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+# 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
+
+- [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
+
+
+
+
+### 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 > 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 < 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
+```
+
+
+---
+
+
+
+
+### 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.
+
+**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:**
+
+
+$$
+L_p=-\tfrac{d^2}{dr^2}+\tfrac{\ell(\ell+1)}{r^2}+\tfrac{C}{r^p}
+$$
+
+
+---
+
+
+
+
+### 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.
+
+**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)
+
+
+---
+
+
+
+
+### 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.
+
+**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)
+
+
+---
+
+
+
+
+### 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.
+
+**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)
+
+
+---
+
+
+
+
+### 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, $pd$.
+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)
+
+
+---
+
+
+
+
+### 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.
+
+**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)
+
+
+---
+
+
+
+
+### 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.
+
+**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)
+
+
+---
+
+
+
+
+### 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.
+
+**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)
+
+
+---
+
+
+
+
+### 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.
+
+**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)
+
+
+---
+
+
+
+
+### Declarations
+
+
+**Claim (verbatim):** \textbf{Funding} The author received no external funding.
+
+**Notes (fidelity):**
+- This is an administrative section; no technical claims beyond reproducibility statements.
+
+**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)
+
+
+---
+
+
+
+
+### 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.
+
+**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)
+
+
+---
+
+
+
+
+### 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).
+
+**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)
+
+
+---
+
+
+
+
+### 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.
+
+**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)
+
+
+---
+
+
+
+
+### 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.
+
+**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)
+
+
+---
+
+
+
+
+### 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.
+
+**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)
+
+
+---
+
+
+
+
+## Formula appendix: all displayed formulas
+
+
+This appendix lists every displayed formula (both `equation` environments and `\[ ... \]` display blocks) in source order.
+
+
+
+
+
+### F001
+
+
+- **Source location:** [Abstract](#annex-abstract)
+
+
+$$
+L_p=-\tfrac{d^2}{dr^2}+\tfrac{\ell(\ell+1)}{r^2}+\tfrac{C}{r^p}
+$$
+
+
+
+
+### F002 (`\label{eq:Lichnerowicz}`)
+
+
+- **Source location:** [1. Introduction](#annex-sec-01-introduction)
+
+
+$$
+L = \nabla^{*}\nabla + V_R,
+\qquad (V_R h)_{ij} = -R^{\;\ell}{}_{i j m}\, h_{\ell}{}^{m},
+\label{eq:Lichnerowicz}
+$$
+
+
+
+
+### F003
+
+
+- **Source location:** [1. Introduction](#annex-sec-01-introduction)
+
+
+$$
+L_p = -\frac{d^2}{dr^2} + \frac{\ell(\ell+1)}{r^2} + \frac{C}{r^p},
+$$
+
+
+
+
+### 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)
+
+
+$$
+g_{ij}=\delta_{ij}+a_{ij},\qquad
+a_{ij}=O(r^{-1}),\quad
+\partial_k a_{ij}=O(r^{-2}),\quad
+\partial_\ell\partial_k a_{ij}=O(r^{-3}),
+$$
+
+
+
+
+### 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)
+
+
+$$
+S_{\text{EH}}=\frac{1}{16\pi G}\int R\sqrt{-g}\,d^4x
+\label{eq:EH_action}
+$$
+
+
+
+
+### 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)
+
+
+$$
+S^{(2)}[h]=\tfrac{1}{2}\!\int_\Sigma h^{ij}L_{ij}{}^{kl}h_{kl}\sqrt{g}\,d^3x,
+$$
+
+
+
+
+### 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)
+
+
+$$
+Lh = \nabla^*\nabla h + V_R h,
+\qquad (V_Rh)_{ij}=-R_i{}^\ell{}_j{}^m h_{\ell m}.
+\label{eq:Lichnerowicz_def}
+$$
+
+
+
+
+### 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)
+
+
+$$
+\Delta_A = -(\nabla_A)^*\nabla_A = -\nabla^*\nabla + {\rm ad}(F_A),
+\label{eq:YMlap}
+$$
+
+
+
+
+### 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)
+
+
+$$
+H^k_\delta(\Sigma;S^2T^*\Sigma)
+=\Bigl\{h\in H^k_{\mathrm{loc}}(\Sigma):\|h\|_{H^k_\delta}<\infty\Bigr\},\qquad
+\|h\|_{H^k_\delta}^2=\sum_{j=0}^k\!\!\int_\Sigma\langle r\rangle^{2(\delta-j)}|\nabla^jh|_g^2\,dV_g.
+$$
+
+
+
+
+### 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)
+
+
+$$
+\langle Lh,k\rangle_{L^2}
+=\langle\nabla h,\nabla k\rangle_{L^2}
++\langle V_Rh,k\rangle_{L^2},
+\label{eq:int_identity}
+$$
+
+
+
+
+### 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)
+
+
+$$
+L:H^2_\delta(\Sigma;S^2T^*\Sigma)\longrightarrow L^2_{\delta-2}(\Sigma;S^2T^*\Sigma)
+\label{eq:L_mapping2}
+$$
+
+
+
+
+### 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)
+
+
+$$
+\sigma_{\mathrm{ess}}(L)=[0,\infty).
+\label{eq:ess_flat}
+$$
+
+
+
+
+### 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)
+
+
+$$
+p_{\mathrm{crit}}=3,
+\label{eq:pcrit}
+$$
+
+
+
+
+### F014 (`\label{eq:vectorlaplacian}`)
+
+
+- **Source location:** [3. Infrared Spectrum and Marginal Modes](#annex-sec-03-infrared-spectrum-and-marginal-modes)
+
+
+$$
+\Delta_V=\nabla^*\nabla+\mathrm{Ric},
+\label{eq:vectorlaplacian}
+$$
+
+
+
+
+### F015 (`\label{eq:vectorfredholm}`)
+
+
+- **Source location:** [3. Infrared Spectrum and Marginal Modes](#annex-sec-03-infrared-spectrum-and-marginal-modes)
+
+
+$$
+\Delta_V: H^2_\delta(\Sigma;T^*\Sigma)\longrightarrow L^2_{\delta-2}(\Sigma;T^*\Sigma)
+\label{eq:vectorfredholm}
+$$
+
+
+
+
+### F016 (`\label{eq:metricdecay}`)
+
+
+- **Source location:** [3. Infrared Spectrum and Marginal Modes](#annex-sec-03-infrared-spectrum-and-marginal-modes)
+
+
+$$
+g_{ij}=\delta_{ij}+O(r^{-1}),\qquad
+\partial g_{ij}=O(r^{-2}),\qquad
+\partial^2 g_{ij}=O(r^{-3}),
+\label{eq:metricdecay}
+$$
+
+
+
+
+### F017 (`\label{eq:lichcritical}`)
+
+
+- **Source location:** [3. Infrared Spectrum and Marginal Modes](#annex-sec-03-infrared-spectrum-and-marginal-modes)
+
+
+$$
+Lh = \nabla^*\nabla h + V_R h,
+\qquad (V_R h)_{ij}=-R_{i\ \ j m}^{\ \ell}h_\ell^{\ m},
+\label{eq:lichcritical}
+$$
+
+
+
+
+### F018 (`\label{eq:weylansatz}`)
+
+
+- **Source location:** [3. Infrared Spectrum and Marginal Modes](#annex-sec-03-infrared-spectrum-and-marginal-modes)
+
+
+$$
+h_n(r,\omega)=A_n\,\phi_n(r)\,r^{-1}H_{ij}(\omega),
+\label{eq:weylansatz}
+$$
+
+
+
+
+### F019
+
+
+- **Source location:** [3. Infrared Spectrum and Marginal Modes](#annex-sec-03-infrared-spectrum-and-marginal-modes)
+
+
+$$
+\|\tilde h_n\|_{L^2}=1,\qquad
+\nabla^j\tilde h_{n,ij}=0,\qquad
+\|L\tilde h_n\|_{L^2}\to0.
+$$
+
+
+
+
+### F020 (`\label{eq:criticalspectrum}`)
+
+
+- **Source location:** [3. Infrared Spectrum and Marginal Modes](#annex-sec-03-infrared-spectrum-and-marginal-modes)
+
+
+$$
+[0,\infty)\subset\sigma_{\mathrm{ess}}(L),\qquad 0\in\sigma_{\mathrm{ess}}(L).
+\label{eq:criticalspectrum}
+$$
+
+
+
+
+### F021
+
+
+- **Source location:** [3. Infrared Spectrum and Marginal Modes](#annex-sec-03-infrared-spectrum-and-marginal-modes)
+
+
+$$
+\Box_g h_{\mu\nu} + 2R_{\mu\ \nu}^{\ \rho\ \sigma}h_{\rho\sigma}=0,
+$$
+
+
+
+
+### F022
+
+
+- **Source location:** [4. Dimensional Scaling and Spectral Phase Structure](#annex-sec-04-dimensional-scaling-and-spectral-phase-structure)
+
+
+$$
+L_d=-\nabla^2 + V(r),
+$$
+
+
+
+
+### F023
+
+
+- **Source location:** [4. Dimensional Scaling and Spectral Phase Structure](#annex-sec-04-dimensional-scaling-and-spectral-phase-structure)
+
+
+$$
+\int_R^\infty r^{\,2(\delta-2)}\,|V(r)|^2\,|h(r)|^2\,r^{\,d-1}dr
+\;\sim\;
+\int_R^\infty r^{\,d-5+2\delta-2p}\,dr.
+$$
+
+
+
+
+### F024 (`\label{eq:pcrit_general}`)
+
+
+- **Source location:** [4. Dimensional Scaling and Spectral Phase Structure](#annex-sec-04-dimensional-scaling-and-spectral-phase-structure)
+
+
+$$
+p_{\mathrm{crit}}=d,
+\label{eq:pcrit_general}
+$$
+
+
+
+
+### F025 (`\label{eq:penalty}`)
+
+
+- **Source location:** [5. Numerical Verification of the Spectral Threshold](#annex-sec-05-numerical-verification-of-the-spectral-threshold)
+
+
+$$
+\mathcal{R}_{\eta,\zeta}[h]
+=
+\frac{\langle h,Lh\rangle
++\eta\,\|\nabla\!\cdot h\|_{L^2(\Omega)}^2
++\zeta\,\|\mathrm{tr}\,h\|_{L^2(\Omega)}^2}
+{\|h\|_{L^2(\Omega)}^2},
+\label{eq:penalty}
+$$
+
+
+
+
+### F026
+
+
+- **Source location:** [5. Numerical Verification of the Spectral Threshold](#annex-sec-05-numerical-verification-of-the-spectral-threshold)
+
+
+$$
+L_p = -\frac{d^2}{dr^2} + \frac{\ell(\ell+1)}{r^2} + \frac{C}{r^p},
+$$
+
+
+
+
+### F027
+
+
+- **Source location:** [5. Numerical Verification of the Spectral Threshold](#annex-sec-05-numerical-verification-of-the-spectral-threshold)
+
+
+$$
+E[\phi]=
+\frac{\int_{R}^{2R}\!\bigl(|\phi'(r)|^2+V_p(r)|\phi(r)|^2\bigr)r^2dr}
+ {\int_{R}^{2R}\!|\phi(r)|^2r^2dr},
+\qquad
+\Delta E(R,p)=E[\phi]-E_{\mathrm{free}}[\phi],
+$$
+
+
+
+
+### F028
+
+
+- **Source location:** [7. Relation to Previous Work and Threshold Phenomena](#annex-sec-07-relation-to-previous-work-and-threshold-phenomena)
+
+
+$$
+\Delta_A = -(\nabla_A)^*\nabla_A = -\nabla^*\nabla + {\rm ad}(F_A),
+$$
+
+
+
+
+### F029
+
+
+- **Source location:** [8. Conclusion](#annex-sec-08-conclusion)
+
+
+$$
+\sigma_{\mathrm{ess}}(L)=[0,\infty).
+$$
+
+
+
+
+### F030
+
+
+- **Source location:** [Appendix A. Gauge Correction and Elliptic Estimates](#annex-app-a-gauge-correction-and-elliptic-estimates)
+
+
+$$
+\Delta_V X = \nabla^*\nabla X + {\rm Ric}(X)
+$$
+
+
+
+
+### F031
+
+
+- **Source location:** [Appendix A. Gauge Correction and Elliptic Estimates](#annex-app-a-gauge-correction-and-elliptic-estimates)
+
+
+$$
+\Delta_V:H^2_\delta(\Sigma;T^*\Sigma)\to L^2_{\delta-2}(\Sigma;T^*\Sigma)
+$$
+
+
+
+
+### F032
+
+
+- **Source location:** [Appendix A. Gauge Correction and Elliptic Estimates](#annex-app-a-gauge-correction-and-elliptic-estimates)
+
+
+$$
+\|\tilde h_n\|_{L^2}=1,\qquad
+\tilde h_n\rightharpoonup0,\qquad
+\|L\tilde h_n\|_{L^2}\to0.
+$$
+
+
+
+
+### F033
+
+
+- **Source location:** [Appendix B. Curvature Structure and the Schwarzschild Example](#annex-app-b-curvature-structure-and-the-schwarzschild-example)
+
+
+$$
+ds^2=-\Bigl(\frac{1-\tfrac{M}{2r}}{1+\tfrac{M}{2r}}\Bigr)^2 dt^2
+ +\Bigl(1+\frac{M}{2r}\Bigr)^4(dr^2+r^2d\omega^2).
+$$
+
+
+
+
+### F034
+
+
+- **Source location:** [Appendix B. Curvature Structure and the Schwarzschild Example](#annex-app-b-curvature-structure-and-the-schwarzschild-example)
+
+
+$$
+|{\rm Riem}(x)|\simeq C\,M\,r^{-3}\qquad(r\to\infty),
+$$
+
+
+
+
+### F035
+
+
+- **Source location:** [Appendix B. Curvature Structure and the Schwarzschild Example](#annex-app-b-curvature-structure-and-the-schwarzschild-example)
+
+
+$$
+Lh = \Delta_0 h - (C M) r^{-3} h + O(r^{-4})h,
+\qquad C M > 0,
+$$
+
+
+
+
+### F036
+
+
+- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)
+
+
+$$
+\|u\|_{H^k_\delta}^2
+ = \sum_{|\alpha|\le k}
+ \int_\Sigma (1+r^2)^{\delta-|\alpha|}
+ |\nabla^\alpha u|^2\,dV_g.
+$$
+
+
+
+
+### F037
+
+
+- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)
+
+
+$$
+P:H^2_\delta\to L^2_{\delta-2}
+$$
+
+
+
+
+### F038
+
+
+- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)
+
+
+$$
+g_{ij} = \delta_{ij} + O(r^{-1}),
+\qquad
+\partial g_{ij} = O(r^{-2}),
+\qquad
+\partial^2 g_{ij} = O(r^{-3}),
+$$
+
+
+
+
+### F039
+
+
+- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)
+
+
+$$
+L = \nabla^{*}\nabla + V_R,
+\qquad
+(V_R h)_{ij} = -R_{i}{}^{\ell}{}_{j}{}^{m}\,h_{\ell m}.
+$$
+
+
+
+
+### F040
+
+
+- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)
+
+
+$$
+V_R : H^2_\delta(\Sigma;S^2T^*\Sigma) \longrightarrow L^2_{\delta-2}(\Sigma;S^2T^*\Sigma),
+$$
+
+
+
+
+### F041
+
+
+- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)
+
+
+$$
+\sigma_{\mathrm{ess}}(L)=\sigma_{\mathrm{ess}}(\nabla^{*}\nabla)=[0,\infty).
+$$
+
+
+
+
+### F042
+
+
+- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)
+
+
+$$
+\|V_R h\|_{L^2_{\delta-2}}^2
+=
+\int_\Sigma \langle r\rangle^{2(\delta-2)}\,|V_R(x)h(x)|^2\,dV_g
+\lesssim
+\int_\Sigma \langle r\rangle^{2(\delta-2)-2p}|h(x)|^2\,dV_g.
+$$
+
+
+
+
+### F043
+
+
+- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)
+
+
+$$
+V_R=\chi_R V_R+(1-\chi_R)V_R
+=:V_R^{\mathrm{(comp)}}+V_R^{\mathrm{(tail)}}.
+$$
+
+
+
+
+### F044
+
+
+- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)
+
+
+$$
+\|V_R^{\mathrm{(tail)}}h\|_{L^2_{\delta-2}}^2
+\lesssim
+\int_{r>R}\langle r\rangle^{2(\delta-2)-2p}|h(x)|^2\,dV_g.
+$$
+
+
+
+
+### F045
+
+
+- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)
+
+
+$$
+\sup_{\|h\|_{H^2_\delta}=1}
+\|V_R^{\mathrm{(tail)}}h\|_{L^2_{\delta-2}}
+\longrightarrow 0
+\quad\text{as }R\to\infty.
+$$
+
+
+
+
+### F046
+
+
+- **Source location:** [Appendix C. Analytical Framework and Weighted Sobolev Spaces](#annex-app-c-analytical-framework-and-weighted-sobolev-spaces)
+
+
+$$
+\sigma_{\mathrm{ess}}(L)=\sigma_{\mathrm{ess}}(\nabla^{*}\nabla)=[0,\infty).
+$$
+
+
+
+
+### F047
+
+
+- **Source location:** [Appendix D. Numerical Validation and Stability Tests](#annex-app-d-numerical-validation-and-stability-tests)
+
+
+$$
+\begin{array}{ll}
+\text{flat:} & \lambda_1 = 0.07386996,\quad \lambda_2 = 0.14713361, \\
+p=2.5: & \lambda_1 = 0.06505331,\quad \lambda_2 = 0.07023744, \\
+p=3.0: & \lambda_1 = 0.07114971,\quad \lambda_2 = 0.07276003, \\
+p=3.5: & \lambda_1 = 0.07298806,\quad \lambda_2 = 0.07351824.
+\end{array}
+$$
+
+
+
+
+### F048
+
+
+- **Source location:** [Appendix D. Numerical Validation and Stability Tests](#annex-app-d-numerical-validation-and-stability-tests)
+
+
+$$
+\begin{array}{ll}
+\text{flat:} & \lambda_1 = 0.07398399,\quad \lambda_2 = 0.14781594, \\
+p=2.5: & \lambda_1 = 0.06505635,\quad \lambda_2 = 0.07029545, \\
+p=3.0: & \lambda_1 = 0.07109678,\quad \lambda_2 = 0.07279109, \\
+p=3.5: & \lambda_1 = 0.07295142,\quad \lambda_2 = 0.07355686.
+\end{array}
+$$
+
+
+
+
+### F049
+
+
+- **Source location:** [Appendix D. Numerical Validation and Stability Tests](#annex-app-d-numerical-validation-and-stability-tests)
+
+
+$$
+\lambda_1(p{=}2.5)
+<
+\lambda_1(p{=}3.0)
+<
+\lambda_1(p{=}3.5)
+\simeq
+\lambda_1(\text{flat})
+$$
+
+
+
+
+### F050
+
+
+- **Source location:** [Appendix D. Numerical Validation and Stability Tests](#annex-app-d-numerical-validation-and-stability-tests)
+
+
+$$
+\lambda_1(R_{\max}{=}6)=0.1965,\quad
+\lambda_1(R_{\max}{=}10)=0.0711,\quad
+\lambda_1(R_{\max}{=}20)=0.0180.
+$$
+
+
+
+
+### 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)
+
+
+$$
+\|h_n\|_{L^2}^2\!\simeq\!A_n^2\!\!\int_{n/2}^{2n}\!r^{-2}r^2dr
+\sim A_n^2 n.
+$$
+
+
+
+
+### 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)
+
+
+$$
+\nabla^*\nabla(\phi_n r^{-1}H)
+= \phi_n\nabla^*\nabla(r^{-1}H)
+ + 2\nabla\phi_n\!\cdot\!\nabla(r^{-1}H)
+ + (\Delta\phi_n)r^{-1}H,
+$$
+
+
+
+
+### 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)
+
+
+$$
+\|[\nabla^*\nabla,\phi_n]h_n\|_{L^2}
+\lesssim n^{-2}.
+$$
+
+
+
+
+### 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)
+
+
+$$
+\|V_R h_n\|_{L^2}^2
+\lesssim A_n^2\!\int_{n/2}^{2n}\!r^{-6}r^2dr
+\sim n^{-4},
+\qquad
+\|V_R h_n\|_{L^2}\lesssim n^{-2}.
+$$
+
+
+
+
+### 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)
+
+
+$$
+\|X_n\|_{H^2_\delta}\lesssim
+\|\nabla\!\cdot h_n\|_{L^2}
+\lesssim n^{-1},
+\qquad -1<\delta<0.
+$$
+
+
+
+
+### 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)
+
+
+$$
+\|L(\mathcal{L}_{X_n}g)\|_{L^2}\lesssim n^{-1},
+$$
+
+
+
+
+### 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)
+
+
+$$
+\|L\tilde h_n\|_{L^2}\to0,
+\qquad
+\|\tilde h_n\|_{L^2}=1.
+$$
+
+
+---
+
+
+
+
+## Annex: original source
+
+
+This annex contains the source material extracted from the provided `arXiv-2511.05345v1.tar.gz` bundle.
+
+
+
+
+
+### 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)
+
+
+---
+
+
+
+
+### 00README.json (verbatim)
+
+
+```json
+{
+ "sources" : [
+ {
+ "usage" : "toplevel",
+ "filename" : "main.tex"
+ }
+ ],
+ "spec_version" : 1,
+ "texlive_version" : "2025",
+ "process" : {
+ "compiler" : "pdflatex"
+ }
+}
+
+```
+
+
+
+---
+
+
+
+
+### 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
+
+
+```
+
+
+
+---
+
+
+
+
+### 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}
+
+
+```
+
+
+
+---
+
+
+
+
+### 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}.
+
+
+```
+
+
+
+---
+
+
+
+
+### 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.
+
+
+```
+
+
+
+---
+
+
+
+
+### 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
+
+### 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.
+
+
+```
+
+
+
+---
+
+
+
+
+### 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}
+
+
+```
+
+
+
+---
+
+
+
+
+### 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.
+
+
+```
+
+
+
+---
+
+
+
+
+### 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}
+
+
+```
+
+
+
+---
+
+
+
+
+### 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}
+
+
+```
+
+
+
+---
+
+
+
+
+### 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
+
+```
+
+
+
+---
+
+
+
+
+### 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}
+
+
+```
+
+
+
+---
+
+
+
+
+### 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}
+
+
+```
+
+
+
+---
+
+
+
+
+### 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}
+
+
+```
+
+
+
+---
+
+
+
+
+### 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.
+
+
+```
+
+
+
+---
+
+
+
+
+### 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