $\Sigma_d$.  
The potential is \emph{short-range} if it defines a compact perturbation of the Laplacian, and \emph{long-range} otherwise. A scaling argument identifies the decay rate separating these regimes.

\begin{proposition}[Dimensional criterion for the critical decay rate]
Let $\Delta$ denote the Laplace-Beltrami operator on a $d$-dimensional asymptotically flat manifold, and let $V(r)\sim r^{-p}$ be a curvature-induced potential.  
Then $V$ is a compact perturbation of $\Delta$ if and only if $p>d$.  
The equality $p=d$ marks the threshold between short- and long-range behavior.
\end{proposition}

\begin{proof}
The naive condition obtained by requiring $\int r^{d-1}|V|^2\,dr<\infty$
tests whether $V$ defines a Hilbert-Schmidt perturbation of $\Delta$,
which is much stronger than compactness.  
On unweighted spaces
$H^2(\mathbb{R}^d)\!\to\!L^2(\mathbb{R}^d)$,
any potential $V(x)\!\to\!0$ at infinity already yields a compact
multiplication operator by the standard Rellich lemma,
so the threshold $p>d$ cannot be inferred from that estimate alone.

In the present setting, however, $L_d$ acts between weighted Sobolev spaces
$H^2_\delta(\Sigma_d)\!\to\!L^2_{\delta-2}(\Sigma_d)$ with $-1<\delta<0$, appropriate to asymptotically flat ends. Compactness can fail at infinity if the decay of
$V(r)\!\sim\!r^{-p}$ is too slow to suppress contributions from large volumes.
For transverse-traceless tensor modes, which behave asymptotically as
$h(r)\!\sim\!r^{-(d-2)/2}$ due to the asymptotically flat falloff conditions,
the weighted $L^2_{\delta-2}$ norm of $Vh$ on $\{r>R\}$ scales as
\[
\int_R^\infty r^{\,2(\delta-2)}\,|V(r)|^2\,|h(r)|^2\,r^{\,d-1}dr
\;\sim\;
\int_R^\infty r^{\,d-5+2\delta-2p}\,dr.
\]
Because $-1<\delta<0$, this integral diverges when $p\!\le\!d$,
showing that for $p\!\le\!d$ the curvature tail continues to couple to
asymptotically flat tensor modes at arbitrarily large radius.
For $p>d$, the contribution from the asymptotic region vanishes and
multiplication by $V$ becomes compact
$H^2_\delta\!\to\!L^2_{\delta-2}$
by the weighted Rellich lemma~\cite{Bartnik1986,Lockhart1985}.
Hence $p=d$ marks the boundary between compact and noncompact behavior.
\end{proof}

\begin{remark}[On sharpness]
The borderline $p=d$ is therefore not merely sufficient but sharp:
for $p\!\le\!d$ one can construct a normalized, transverse-traceless
Weyl sequence supported at large $r$ with $\|L_d h_n\|\!\to\!0$,
placing $0$ in $\sigma_{\mathrm{ess}}(L_d)$.
\end{remark}

This reproduces the inverse-cube decay in three dimensions and extends it to arbitrary $d$.  
The resulting scaling law,
\begin{equation}
p_{\mathrm{crit}}=d,
\label{eq:pcrit_general}
\end{equation}
expresses a dimensional balance between kinetic dispersion $\nabla^2\!\sim\!r^{-2}$ and curvature coupling $V(r)\!\sim\!r^{-p}$.  
When $p>d$, curvature effects are integrable and the spectrum remains purely continuous; when $p\le d$, the potential becomes marginal or long-range, introducing infrared correlations.

\subsection{Interpretation and structural implications}

Equation~\eqref{eq:pcrit_general} has a clear geometric meaning.  
In any spatial dimension, the decay rate $r^{-d}$ represents the marginal case where background curvature fails to decay fast enough to ensure spectral transparency.  
For both spin-1 and spin-2 fields, whose quadratic operators share the same Laplace-type structure, this scaling marks the onset of infrared sensitivity.  
The equality $p_{\mathrm{crit}}=3$ in three dimensions is thus one instance of a general relation between dimensionality and the asymptotic behavior of curvature-coupled Laplace-type operators.

Although this correspondence arises from dimensional rather than dynamical analysis, it provides a coherent geometric framework for comparing gauge and gravitational fields across dimensions. 
It also supplies a natural language for describing the transition between dispersive and marginally bound regimes, the spectral phase structure of curvature-coupled Laplace operators.


```



---


<a id="annex-sec-05-numerical-verification-of-the-spectral-threshold"></a>

### main.tex — 5. Numerical Verification of the Spectral Threshold (verbatim)


```tex
\section{Numerical Verification of the Spectral Threshold}
\label{sec:numerics}

The analytic results of Sections~\ref{sec:spectral_universality}–\ref{sec:dimensional_scaling}
predict that when the background curvature decays faster than $r^{-3}$, 
the spectrum of the spatial Lichnerowicz operator remains purely continuous,
while the inverse–cube decay marks the onset of marginal long–range coupling. This section provides numerical evidence for that threshold
through two complementary diagnostics:
(i) a Rayleigh–quotient scaling analysis of an asymptotic radial model, and 
(ii) a computation of the low–lying spectrum of the full three–dimensional
discretized tensor operator.

\subsection{Numerical setup and nondimensionalization}

The tensor operator $L=\nabla^{*}\nabla+V_R$ is discretized
on a uniform Cartesian grid $\Omega=[-R_{\max},R_{\max}]^3$ 
with spacing $h=\Delta x/L_0$ after nondimensionalizing by a fixed
asymptotic length scale $L_0$.
Centered finite differences approximate $\nabla$ and $\nabla^{*}\nabla$.
Dirichlet boundaries $h|_{\partial\Omega}=0$ define a finite–volume eigenproblem; 
the approach to the continuum spectrum is monitored by extrapolation in $R_{\max}$.

To enforce the transverse–traceless constraint we use the penalty functional
\begin{equation}
\mathcal{R}_{\eta,\zeta}[h]
=
\frac{\langle h,Lh\rangle
+\eta\,\|\nabla\!\cdot h\|_{L^2(\Omega)}^2
+\zeta\,\|\mathrm{tr}\,h\|_{L^2(\Omega)}^2}
{\|h\|_{L^2(\Omega)}^2},
\label{eq:penalty}
\end{equation}
whose stationary points satisfy
\(
(K+\eta D^\top D+\zeta T^\top T)u
=\lambda M u,
\)
where $K$ and $M$ are the stiffness and mass matrices
and $D$, $T$ the discrete divergence and trace operators.
Varying the penalties $\eta,\zeta$ by factors of $2$–$4$
changes $\lambda_1$ by less than $10^{-3}$,
indicating that the lowest modes lie in the TT subspace to numerical accuracy.
An explicit TT projection check is reported in the Supplementary Material~(Sec.~S1).

Representative nondimensional parameters are 
$h\in\{1.0,0.75,0.5\}$,
$R_{\max}\in\{6,10,14,18,20\}$,
and $C=-1$.
Each grid contains $N^3$ points 
($N=21$–$41$, up to $3.6\times10^5$ degrees of freedom).
All runs use double precision and converge within relative error $10^{-5}$.

Dirichlet boundaries discretize the near–threshold continuum into
box modes with $\lambda_1(R_{\max})\propto R_{\max}^{-2}$;
the observed scaling and small residuals confirm that
the computed eigenvalues track the physical continuum edge
rather than artificial confinement.

\begin{definition}[Numerical operators]
\label{def:numericaloperator}
The asymptotic radial model operator is
\[
L_p = -\frac{d^2}{dr^2} + \frac{\ell(\ell+1)}{r^2} + \frac{C}{r^p},
\]
representing a single angular channel of the tensor operator.
The three–dimensional discretized $L=\nabla^{*}\nabla+V_R$
includes the full curvature coupling and constraint enforcement.
\end{definition}

\begin{remark}[Radial correspondence]
Projecting $L$ onto tensor harmonics of order $\ell$
yields an effective potential $V_{\mathrm{eff}}(r)=\ell(\ell+1)/r^2+C/r^p+O(r^{-p-1})$,
so the radial model reproduces the asymptotic channel structure of the full operator.
For the Schwarzschild tail $R_{i}{}^{k}{}_{j}{}^{\ell}h_{k\ell}$, one has $C<0$,
corresponding to an attractive potential in the far field.
\end{remark}

\subsection{Rayleigh–quotient scaling test}

The Rayleigh–quotient method directly probes the predicted scaling 
$\Delta E(R,p)\!\sim\!R^{-(p-2)}$, expressing the competition between
the Laplacian and the curvature potential.
For
\(
V_p(r)=\ell(\ell+1)/r^2+C/r^p,
\)
define
\[
E[\phi]=
\frac{\int_{R}^{2R}\!\bigl(|\phi'(r)|^2+V_p(r)|\phi(r)|^2\bigr)r^2dr}
     {\int_{R}^{2R}\!|\phi(r)|^2r^2dr},
\qquad
\Delta E(R,p)=E[\phi]-E_{\mathrm{free}}[\phi],
\]
using a normalized bump function $\phi(r)$ on $[R,2R]$.
The flat case $C=0$ defines $E_{\mathrm{free}}$.
Table~\ref{tab:rayleigh} lists the results; the fitted slopes 
$\alpha(p)\approx-(p-2)$ confirm the analytic scaling.

\begin{longtable}{r r r r r}
\caption{Rayleigh-quotient energy shift $\Delta E(R,p)$ for $C=-1$.
The power-law scaling with $R$ follows $\Delta E\!\sim\!R^{-(p-2)}$,
confirming that p=3 behaves as the marginal case separating decaying from saturating behavior.}
\label{tab:rayleigh} \\
\hline
$p$ & $R$ & $\Delta E(R,p)$ & $E_{\text{full}}$ & $E_{\text{free}}$ \\
\hline
\endfirsthead
\caption{Rayleigh-quotient energy shift $\Delta E(R,p)$ for $C=-1$ (continued).} \\
\hline
$p$ & $R$ & $\Delta E(R,p)$ & $E_{\text{full}}$ & $E_{\text{free}}$ \\
\hline
\endhead
\hline
\multicolumn{5}{r}{\textit{Continued on next page}} \\
\hline
\endfoot
\hline
\endlastfoot
% (table entries unchanged)
2.00 & 10  & $-2.85\times10^{-2}$ & $1.50\times10^{-1}$ & $1.54\times10^{-1}$ \\
     & 20  & $-2.85\times10^{-2}$ & $3.74\times10^{-2}$ & $3.85\times10^{-2}$ \\
     & 40  & $-2.85\times10^{-2}$ & $9.36\times10^{-3}$ & $9.63\times10^{-3}$ \\
     & 80  & $-2.85\times10^{-2}$ & $2.34\times10^{-3}$ & $2.41\times10^{-3}$ \\
     & 160 & $-2.85\times10^{-2}$ & $5.85\times10^{-4}$ & $6.02\times10^{-4}$ \\
     & 320 & $-2.85\times10^{-2}$ & $1.46\times10^{-4}$ & $1.50\times10^{-4}$ \\
     & 640 & $-2.85\times10^{-2}$ & $3.65\times10^{-5}$ & $3.76\times10^{-5}$ \\
\hline
2.50 & 10  & $-7.39\times10^{-3}$ & $1.53\times10^{-1}$ & $1.54\times10^{-1}$ \\
     & 20  & $-5.23\times10^{-3}$ & $3.83\times10^{-2}$ & $3.85\times10^{-2}$ \\
     & 40  & $-3.69\times10^{-3}$ & $9.59\times10^{-3}$ & $9.63\times10^{-3}$ \\
     & 80  & $-2.61\times10^{-3}$ & $2.40\times10^{-3}$ & $2.41\times10^{-3}$ \\
     & 160 & $-1.85\times10^{-3}$ & $6.01\times10^{-4}$ & $6.02\times10^{-4}$ \\
     & 320 & $-1.31\times10^{-3}$ & $1.50\times10^{-4}$ & $1.50\times10^{-4}$ \\
     & 640 & $-9.23\times10^{-4}$ & $3.76\times10^{-5}$ & $3.76\times10^{-5}$ \\
\hline
3.00 & 10  & $-1.92\times10^{-3}$ & $1.54\times10^{-1}$ & $1.54\times10^{-1}$ \\
     & 20  & $-9.62\times10^{-4}$ & $3.85\times10^{-2}$ & $3.85\times10^{-2}$ \\
     & 40  & $-4.81\times10^{-4}$ & $9.63\times10^{-3}$ & $9.63\times10^{-3}$ \\
     & 80  & $-2.40\times10^{-4}$ & $2.41\times10^{-3}$ & $2.41\times10^{-3}$ \\
     & 160 & $-1.20\times10^{-4}$ & $6.02\times10^{-4}$ & $6.02\times10^{-4}$ \\
     & 320 & $-6.01\times10^{-5}$ & $1.50\times10^{-4}$ & $1.50\times10^{-4}$ \\
     & 640 & $-3.01\times10^{-5}$ & $3.76\times10^{-5}$ & $3.76\times10^{-5}$ \\
\hline
3.50 & 10  & $-5.02\times10^{-4}$ & $1.54\times10^{-1}$ & $1.54\times10^{-1}$ \\
     & 20  & $-1.78\times10^{-4}$ & $3.85\times10^{-2}$ & $3.85\times10^{-2}$ \\
     & 40  & $-6.28\times10^{-5}$ & $9.63\times10^{-3}$ & $9.63\times10^{-3}$ \\
     & 80  & $-2.22\times10^{-5}$ & $2.41\times10^{-3}$ & $2.41\times10^{-3}$ \\
     & 160 & $-7.85\times10^{-6}$ & $6.02\times10^{-4}$ & $6.02\times10^{-4}$ \\
     & 320 & $-2.78\times10^{-6}$ & $1.50\times10^{-4}$ & $1.50\times10^{-4}$ \\
     & 640 & $-9.81\times10^{-7}$ & $3.76\times10^{-5}$ & $3.76\times10^{-5}$ \\
\hline
4.00 & 10  & $-1.32\times10^{-4}$ & $1.54\times10^{-1}$ & $1.54\times10^{-1}$ \\
     & 20  & $-3.29\times10^{-5}$ & $3.85\times10^{-2}$ & $3.85\times10^{-2}$ \\
     & 40  & $-8.23\times10^{-6}$ & $9.63\times10^{-3}$ & $9.63\times10^{-3}$ \\
     & 80  & $-2.06\times10^{-6}$ & $2.41\times10^{-3}$ & $2.41\times10^{-3}$ \\
     & 160 & $-5.14\times10^{-7}$ & $6.02\times10^{-4}$ & $6.02\times10^{-4}$ \\
     & 320 & $-1.29\times10^{-7}$ & $1.50\times10^{-4}$ & $1.50\times10^{-4}$ \\
     & 640 & $-3.21\times10^{-8}$ & $3.76\times10^{-5}$ & $3.76\times10^{-5}$ \\
\hline
\end{longtable}

\begin{figure}[H]
\centering
\includegraphics[width=0.65\linewidth]{rayleigh_scaling}
\caption{Log–log scaling of $|\Delta E(R,p)|$ for representative $p$.
Measured slopes $\alpha(p)\!\approx\!-(p-2)$ agree with the analytic prediction.
}
\label{fig:rayleigh_scaling}
\end{figure}

\subsection{Three–dimensional eigenvalue analysis}

To test the full tensor operator, we compute the lowest eigenvalues
$\lambda_1$ of the discretized model for several $p$ and $R_{\max}$.
All runs use $C=-1$ and the TT–penalty enforcement
of Eq.~\eqref{eq:penalty}.
The results are shown in Table~\ref{tab:3d_eigenvalues} and 
Fig.~\ref{fig:3d_eigenvalues}.
The systematic decrease of $\lambda_1$ with increasing $R_{\max}$
confirms convergence toward the continuum limit.
A least–squares fit 
$\lambda_1(R_{\max})=aR_{\max}^{-2}+bR_{\max}^{-3}$ 
yields small residuals, consistent with the finite–volume interpretation.

\begin{longtable}{r r r}
\caption{Lowest eigenvalue $\lambda_1$ of the discretized tensor operator
for several decay exponents $p$ and outer radii $R_{\max}$.
Decreasing $\lambda_1$ with larger $R_{\max}$ confirms convergence toward the
infinite-volume limit.}
\label{tab:3d_eigenvalues}\\
\hline
$p$ & $R_{\max}$ & $\lambda_1$ \\
\hline
\endfirsthead
\caption[]{Lowest eigenvalue $\lambda_1$ (continued).}\\
\hline
$p$ & $R_{\max}$ & $\lambda_1$ \\
\hline
\endhead
\hline
\multicolumn{3}{r}{\textit{Continued on next page}}\\
\hline
\endfoot
\hline
\endlastfoot

% === flat reference ===
\textit{flat} & 6  & 0.2044 \\
              & 10 & 0.0739 \\
              & 14 & 0.0377 \\
              & 18 & 0.0228 \\
              & 20 & 0.0185 \\
\hline
% === p = 2.0 ===
2.0 & 6  & 0.1466 \\
    & 10 & 0.0435 \\
    & 14 & 0.0192 \\
    & 18 & 0.0102 \\
    & 20 & 0.0078 \\
\hline
% === p = 2.5 ===
2.5 & 6  & 0.1828 \\
    & 10 & 0.0651 \\
    & 14 & 0.0334 \\
    & 18 & 0.0203 \\
    & 20 & 0.0166 \\
\hline
% === p = 3.0 ===
3.0 & 6  & 0.1965 \\
    & 10 & 0.0711 \\
    & 14 & 0.0365 \\
    & 18 & 0.0222 \\
    & 20 & 0.0180 \\
\hline
% === p = 3.5 ===
3.5 & 6  & 0.2016 \\
    & 10 & 0.0730 \\
    & 14 & 0.0374 \\
    & 18 & 0.0227 \\
    & 20 & 0.0184 \\
\hline
% === p = 4.0 ===
4.0 & 6  & 0.2035 \\
    & 10 & 0.0736 \\
    & 14 & 0.0376 \\
    & 18 & 0.0228 \\
    & 20 & 0.0185 \\
\hline
\end{longtable}

\noindent
The systematic decrease of $\lambda_1$ with increasing $R_{\max}$
confirms convergence toward the continuum limit.
Intermediate values at $R_{\max}=18$ demonstrate a smooth monotonic
approach to the asymptotic regime, ensuring that the lowest eigenvalues
stabilize well before boundary effects dominate.
These eigenvalue trends complement the Rayleigh-quotient scaling test,
both identifying $p=3$ as the marginal decay rate where curvature transitions
from spectrally relevant to effectively negligible. \footnote{
We did not explore values $p<2$ in detail, since such slow falloff
is incompatible with the asymptotic behavior of isolated vacuum solutions
in general relativity (cf. the Supplementary Material).
Nevertheless, extrapolating the monotone suppression of $\lambda_1$
between $p=3.0$, $2.5$, and $2.0$ suggests that even slower decay
($p\lesssim2$) would further reduce the infrared eigenvalues,
producing a more strongly gapped, bound–state–like spectrum.
This is consistent with interpreting the regime $p<3$
as genuinely long–range in the spectral sense.
}

\begin{figure}[H]
\centering
\includegraphics[width=0.75\textwidth]{3d_eigenvalues}
\caption{
Convergence of $\lambda_1$ with domain size $R_{\max}$ for several decay exponents $p$.
The flattening of $\lambda_1(R_{\max})$ for $p\!\ge\!3$ shows that curvature becomes spectrally negligible beyond the inverse–cube rate, while slower decay $(p<3)$ yields progressively deeper infrared shifts.}
\label{fig:3d_eigenvalues}
\end{figure}

\begin{remark}[Interpretation]
The continuous approach of $\lambda_1(p)$ to its flat–space value as $p$ increases
demonstrates a smooth transition between confining and radiative regimes.
For $p\!\le\!2.5$, curvature remains spectrally significant; at $p\!=\!3$, curvature and dispersion balance; and for $p\!>\!3$, the spectrum becomes indistinguishable from flat space.
No discrete bound states appear, in agreement with the analytic prediction of a purely continuous essential spectrum beyond the inverse–cube threshold.
Additional convergence and stability checks, including
penalty sweeps, grid refinement, and mode localization, are presented
in the Supplementary Material (Sec.~S3).
\end{remark}

\begin{lemma}[Asymptotic behavior of the lowest eigenvalue]
\label{lem:eigenvalueconvergence}
For $p\ge3$, $\lambda_1(L_p)\!\sim\!R_{\max}^{-2}$ and approaches $0^+$ as
$R_{\max}\!\to\!\infty$, consistent with approach to the continuum threshold
and the absence of bound states. 
For $p<3$, the slower decay of curvature increases the infrared coupling,
yielding smaller $\lambda_1$ but no discrete negative modes.
\end{lemma}

\begin{remark}[Numerical consistency with the analytic threshold]
Both diagnostics reproduce the qualitative behavior predicted by the analytic theory:
power–law Rayleigh scaling with slope $\alpha(p)=-(p-2)$ and a continuous
spectral transition centered at $p=3$. The results therefore confirm that the inverse–cube decay constitutes a sharp spectral threshold, marking the exact boundary between compact and noncompact curvature perturbations in the spin–2 sector. A detailed convergence and stability analysis verifying that these features are not numerical artifacts of discretization, boundary
conditions, or constraint penalties is provided in
the Supplementary Material.
\end{remark}


```



---


<a id="annex-sec-06-physical-interpretation-and-implications"></a>

### main.tex — 6. Physical Interpretation and Implications (verbatim)


```tex
\section{Physical Interpretation and Implications}
\label{sec:physical_interpretation}

The analytic and numerical analyses of Sections~\ref{sec:critical_decay}-\ref{sec:numerics}
identify a sharp transition between two qualitatively distinct spectral regimes of
gravitational perturbations on asymptotically flat manifolds.
This section interprets the threshold $p=3$ in physical terms and connects it to known
infrared phenomena of general relativity, including radiative behavior, gravitational
memory, and asymptotic symmetry.
Beyond the classical setting, the same spectral structure underlies the infrared
behavior of the quantized linearized field, providing a geometric origin for the
soft sector of quantum gravity.

\subsection{Infrared Structure of the Quantized Field}
Although the present analysis is entirely classical, the spectral properties of the
spatial Lichnerowicz operator determine the infrared structure of the quantized
linearized gravitational field.  
In canonical quantization, the equal-time two-point function in harmonic gauge is
the inverse of $L$, so the large-distance correlations of the graviton field are
governed by the same spectral threshold identified here.
For curvature decaying faster than $r^{-3}$, $L^{-1}$ remains a short-range operator
and defines a regular Fock vacuum with finite infrared correlations.
At the marginal rate $r^{-3}$, however, the Green's function develops a slow
algebraic tail, corresponding to the emergence of zero-frequency, spatially
extended modes.
These modes represent the static counterparts of the soft-graviton excitations
that appear dynamically in the full quantum theory.
From this viewpoint, the $r^{-3}$ decay marks the geometric origin of the
infrared enhancement familiar from the Weinberg soft-graviton theorem
and its modern extensions in the asymptotic symmetry framework
\cite{Weinberg1965,Strominger2018}.

\begin{remark}[Radiative and confining regimes]
When the curvature potential decays faster than $r^{-3}$, gravitational perturbations
propagate as freely radiating modes whose energy escapes to infinity,
leaving the essential spectrum continuous and gapless.
At the borderline decay rate $r^{-3}$, dispersion and curvature balance precisely,
producing marginally bound tensor configurations that are spatially extended but
nonlocalized.
These modes neither dissipate completely nor remain compactly confined, representing
the static analogue of zero-energy resonances in potential scattering.
For $p>3$, the curvature becomes spectrally negligible and the field is fully radiative;
for $p<3$, curvature acts as a long-range potential that enhances infrared coupling and could support quasi-bound behavior.
\end{remark}

\subsection{Connection with Gravitational Memory and Soft Modes}

The marginally bound tensor modes at the $r^{-3}$ threshold are consistent with a
spectral interpretation of the gravitational memory effect.
In the nonlinear theory, the Christodoulou memory corresponds to a permanent displacement
of test particles due to the flux of gravitational radiation through null infinity.
Within the present linear, time-independent framework, zero-frequency, spatially
extended modes sustain correlations that do not fully decay.

\begin{remark}[Static precursors of memory]
If such stationary correlations were evolved within a dynamical setting,
their time-integrated imprint would resemble the displacement produced by nonlinear
memory.
In this sense, the marginal modes act as static precursors of the soft sector familiar
from the infrared triangle connecting asymptotic symmetries, soft graviton theorems,
and memory effects.
They indicate that the spatial Lichnerowicz operator already encodes the seeds of the
infrared structure that reemerges dynamically at null infinity.
A complete treatment of this correspondence would require coupling the elliptic analysis
to the time-dependent linearized Einstein equations near $\mathscr{I}^+$, which we leave
for future work.
\end{remark}

\subsection{Relation to Late-Time Tails}

Price’s law for black-hole perturbations, $\psi\!\sim\!t^{-2\ell-3}$,
arises from an effective potential decaying as $r^{-3}$.
The same inverse-cube scaling governs the onset of noncompactness for the
spatial Lichnerowicz operator, suggesting that both results stem from a common geometric
mechanism: curvature of order $r^{-3}$ produces algebraic energy leakage and slow
relaxation, defining the boundary between exponential and power-law decay.

\begin{remark}[Unified interpretation of tails]
The spectral perspective presented here identifies the stationary origin of the
temporal tail: marginally bound spatial modes correspond to low-frequency perturbations
whose gradual radiative leakage produces the late-time decay law.
Thus, the spatial and temporal manifestations of the inverse-cube scaling
represent two aspects of the same infrared structure.
\end{remark}

\subsection{Asymptotic Symmetries and Infrared Structure}

At the critical decay rate, the geometry admits zero-frequency tensor excitations
associated with emergent asymptotic diffeomorphisms.
These marginal modes can be viewed as the linearized precursors of BMS
supertranslations, encoding conserved charges at spatial infinity.

\begin{remark}[Spectral interpretation of asymptotic symmetry]
The appearance of an extended zero mode at the inverse-cube threshold
signals the enlargement of the asymptotic symmetry algebra to include
nontrivial diffeomorphisms acting at infinity.
The spectral transition at $p=3$ thus links the mathematical onset of noncompactness
to the physical emergence of memory and soft graviton behavior within a common framework.
\end{remark}

\paragraph{Universality of the spectral threshold.}
The coincidence of analytic, numerical, and geometric evidence suggests that the
inverse-cube decay represents a universal spectral boundary for long-range
field theories on $\mathbb{R}^3$.
In both gauge and gravitational contexts, curvature or field strength decaying as
$r^{-3}$ marks the transition between compact and noncompact spectral behavior,
where the Laplacian ceases to dominate the asymptotic dynamics.
This threshold thus defines a geometric law of infrared structure:
it separates the radiative regime, characterized by freely propagating modes,
from the marginal regime in which curvature produces algebraic tails and
soft correlations that persist to infinity.


```



---


<a id="annex-sec-07-relation-to-previous-work-and-threshold-phenomena"></a>

### main.tex — 7. Relation to Previous Work and Threshold Phenomena (verbatim)


```tex
\section{Relation to Previous Work and Threshold Phenomena}
\label{sec:relation_previous}

The spectral threshold established in Sections~\ref{sec:critical_decay}-\ref{sec:numerics}
connects several independent developments in spectral geometry, mathematical relativity,
and gauge theory.
It refines classical results on elliptic operators on asymptotically flat manifolds,
relates to long-range scattering and tail phenomena in black-hole perturbation theory,
and parallels curvature-controlled infrared thresholds first identified in non-Abelian
gauge theory.

\subsection{Spectral theory of elliptic operators on asymptotically flat ends}

The analytic foundation for elliptic operators on noncompact manifolds with Euclidean ends
was laid by Lockhart and McOwen~\cite{Lockhart1985}.
They proved Fredholm and isomorphism properties for elliptic operators
acting between weighted Sobolev spaces
$H^2_\delta \to L^2_{\delta-2}$,
identifying indicial roots as the precise obstructions to invertibility.
These results underlie much of geometric analysis on asymptotically flat manifolds,
including the constraint equations of general relativity
and deformation theory of vacuum initial data.

\begin{remark}[Extension of classical elliptic results]
Earlier analyses treated curvature terms as short-range perturbations of the Laplacian
without identifying a quantitative boundary between short- and long-range behavior.
The present result provides a sharp criterion:
if $|{\rm Riem}(x)|\le C\,r^{-p}$ with $p>3$, the curvature potential $V_R$
is relatively compact with respect to $\Delta_T$ and
$\sigma_{\mathrm{ess}}(L)=[0,\infty)$;
when $p=3$, compactness fails and a normalized Weyl sequence appears at zero energy.
This refines the Lockhart-McOwen framework by isolating the exact curvature decay rate
at which the Fredholm property transitions to noncompact spectral behavior.
\end{remark}

\subsection{Thresholds in long-range potentials}

The $r^{-3}$ boundary identified here is the tensorial analogue of the classical
long-range threshold in Schrödinger theory.
In three dimensions, Simon~\cite{Simon1976} proved that potentials decaying faster than
$r^{-3}$ yield purely absolutely continuous spectrum on $[0,\infty)$,
whereas slower decay allows resonances or threshold states near zero energy.
The gravitational case exhibits an analogous structure,
with the Riemann curvature playing the role of an effective potential.

\begin{remark}[Connection with Schrödinger thresholds]
The Lichnerowicz operator realizes, at the tensorial level,
the same balance between dispersion and long-range coupling
that governs Schrödinger operators.
The inverse-cube decay marks the onset of marginally extended configurations,
beyond which curvature ceases to influence the spectrum.
This correspondence illustrates the structural similarity of infrared thresholds
across scalar, vector, and tensor field equations.
\end{remark}

\subsection{Relation to late-time tails in black-hole spacetimes}

Price~\cite{Price1972} and Ching \emph{et al.}~\cite{Ching1995}
showed that perturbations of the Schwarzschild spacetime decay as $t^{-2\ell-3}$,
with the algebraic tail arising from an effective potential proportional to $r^{-3}$.
The same scaling controls the marginal behavior of the spatial Lichnerowicz operator.

\begin{remark}[Spatial origin of temporal tails]
The elliptic analysis here provides the stationary counterpart
of the dynamical late-time decay law.
At the inverse-cube decay, the spatial operator supports near-zero modes
that correspond to the low-frequency enhancement responsible for
algebraic relaxation.
The threshold $p=3$ thus unifies the stationary and dynamical
manifestations of gravitational infrared behavior.
\end{remark}

\subsection{Infrared structure and memory}

In the context of asymptotic symmetry and gravitational memory,
zero-frequency perturbations encode residual deformations between
radiative vacua at null infinity.
The existence of an $L^2$-normalized Weyl sequence at zero energy
provides a spatial realization of these soft configurations.

\begin{remark}[Spectral characterization of the infrared sector]
At curvature decay $|{\rm Riem}|\!\sim\!r^{-3}$,
the Lichnerowicz operator develops marginally extended tensor modes
that remain spatially nonlocal yet finite in energy.
These represent the elliptic, spatial manifestation of soft graviton modes
and delineate the precise geometric condition for the emergence
of an infrared sector in linearized gravity.
\end{remark}

\subsection{Parallel thresholds across gauge and gravitational systems}

A closely related threshold occurs in non-Abelian gauge theory.
For the covariant Laplacian
\[
\Delta_A = -(\nabla_A)^*\nabla_A = -\nabla^*\nabla + {\rm ad}(F_A),
\]
acting on adjoint-valued fields with curvature $F_A$,
the decay $|F_A|\!\sim\!r^{-3}$ separates spectrally radiative behavior
from the onset of infrared sensitivity.

\begin{proposition}[Parallel inverse-cube threshold]
For Laplace–type operators on bundles over $\mathbb{R}^3$,
a curvature decay of order $r^{-3}$ marks the transition between
short-range, radiative behavior and long-range, infrared coupling.
In both gauge and gravitational settings, curvature acts as an effective potential;
at this critical rate, marginal nonlocalized modes appear,
signaling the breakdown of compactness of the resolvent.
\end{proposition}

\subsection{Summary and outlook}

Taken together, these results connect several domains of spectral analysis
and physical theory.
For $p>3$, curvature perturbations are short-range and the spectrum purely radiative.
At $p=3$, curvature ceases to be compact, marginal modes emerge,
and the infrared sector appears continuously but sharply.
For $p<3$, curvature strengthens the coupling further, but without forming
discrete bound states in the present tensorial setting.

\begin{remark}[Open directions]
Future work should establish a full limiting absorption principle
at the critical rate and extend the analysis to coupled
gravity–gauge systems.
Such results would provide a rigorous foundation
for the infrared correspondence between soft sectors,
asymptotic symmetries, and spectral transitions.
\end{remark}


```



---


<a id="annex-sec-08-conclusion"></a>

### main.tex — 8. Conclusion (verbatim)


```tex
\section{Conclusion}
\label{sec:conclusion}

The analyses presented here establish a sharp spectral threshold
for the spatial Lichnerowicz operator on asymptotically flat three–manifolds.
The results integrate geometric analysis, spectral theory, and physical interpretation
within a unified framework.

\begin{theorem}[Spectral threshold for linearized gravity]
\label{thm:main_threshold}
Let $(\Sigma,g)$ be a smooth asymptotically flat three–manifold with curvature decay
$|{\rm Riem}(x)|\!\le\!C\,r^{-p}$.
Then:
\begin{enumerate}
\item For $p>3$, the curvature potential $V_R$ is relatively compact with respect to
$\nabla^*\nabla$, and the essential spectrum is purely continuous:
\[
\sigma_{\mathrm{ess}}(L)=[0,\infty).
\]
\item At the critical rate $p=3$, compactness fails and a normalized Weyl sequence
appears at zero energy, producing marginally extended tensor configurations
that remain spatially nonlocal yet finite in energy.
\item For $p<3$, curvature acts as a long-range potential that enhances infrared coupling,
but without producing isolated bound states in the tensorial sector.
\end{enumerate}
\end{theorem}

\begin{remark}[Physical regimes]
The three regimes delineated above correspond respectively to radiative propagation
($p>3$), marginal persistence ($p=3$), and enhanced infrared coupling ($p<3$).
The inverse–cube decay therefore represents the sharp geometric boundary between
short-range and long-range gravitational behavior in three spatial dimensions.
\end{remark}

These findings extend classical spectral theory to curvature–coupled tensor operators
and reveal a structural parallel with non–Abelian gauge fields and Schrödinger operators. In all three settings, the inverse–cube decay marks the point at which the potential ceases to be spectrally negligible and marginal modes first appear. Numerical analysis of the reduced model confirms that this transition occurs continuously but sharply at $p=3$, with no evidence of discrete confinement, validating the analytic predictions of Sections~\ref{sec:critical_decay}-\ref{sec:numerics}.

\begin{remark}[Future directions]
Several open problems arise naturally from this work:
establishing a limiting absorption principle at the critical rate;
extending the spectral analysis to Schwarzschild and Kerr slices;
and developing the full dynamical correspondence between marginal spatial modes
and the soft–memory sector at null infinity.
Such results would further clarify the geometric and spectral unity of the infrared
structure in gauge and gravitational theories.
\end{remark}


```



---


<a id="annex-sec-09-declarations"></a>

### main.tex — Declarations (verbatim)


```tex
\section*{Declarations}

\textbf{Funding} The author received no external funding.

\textbf{Conflict of interest} The author declares no conflict of interest.

\textbf{Data availability} All data supporting the conclusions of this work are contained within the article and its Supplementary Material. No external datasets were used. 
Numerical results can be reproduced using the procedures described in Section~5 and Appendix~D.

\appendix

```



---


<a id="annex-app-a-gauge-correction-and-elliptic-estimates"></a>

### main.tex — Appendix A. Gauge Correction and Elliptic Estimates (verbatim)


```tex
\section{Gauge Correction and Elliptic Estimates}
\label{appendix:gauge}

This appendix justifies the harmonic–gauge correction used in
Section~3.  
Under the hypotheses of Section~5, namely, $(\Sigma,g)$ smooth,
asymptotically flat, simply connected, and with ${\rm Ric}=O(r^{-3})$, 
the vector Laplacian
\[
\Delta_V X = \nabla^*\nabla X + {\rm Ric}(X)
\]
is uniformly elliptic and symmetric on $L^2(\Sigma;T^*\Sigma)$.

\begin{lemma}[Isomorphism property]
For weights $-1<\delta<0$, the mapping
\[
\Delta_V:H^2_\delta(\Sigma;T^*\Sigma)\to L^2_{\delta-2}(\Sigma;T^*\Sigma)
\]
is Fredholm of index zero and an isomorphism whenever
$H^1_{\mathrm{dR}}(\Sigma)=0$.
\end{lemma}

\begin{proof}[Sketch]
A direct consequence of Lockhart-McOwen theory
(\emph{Comm.\ Pure Appl.\ Math.} \textbf{38}, 603 (1985)),
since the Euclidean indicial roots are $\{0,-2\}$.
\end{proof}

\begin{lemma}[Gauge correction]
For each $h\in H^2_\delta(\Sigma;S^2T^*\Sigma)$ with $-1<\delta<0$, there exists a
unique $X\in H^2_\delta(\Sigma;T^*\Sigma)$ satisfying
$\Delta_V X=\nabla\!\cdot h$ and
$\|X\|_{H^2_\delta}\le C\|\nabla\!\cdot h\|_{L^2_{\delta-2}}$.
\end{lemma}

\begin{proposition}[Corrected Weyl sequence]
Let $\{h_n\}$ be the approximate sequence of Section~3.
Defining $\tilde h_n=h_n-\mathcal{L}_{X_n}g$ with
$X_n=\Delta_V^{-1}(\nabla\!\cdot h_n)$ yields
\[
\|\tilde h_n\|_{L^2}=1,\qquad
\tilde h_n\rightharpoonup0,\qquad
\|L\tilde h_n\|_{L^2}\to0.
\]
Thus $0\in\sigma_{\mathrm{ess}}(L)$ in harmonic gauge.
\end{proposition}


```



---


<a id="annex-app-b-curvature-structure-and-the-schwarzschild-example"></a>

### main.tex — Appendix B. Curvature Structure and the Schwarzschild Example (verbatim)


```tex
\section{Curvature Structure and the Schwarzschild Example}
\label{appendix:schwarzschild}

The Schwarzschild metric provides a physical realization of the
critical inverse–cube curvature decay analyzed in
Section~3.
\begin{definition}[Spatial metric]
In isotropic coordinates $(t,r,\omega)$, the Schwarzschild line element is
\[
ds^2=-\Bigl(\frac{1-\tfrac{M}{2r}}{1+\tfrac{M}{2r}}\Bigr)^2 dt^2
     +\Bigl(1+\frac{M}{2r}\Bigr)^4(dr^2+r^2d\omega^2).
\]
On a time-symmetric slice $t=\mathrm{const.}$,
the spatial metric is $g_{ij}=\psi^4\delta_{ij}$ with
$\psi(r)=1+\tfrac{M}{2r}$.
\end{definition}

\begin{lemma}[Asymptotic curvature]
For this metric,
\[
|{\rm Riem}(x)|\simeq C\,M\,r^{-3}\qquad(r\to\infty),
\]
so the curvature saturates the inverse–cube decay assumed in
Theorem~2.
\end{lemma}

\begin{proposition}[Effective potential]
The spatial Lichnerowicz operator on the Schwarzschild background satisfies
\[
Lh = \Delta_0 h - (C M) r^{-3} h + O(r^{-4})h,
\qquad C M > 0,
\]
showing that the Schwarzschild geometry realizes, in its far-field limit,
the same attractive $r^{-3}$ potential analyzed in the numerical model
of Section~5. \footnote{The overall minus sign arises from the definition 
$(V_R h)_{ij} = -R_{i}{}^{k}{}_{j}{}^{\ell} h_{k\ell}$, which makes the curvature coupling attractive for positive mass $M>0$.}
\end{proposition}


```



---


<a id="annex-app-c-analytical-framework-and-weighted-sobolev-spaces"></a>

### main.tex — Appendix C. Analytical Framework and Weighted Sobolev Spaces (verbatim)


```tex
\section{Analytical Framework and Weighted Sobolev Spaces}
\label{appendix:sobolev}

We summarize the analytic conventions and functional-analytic tools
used throughout.

\begin{definition}[Weighted Sobolev spaces]
For a smooth radius function $r$ on an asymptotically flat
three–manifold $(\Sigma,g)$ and $\delta\in\mathbb{R}$,
\[
\|u\|_{H^k_\delta}^2
   = \sum_{|\alpha|\le k}
     \int_\Sigma (1+r^2)^{\delta-|\alpha|}
     |\nabla^\alpha u|^2\,dV_g.
\]
Then $H^k_\delta(\Sigma;E)$ is the completion of
$C_c^\infty(\Sigma;E)$ under this norm.
\end{definition}

\begin{lemma}[Fredholm property]
If $P$ is a uniformly elliptic operator approaching constant coefficients
at infinity, then
\[
P:H^2_\delta\to L^2_{\delta-2}
\]
is Fredholm for all $\delta$ not equal to an indicial root
\cite{Lockhart1985}.
\end{lemma}

\begin{proposition}[Self-adjointness and essential spectrum]
For $\delta\in(-1,0)$ and $V=O(r^{-p})$ with $p>2$,
operators of the form $L=\nabla^*\nabla+V$
are self–adjoint on $L^2(\Sigma;E)$.
The essential spectrum $\sigma_{\mathrm{ess}}(L)$
is determined by the existence of Weyl sequences as in
Weyl’s criterion.
\end{proposition}

\begin{lemma}[Compactness of the curvature potential for $p>3$]
\label{lem:compact_VR}
Let $(\Sigma,g)$ be a smooth asymptotically flat three-manifold with a single Euclidean end, and assume
\[
g_{ij} = \delta_{ij} + O(r^{-1}), 
\qquad 
\partial g_{ij} = O(r^{-2}), 
\qquad 
\partial^2 g_{ij} = O(r^{-3}),
\]
so that $|{\rm Riem}(x)| \le C\,\langle r\rangle^{-p}$ for some $p>3$.
Fix a weight $-1<\delta<0$, and let
\[
L = \nabla^{*}\nabla + V_R,
\qquad
(V_R h)_{ij} = -R_{i}{}^{\ell}{}_{j}{}^{m}\,h_{\ell m}.
\]
Then the curvature term defines a compact operator
\[
V_R : H^2_\delta(\Sigma;S^2T^*\Sigma) \longrightarrow L^2_{\delta-2}(\Sigma;S^2T^*\Sigma),
\]
and therefore $L$ is a compact perturbation of $\nabla^{*}\nabla$ on $L^2(\Sigma;S^2T^*\Sigma)$.
In particular,
\[
\sigma_{\mathrm{ess}}(L)=\sigma_{\mathrm{ess}}(\nabla^{*}\nabla)=[0,\infty).
\]
\end{lemma}

\begin{proof}
The curvature bound $|{\rm Riem}(x)|\le C\langle r\rangle^{-p}$ implies
$|V_R(x)|\le C\langle r\rangle^{-p}$, so that for any 
$h\in H^2_\delta(\Sigma;S^2T^*\Sigma)$,
\[
\|V_R h\|_{L^2_{\delta-2}}^2
=
\int_\Sigma \langle r\rangle^{2(\delta-2)}\,|V_R(x)h(x)|^2\,dV_g
\lesssim
\int_\Sigma \langle r\rangle^{2(\delta-2)-2p}|h(x)|^2\,dV_g.
\]
Since $-1<\delta<0$ and $p>3$, the exponent $2(\delta-2)-2p$ is less than $-6$, making the weight $\langle r\rangle^{2(\delta-2)-2p}$ integrable at infinity in three dimensions. 
Hence $V_R:H^2_\delta\to L^2_{\delta-2}$ is bounded.

To verify compactness, let $\chi_R$ be a smooth cutoff equal to $1$ on $\{r\le R\}$ and supported in $\{r\le 2R\}$.  
Decompose
\[
V_R=\chi_R V_R+(1-\chi_R)V_R
=:V_R^{\mathrm{(comp)}}+V_R^{\mathrm{(tail)}}.
\]
On the bounded region $\{r\le2R\}$, the metric is smooth and the weighted norms are equivalent to the unweighted ones.  
By the classical Rellich compactness theorem on precompact domains, the embedding
$H^2_\delta(\{r\le2R\})\hookrightarrow L^2_{\delta-2}(\{r\le2R\})$
is compact; since $V_R$ is smooth there, the multiplication operator
$V_R^{\mathrm{(comp)}}$ is compact.

The remainder $V_R^{\mathrm{(tail)}}$ is supported in $\{r>R\}$, where the curvature decay dominates.  
For $h$ with $\|h\|_{H^2_\delta}=1$,
\[
\|V_R^{\mathrm{(tail)}}h\|_{L^2_{\delta-2}}^2
\lesssim
\int_{r>R}\langle r\rangle^{2(\delta-2)-2p}|h(x)|^2\,dV_g.
\]
Because $2(\delta-2)-2p<-6$, the weight decays faster than $r^{-6}$; 
combined with the uniform bound $|h|^2\in L^2_\delta$, dominated convergence implies that
\[
\sup_{\|h\|_{H^2_\delta}=1}
\|V_R^{\mathrm{(tail)}}h\|_{L^2_{\delta-2}}
\longrightarrow 0
\quad\text{as }R\to\infty.
\]
Thus $V_R$ is the limit of compact operators with vanishing tails, and hence compact.

Finally, since $L=\nabla^{*}\nabla+V_R$ is a compact perturbation of the nonnegative, self-adjoint operator $\nabla^{*}\nabla$, Weyl’s theorem ensures that the essential spectra coincide:
\[
\sigma_{\mathrm{ess}}(L)=\sigma_{\mathrm{ess}}(\nabla^{*}\nabla)=[0,\infty).
\]
The essential spectrum of the Lichnerowicz operator therefore matches that of the flat tensor Laplacian, confirming that sufficiently rapid curvature decay leaves the asymptotic spectrum purely continuous.
\end{proof}

\begin{remark}
These weighted spaces and mapping properties justify the estimates
and compactness arguments used in
Sections~2–3.
\end{remark}


```



---


<a id="annex-app-d-numerical-validation-and-stability-tests"></a>

### main.tex — Appendix D. Numerical Validation and Stability Tests (verbatim)


```tex
\section{Numerical Validation and Stability Tests}
\label{appendix:stability}

This appendix documents three numerical consistency checks supporting
the eigenvalue results reported in Section~5:
(i) grid-spacing convergence,
(ii) finite-volume convergence in $R_{\max}$,
and (iii) robustness under constraint enforcement.
These tests show that the observed spectral transition near $p=3$
is not a numerical artifact of discretization, boundary conditions,
or gauge penalties.

\subsection{Grid refinement at fixed physical volume}

To verify that the numerical results reported in Section~5
are free from discretization artifacts, we repeated the computations at two
grid resolutions.  The coarse run used spacing $h\!\approx\!1.0$ 
($N=21$ points per axis, $\sim4.1\times10^4$ degrees of freedom),
and the refined run used $h\!\approx\!0.5$
($N=41$, $\sim3.6\times10^5$ degrees of freedom).
Converged eigenvalues at both resolutions are listed below.
The near-invariance of $\lambda_1$ and $\lambda_2$ confirms that the
numerical spectrum is stable with respect to grid refinement and that the
observed scaling behavior in Section~5 is not a numerical artifact.

\noindent
\emph{Coarse resolution ($h\!\approx\!1.0$, $N=21$ points/axis,
$\sim 4.1\times 10^4$ DOFs).}
\[
\begin{array}{ll}
\text{flat:} & \lambda_1 = 0.07386996,\quad \lambda_2 = 0.14713361, \\
p=2.5:       & \lambda_1 = 0.06505331,\quad \lambda_2 = 0.07023744, \\
p=3.0:       & \lambda_1 = 0.07114971,\quad \lambda_2 = 0.07276003, \\
p=3.5:       & \lambda_1 = 0.07298806,\quad \lambda_2 = 0.07351824.
\end{array}
\]

\noindent
\emph{Refined resolution ($h\!\approx\!0.5$, $N=41$ points/axis,
$\sim 3.6\times 10^5$ DOFs).}
\[
\begin{array}{ll}
\text{flat:} & \lambda_1 = 0.07398399,\quad \lambda_2 = 0.14781594, \\
p=2.5:       & \lambda_1 = 0.06505635,\quad \lambda_2 = 0.07029545, \\
p=3.0:       & \lambda_1 = 0.07109678,\quad \lambda_2 = 0.07279109, \\
p=3.5:       & \lambda_1 = 0.07295142,\quad \lambda_2 = 0.07355686.
\end{array}
\]
\medskip

Halving the grid spacing changes $\lambda_1$ by less than $0.2\%$
across all $p$, demonstrating numerical convergence in $h$.
Crucially, the ordering
\[
\lambda_1(p{=}2.5)
<
\lambda_1(p{=}3.0)
<
\lambda_1(p{=}3.5)
\simeq
\lambda_1(\text{flat})
\]
is preserved under refinement.
This ordering encodes the physical trend reported in
Section~5:
slower curvature decay (smaller $p$) produces a stronger infrared
distortion of the lowest mode, while faster decay $(p>3)$ becomes
spectrally indistinguishable from flat space.
The persistence of this structure under refinement shows that it is
not a coarse-grid artifact.

\subsection{Finite-volume convergence in $R_{\max}$}

We also varied the outer box size
$R_{\max}\in\{6,10,14,18,20\}$
at fixed grid spacing $h\approx 1.0$
(so that increasing $R_{\max}$ increases the number of unknowns)
and observed that $\lambda_1$ decreases monotonically with $R_{\max}$
for every $p$ tested.
For example, at $p=3.0$ we find
\[
\lambda_1(R_{\max}{=}6)=0.1965,\quad
\lambda_1(R_{\max}{=}10)=0.0711,\quad
\lambda_1(R_{\max}{=}20)=0.0180.
\]
The approximate scaling $\lambda_1 \sim R_{\max}^{-2}$
agrees with the interpretation of $\lambda_1$ as the lowest
discrete ``box mode'' approaching the continuum threshold.
This confirms that the small eigenvalues reported in
Table~2
are controlled by the infrared volume scale and not spurious
numerical locking to the boundary.

\paragraph{Penalty strength studies and TT projection cross–check.}
We sweep $\eta,\zeta$ over two decades (e.g. $\eta=\zeta\in\{10, 40, 160, 640\}$ in nondimensional units);
for $p\in\{2.5,3.0,3.5\}$ the relative shifts in $\lambda_{1,2}$ between the two largest penalties are $<10^{-3}$.
We also compute the lowest eigenpair using an explicit TT projection:
given any $u$, let $u^{\mathrm{TT}}=\Pi_{\mathrm{TT}} u$ via Helmholtz–Hodge decomposition
(solve $\Delta_V X=\nabla\!\cdot u$, then set $u^{\mathrm{TT}}=u-\mathcal{L}_X g - \frac{1}{3}(\mathrm{tr}\,u)g$).
The projected–eigenpair agrees with the penalty result within the grid error.
(Algorithmic details in Sec.~S1; linear solves via conjugate gradients with algebraic multigrid preconditioner.)

\begin{remark}[Conclusion]
The convergence in grid spacing, the $R_{\max}$ scaling, 
and the robustness under transverse-traceless enforcement
all support a single interpretation:
the trends reported in Section~5
reflect genuine infrared properties of the Lichnerowicz operator.
In particular, they provide independent numerical evidence that
the inverse-cube decay of curvature ($p=3$)
marks the threshold between spectrally relevant long-range curvature
and spectrally negligible curvature.
\end{remark}

\paragraph{Code availability.}
The complete Python script reproducing Table~2
is provided as \texttt{3d\_tensor\_operator.py} in the Supplementary Reproducibility Package (ResearchGate upload).
It implements the finite–difference Laplacian, the regularized tidal curvature
term $E_{ij}\!\sim\!r^{-p}(n_i n_j-\delta_{ij}/3)$, and the
trace–penalty enforcement of the TT constraint.


```



---


<a id="annex-app-e-weyl-sequence-construction-and-verification-of-the-critical-spectrum"></a>

### main.tex — Appendix E. Weyl Sequence Construction and Verification of the Critical Spectrum (verbatim)


```tex
\section{Weyl Sequence Construction and Verification of the Critical Spectrum}
\label{appendix:weyl}

This appendix provides quantitative estimates completing
the proof of Lemma~2 and
Theorem~1.
Throughout, $(\Sigma,g)$ satisfies the asymptotic flatness
conditions of eq. 11 with $|{\rm Riem}(x)|\le C r^{-3}$.

\subsection*{Normalization and scaling}
Let $H_{ij}(\omega)$ be a trace-free, divergence-free harmonic on $S^2$
and define $h_n=A_n\phi_n(r)r^{-1}H_{ij}(\omega)$ with $\phi_n$ supported
on $A_n=\{n/2<r<2n\}$.
The metric volume element satisfies
$dV_g=(1+O(r^{-1}))r^2dr\,d\omega$,
so that
\[
\|h_n\|_{L^2}^2\!\simeq\!A_n^2\!\!\int_{n/2}^{2n}\!r^{-2}r^2dr
\sim A_n^2 n.
\]
Choosing $\|h_n\|_{L^2}=1$ gives $A_n\simeq n^{-1/2}$.
Derivatives of $\phi_n$ are supported on the thin shells
$S_n=\{n/2<r<n\}\cup\{3n/2<r<2n\}$
where $|\nabla\phi_n|\lesssim n^{-1}$.

\subsection*{Commutator and curvature estimates}
Expanding
\[
\nabla^*\nabla(\phi_n r^{-1}H)
= \phi_n\nabla^*\nabla(r^{-1}H)
 + 2\nabla\phi_n\!\cdot\!\nabla(r^{-1}H)
 + (\Delta\phi_n)r^{-1}H,
\]
the first term vanishes identically in flat space since
$\nabla^*\nabla(r^{-1}H)=0$ for tensor harmonics of order $\ell=2$.
The commutator terms are supported on $S_n$
and satisfy
\[
\|[\nabla^*\nabla,\phi_n]h_n\|_{L^2}
\lesssim n^{-2}.
\]
The curvature potential obeys
$|V_R|\le C r^{-3}$,
so that
\[
\|V_R h_n\|_{L^2}^2
\lesssim A_n^2\!\int_{n/2}^{2n}\!r^{-6}r^2dr
\sim n^{-4},
\qquad
\|V_R h_n\|_{L^2}\lesssim n^{-2}.
\]
Together these give $\|Lh_n\|_{L^2}\lesssim n^{-2}$.

\subsection*{Gauge correction}
Solving $\Delta_V X_n=\nabla\!\cdot h_n$
with the Fredholm estimate of
Lemma~1 yields
\[
\|X_n\|_{H^2_\delta}\lesssim
\|\nabla\!\cdot h_n\|_{L^2}
\lesssim n^{-1},
\qquad -1<\delta<0.
\]
Hence
$\|\mathcal{L}_{X_n}g\|_{L^2}\lesssim n^{-1}$
and the corrected fields
$\tilde h_n=h_n-\mathcal{L}_{X_n}g$
satisfy $\nabla^j\tilde h_{n,ij}=0$.
Since $L(\mathcal{L}_{X_n}g)$ involves at most two derivatives of $X_n$,
\[
\|L(\mathcal{L}_{X_n}g)\|_{L^2}\lesssim n^{-1},
\]
which is subleading relative to $\|Lh_n\|_{L^2}\lesssim n^{-2}$.
Thus
\[
\|L\tilde h_n\|_{L^2}\to0,
\qquad
\|\tilde h_n\|_{L^2}=1.
\]

\subsection*{Conclusion}
The sequence $\{\tilde h_n\}$ is normalized,
divergence-free, and supported on disjoint annuli
escaping to infinity, implying
$\tilde h_n\rightharpoonup0$ weakly in $L^2$.
By Weyl’s criterion, $0\in\sigma_{\mathrm{ess}}(L)$,
establishing Theorem~1.

\textbf{Author contributions} The author is solely responsible for all aspects of this work.

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```



---