# The r⁻³ Curvature Decay Threshold in Linearized Gravity
## A receipt-first, definition-preserving rewrite (readable without a GR/QC background)

**Source (arXiv abstract page):** https://arxiv.org/abs/2511.05345

**Source (PDF):** https://arxiv.org/pdf/2511.05345.pdf

**Local PDF SHA-256:** `992e50be5f84477fdbcf72ddea26bec88e6e1b756d948161af1b71bf03fc6742`

**Author (paper):** Michael Wilson (University of Arkansas at Little Rock)

**Date (paper):** November 10, 2025

---

> Most dossiers read as if physics is optional: eternal ROI, frictionless execution, boundless optimism. We strip away the gloss to reveal the operational reality.

This is a clean-room rewrite of the paper’s *technical spine*.

- It preserves the paper’s **definitions**, **operators**, and **scaling statements**.
- It replaces local jargon with **plain-language interpretations**, so you can track what the math says without already being fluent in the field.
- It keeps a strict separation between **what the paper states** and **what that would imply if true**.

---

## Quick digest (for people who do not want to read 13 pages)

**Context:** Linearized gravity is the “small perturbations” approximation. The paper asks a narrow but important question: what *asymptotic* geometric decay rate separates “fully radiative” behavior from “infrared / memory-like” behavior on a spatial slice.

> The claim is that `|Riem| ~ r^-3` is the sharp border.

### What the paper claims
- If curvature decays faster than `r^-3`, the relevant spatial operator behaves like the flat operator: extended tensor modes disperse and there is no special zero-energy sector.
- At exactly `r^-3`, compactness fails and **zero energy enters the essential spectrum**, producing **marginally extended, finite-energy** modes.
- A simplified radial model reproduces the same transition at `p = 3`, and a dimensional argument suggests a general rule `p_crit = d`.

### Why this matters (without invoking mysticism)
If you can tie “long-range gravitational memory / soft modes” to a concrete spatial decay threshold, you get a clean structural statement:

- below the threshold: the far field can keep “persistent correlations,”
- above the threshold: the far field behaves like ordinary radiating waves.

That is not a claim about all of nonlinear gravity. It is a claim about the linear operator that controls stationary perturbations in harmonic gauge.

### Two ways to read this rewrite (pick one)

1) **Fast comprehension path (skip proofs, keep definitions):** Read Sections 1, 3.3, 6, and the “What to do with this” checklist. You will still see every operator and hypothesis.

2) **Definition-preserving path (follow the logic):** Read Sections 1–5 in order. This is the “operator → spectrum → threshold → numerical checks” route.

### Key objects (definitions you must know to follow the argument)

| Object | Definition (paper) | What it controls | What can go wrong
|---|---|---|---|
| Spatial Lichnerowicz operator | `L = ∇*∇ + V_R` | Stationary (time-independent) harmonic-gauge perturbations | Threshold behavior depends on decay of `V_R`
| Curvature potential | `(V_R h)_{ij} = -R^\ell_{ijm} h_{\ell m}` | How background curvature couples into the spin-2 operator | Compactness can fail at infinity
| Essential spectrum | `σ_ess(L)` | Extended (non-localized) spectral behavior | Whether `0` lies in `σ_ess(L)` is the pivot
| Weyl sequence | `{h_n}` with `||h_n||=1`, `h_n ⇀ 0`, `||L h_n|| → 0` | A way to prove `0 ∈ σ_ess(L)` | Requires careful gauge correction
| Radial model | `L_p = -d^2/dr^2 + ℓ(ℓ+1)/r^2 + C/r^p` | A tractable “one channel” proxy for scaling | Must not be oversold as the full operator

---

## 1. Introduction (what problem is being solved)

> The paper is trying to turn “infrared structure” from a slogan into a spectral criterion.

The starting tension is simple:

- There are well-known long-range gravitational phenomena (memory, tails, soft modes).
- Most of the clean theory is phrased at null infinity (asymptotic symmetry frameworks).
- It is less clear what a purely spatial (Cauchy slice) mechanism is, and specifically how **curvature decay** controls the existence of persistent low-frequency structure.

The paper proposes a criterion based on a single operator:

```text
L = ∇*∇ + V_R,

(V_R h)_{ij} = -R^\ell_{ijm} h_{\ell m}.
```

It is explicit about the intended interpretation:

- `σ_ess(L) = [0,∞)` corresponds to freely propagating tensor modes.
- `0 ∈ σ_ess(L)` corresponds to marginally bound, spatially extended, finite-energy modes.

The paper’s central threshold claim (already in the abstract) is:

```text
|Riem| ~ r^-3  is the sharp spectral threshold in 3 spatial dimensions.
```

### How the paper says it will proceed (section map)

The paper itself says the structure is:

- Section 2: analytic framework and the scaling principle shared by gauge and gravity operators.
- Section 3: the critical decay regime and the “zero enters the essential spectrum” result.
- Section 4: general dimension scaling and `p_crit = d`.
- Section 5: numerical verification: a radial scaling test plus a full 3D discretized tensor eigenvalue calculation.

---

## 2. Spectral scaling and structural parallels across spin‑1 and spin‑2 fields

> Two different theories can share the same infrared scaling mechanism because the geometry forces it.

### 2.1 Geometric setup and harmonic gauge (paper definitions)

The spatial manifold `(Σ, g)` is asymptotically flat. In an asymptotic coordinate chart:

```text
g_{ij} = δ_{ij} + a_{ij}

a_{ij} = O(r^-1)
∂_k a_{ij} = O(r^-2)
∂_ℓ ∂_k a_{ij} = O(r^-3)
```

The paper notes that these falloff conditions imply:

- `Γ^k_{ij} = O(r^-2)`
- `|Riem| = O(r^-3)`

The operator `L` is defined to act on symmetric trace-free tensor fields `h_{ij}`. (The paper works in harmonic gauge; the details are enforced analytically via a vector Laplacian and gauge correction later.)

### 2.2 Shared scaling structure (spin‑1 analogue)

The paper draws a structural parallel to Yang–Mills:

```text
Δ_A = -(∇_A)* ∇_A = -∇*∇ + ad(F_A).
```

Both the spin‑1 and spin‑2 operators have the same schematic form:

- Laplace term
- plus a curvature-induced potential term that decays like `r^-p`

The claim is not “the theories are the same.” It is “the *spectral scaling* is governed by the same dimensional mechanism.”

### 2.3 Weighted Sobolev spaces + symmetry identity (what gets used later)

The analysis is carried out on weighted spaces with weight `δ`:

```text
H^k_δ(Σ; S^2 T*Σ) = { h in H^k_loc : ||h||_{H^k_δ} < ∞ }

||h||^2_{H^k_δ} = Σ_{j=0..k} ∫_Σ <r>^{2(δ-j)} |∇^j h|^2_g dV_g.
```

Where the shorthand `<r>` is the standard “one plus radius” weight:

```text
<r> = (1 + r^2)^{1/2}.
```

An integration-by-parts identity makes `L` explicitly symmetric on compactly supported smooth tensors:

```text
⟨Lh, k⟩_{L^2} = ⟨∇h, ∇k⟩_{L^2} + ⟨V_R h, k⟩_{L^2}.
```

### 2.4 Self-adjoint realization + essential spectrum in the fast-decay regime

The paper states (for `-1 < δ < 0`) that the mapping

```text
L : H^2_δ → L^2_{δ-2}
```

is bounded, and that `L = ∇*∇ + V_R` is self-adjoint on the same domain as `∇*∇` under decay assumptions.

If curvature decays faster than `r^-3` (i.e. `p > 3`), `V_R` is compact in the relevant mapping, and Weyl’s theorem gives:

```text
σ_ess(L) = [0, ∞).
```

### 2.5 Why “3” appears (the core scaling argument)

The borderline value is obtained by comparing:

- the `r^-2` angular term induced by the Laplacian after spherical harmonic decomposition,
- to a curvature tail `V_R ~ r^-p`.

The paper summarizes the outcome as:

```text
p_crit = 3  (in three spatial dimensions).
```

This is the first place where the narrative should snap into focus:

- above 3: curvature tail is spectrally negligible in the weighted setting,
- at 3: curvature and dispersion balance,
- below 3: curvature becomes long-range in the spectral sense.

---

## 3. Infrared spectrum and marginal modes (what “zero in the essential spectrum” means)

> This is the paper’s most concrete mathematical assertion: it constructs the mechanism that makes `0` appear.

### 3.1 Fredholm + gauge framework (so the “harmonic gauge” assumption is not hand-waved)

A key analytic tool is the vector Laplacian acting on one-forms:

```text
Δ_V = ∇*∇ + Ric.
```

The paper states a weighted Fredholm property:

```text
Δ_V : H^2_δ(Σ; T*Σ) → L^2_{δ-2}(Σ; T*Σ)

is Fredholm and invertible for -1 < δ < 0.
```

Operationally, this gives a well-posed “gauge correction” step: you can solve for a vector field `X` to correct a candidate tensor field so it satisfies the divergence constraint.

#### Lemma 1 (paper statement; verbatim structure retained)

For `−1 < δ < 0`:

```text
Δ_V : H^2_δ(Σ; T*Σ) → L^2_{δ-2}(Σ; T*Σ)
```

is Fredholm with bounded inverse. The paper uses this to justify solving `Δ_V X = f` with the estimate `||X||_{H^2_δ} ≤ C ||f||_{L^2_{δ-2}}`.

### 3.2 Weyl sequence at the critical decay (the construction)

Assume asymptotic flatness with falloff

```text
g_{ij} = δ_{ij} + O(r^-1)
∂g_{ij} = O(r^-2)
∂^2 g_{ij} = O(r^-3)
```

and curvature tail

```text
|Riem(x)| ≃ C r^-3   as r → ∞.
```

Then `L` is written again on `L^2(Σ; S^2 T*Σ)`:

```text
L h = ∇*∇ h + V_R h

(V_R h)_{ij} = -R_{i\ell jm} h_{\ell m}.
```

The approximate zero-mode ansatz is (paper definition):

```text
h_n(r, ω) = A_n  φ_n(r)  r^-1  H_{ij}(ω)

φ_n(r) = cutoff equal to 1 on [n, 3n/2] and vanishing outside [n/2, 2n]
||h_n||_{L^2} = 1  ⇒  A_n ≃ n^-1/2.
```

#### Lemma 2 (what the paper is asserting, operationally)

The paper’s “approximate zero mode” claim is:

- If you build `h_n` supported on a large annulus and normalize it (`||h_n||=1`),
- then the action of `L` on it becomes small (`||L h_n|| → 0`) as the annulus goes to infinity,
- and you can fix the divergence constraint by solving `Δ_V X_n = ∇·h_n` and subtracting a Lie derivative term `L_{X_n} g`.

The algebraic content is the triple condition:

```text
||\tilde h_n||_{L^2} = 1,
∇^j \tilde h_{n,ij} = 0,
||L \tilde h_n||_{L^2} → 0.
```

The paper’s proof sketch (as extracted) assigns the scale:

- derivatives of `φ_n` contribute factors `n^-1`,
- giving `||∇·h_n||_{L^2} ≲ n^-1` and `||L h_n||_{L^2} ≲ n^-2`,
- and then the gauge correction yields `||L_{X_n} g||_{L^2} ≲ n^-1`.

Then the paper performs a gauge correction:

- solve `Δ_V X_n = ∇·h_n`
- define a corrected sequence `\tilde h_n = h_n - L_{X_n} g`.

The conclusion is the Weyl sequence criterion:

```text
||\tilde h_n||_{L^2} = 1,
∇^j \tilde h_{n,ij} = 0,
||L \tilde h_n||_{L^2} → 0.
```

This implies:

```text
0 ∈ σ_ess(L)
```

in the critical decay regime.

#### Theorem 1 (paper statement; what it buys you)

The paper’s formal conclusion for the critical decay regime is:

```text
[0, ∞) ⊂ σ_ess(L),
0 ∈ σ_ess(L).
```

This is the paper’s internal definition of “infrared continuum onset”: `0` is no longer excluded from the essential spectrum when the curvature tail decays like `r^-3`.

### 3.3 A plain-English interpretation of the Weyl sequence claim (what should stick)

You can read “`0 ∈ σ_ess(L)`” as:

- You can build a sequence of increasingly far-out tensor configurations,
- each with finite (normalized) `L^2` energy,
- whose “operator energy” `||L h||` goes to zero,
- meaning the operator behaves like it has “almost-static” extended modes at arbitrarily large radius.

This is not the same as saying “there is a normalizable bound state.” The paper is describing a marginal, continuum-edge phenomenon, not a discrete eigenvalue below zero.

---

## 4. Dimensional scaling and spectral phase structure (generalization)

> The paper claims the “3” is not accidental; it is `d`.

The paper introduces a dimensional generalization:

```text
L_d = -∇^2 + V(r),   with  V(r) ~ r^-p  on a d-dimensional asymptotically flat manifold.
```

### Proposition (paper statement)

The key statement is:

```text
V is a compact perturbation of Δ  ⇔  p > d.

p = d is the threshold between short- and long-range behavior.
```

The paper explains why a naive `L^2` or Hilbert–Schmidt test is too strong, and why the weighted mapping and asymptotic mode scaling matter.

#### The proof mechanism (as stated in the paper)

The paper’s proof outline has two key moves:

1) “`V → 0` at infinity implies compact multiplication on unweighted spaces” is **not** the right test here, because the operator is being controlled on weighted mappings appropriate to asymptotically flat ends.

2) In the weighted space, compactness fails exactly when the tail contributions fail to vanish at large radius. For the asymptotic tensor mode scaling, the paper states:

```text
h(r) ~ r^{-(d-2)/2}
```

and then estimates the weighted norm of `V h` on `{r > R}` by an integral of the form:

```text
∫_R^∞ r^{d - 5 + 2δ - 2p} dr
```

Because `−1 < δ < 0`, the paper concludes the integral diverges when `p ≤ d`, implying noncompact behavior in the curvature tail.

If you only keep one operational takeaway from Section 4, it is:

- the threshold depends on how the asymptotic modes scale in the weighted space,
- and that scaling drives the integral that converges or diverges in the tail.

In 3 dimensions, this reduces back to `p_crit = 3`.

---

## 5. Numerical verification of the spectral threshold (what was actually computed)

> The numerical section is not “just plots.” It has explicit discretization choices and convergence targets.

### 5.1 Numerical setup and nondimensionalization (paper parameters)

The full tensor operator `L = ∇*∇ + V_R` is discretized on a uniform Cartesian grid:

```text
Ω = [-R_max, R_max]^3
```

with spacing `h = Δx / L_0` after nondimensionalization by a fixed length scale `L_0`.

- Centered finite differences approximate `∇` and `∇*∇`.
- Dirichlet boundary conditions impose `h|_{∂Ω} = 0`.
- The continuum approach is monitored by extrapolation in `R_max`.

To enforce transverse–traceless constraints, the paper uses a penalty functional:

```text
⟨h, Lh⟩ + η ||∇·h||^2_{L^2(Ω)} + ζ ( ... )
```

and reports that varying penalties `η, ζ` by factors `2–4` shifts the lowest eigenvalue by less than `10^-3`.

#### The exact penalty functional (paper equation (16))

The paper’s explicit form of the penalized Rayleigh quotient is:

```text
R_{η,ζ}[h] = ( ⟨h, Lh⟩ + η ||∇· h||^2_{L^2(Ω)} + ζ ||tr h||^2_{L^2(Ω)} ) / ||h||^2_{L^2(Ω)}.
```

It then states the associated discrete generalized eigenproblem:

```text
(K + η D^⊤ D + ζ T^⊤ T) u = λ M u,
```

where `K` and `M` are stiffness and mass matrices and `D, T` are discrete divergence and trace operators.

Representative nondimensional parameters (paper list):

- `h ∈ {1.0, 0.75, 0.5}`
- `R_max ∈ {6, 10, 14, 18, 20}`
- `C = -1`
- grid sizes `N = 21–41` (up to `3.6 × 10^5` degrees of freedom)
- convergence target: relative error `10^-5`.

The paper explicitly notes that Dirichlet boundaries discretize a near-threshold continuum into “box modes” with scaling

```text
λ_1(R_max) ∝ R_max^-2.
```

This matters because it is an explicit check that the numerics are tracking continuum-edge behavior rather than a false “confinement.”

### 5.1 Definition (radial operator vs full 3D operator)

The paper defines a radial model operator (single angular channel):

```text
L_p = -d^2/dr^2 + ℓ(ℓ+1)/r^2 + C/r^p.
```

and describes the full 3D discretized operator as including curvature coupling and constraint enforcement.

It also states a channel correspondence:

```text
V_eff(r) = ℓ(ℓ+1)/r^2 + C/r^p + O(r^{-p-1}).
```

### 5.2 Rayleigh–quotient scaling test (explicit energy functional)

The paper uses an energy functional on a normalized bump function `φ(r)` supported on `[R, 2R]`:

```text
E[φ] =  ( ∫_R^{2R} ( |φ'(r)|^2 + V_p(r)|φ(r)|^2 ) r^2 dr ) / ( ∫_R^{2R} |φ(r)|^2 r^2 dr )

ΔE(R, p) = E[φ] - E_free[φ].
```

It reports a predicted scaling relationship:

```text
ΔE(R, p) ~ R^{-(p-2)}.
```

and states that fitted slopes `α(p) ≈ -(p-2)` match the analytic scaling.

### What the numerics are *actually used for*

The numerics support two claims:

1) In the radial model, the lowest eigenvalue approaches the continuum threshold near `p = 3`.

2) In the full 3D discretized tensor operator, the same transition appears without evidence of discrete bound-state formation.

The important “interpretation constraint” is:

- the numerics validate a **continuous spectral crossover** centered at `p = 3`, not a dramatic phase change.

---

## 6. Physical interpretation and implications (what “infrared structure” means here)

> The paper is careful: it is still a classical, stationary analysis. It argues the operator already contains the seeds of the soft sector.

The paper links the operator’s spectrum to a quantized two-point function: in canonical quantization (harmonic gauge), the equal-time two-point function is the inverse of `L`, so large-distance correlations are governed by the same threshold.

It also states a regime interpretation:

- for `p > 3`: radiative propagation, fully dispersive
- for `p = 3`: marginal persistence (extended but finite-energy)
- for `p < 3`: enhanced infrared coupling (without discrete confinement in the tensor sector).

The paper draws a conceptual link to soft graviton theorems and gravitational memory, but it flags that a complete correspondence would require coupling to the time-dependent linearized Einstein equations near null infinity.

---

## 7. Relation to previous work and threshold alignment (why this is not an isolated curiosity)

> The paper’s “parallel threshold” claim is what turns a one-off scaling observation into a pattern.

The paper states an analogous threshold in non-Abelian gauge theory:

```text
Δ_A = -(∇_A)* ∇_A = -∇*∇ + ad(F_A),

|F_A| ~ r^-3  is the analogue threshold.
```

It claims that for Laplace-type operators on bundles over `R^3`, a curvature tail of order `r^-3` separates short-range radiative behavior from long-range infrared coupling.

It also ties `r^-3` to known scattering thresholds in Schrödinger-type operators.

---

## 8. Conclusion (the paper’s end state)

> If you accept the operator model, then `r^-3` is the boundary between “radiative only” and “infrared sector present” on a spatial slice.

The paper’s concluding theorem is explicit:

```text
Theorem (paper): Let |Riem(x)| ≤ C r^-p on an asymptotically flat 3-manifold.

1) p > 3  ⇒  σ_ess(L) = [0,∞).
2) p = 3  ⇒  compactness fails and a normalized Weyl sequence appears at zero energy.
3) p < 3  ⇒  curvature enhances infrared coupling (without isolated bound states in the tensor sector).
```

It adds explicit future directions: limiting absorption principle at critical rate; extension to Schwarzschild and Kerr; dynamical correspondence between spatial modes and soft/memory sector at null infinity.

---

## What to do with this (if you are not a gravitational physicist)

> Treat this as a structural result about a linear operator under asymptotic decay assumptions.

If you want to use this paper responsibly, the minimum checklist is:

1) You can restate the operator definitions (`L`, `V_R`, radial `L_p`) without distortion.
2) You can state the threshold claim with its scope (“linearized,” “harmonic gauge,” “asymptotically flat,” “spatial slice”).
3) You can distinguish “operator has marginal zero-energy Weyl sequence” from “there are bound states” (the paper says no discrete confinement is observed).
4) You can keep the generalization `p_crit = d` separate from the physically interpreted 3D statement.

*If you can’t do those four things, you are not citing the paper. You are citing the vibe.*

---

## Appendix A — Formal statements (paper text; structure preserved)

This appendix exists so you can see the “hard edges” in one place: the operator definitions and the main threshold statements.

### A.1 Operator definitions (paper equations (1), (9), (12))

```text
L = ∇*∇ + V_R,
(V_R h)_{ij} = -R^\ell_{ijm} h_{\ell m}.

Δ_V = ∇*∇ + Ric.

L h = ∇*∇ h + V_R h,
(V_R h)_{ij} = -R_{i\ell jm} h_{\ell m}.
```

### A.2 Lemma 1 (Fredholm property; as extracted)

```text
For −1 < δ < 0:
Δ_V : H^2_δ(Σ; T*Σ) → L^2_{δ-2}(Σ; T*Σ)
is Fredholm with bounded inverse.
```

### A.3 Lemma 2 (Approximate zero modes; as extracted)

```text
Let H_{ij}(ω) be symmetric, trace-free, divergence-free on S^2 and define
h_n(r, ω) = A_n ϕ_n(r) r^-1 H_{ij}(ω),
where ϕ_n = 1 on [n, 3n/2] and 0 outside [n/2, 2n].
After divergence correction using Δ_V X_n = ∇·h_n and h̃_n = h_n − L_{X_n} g:
||h̃_n||_{L^2} = 1,  ∇^j h̃_{n,ij} = 0,  ||L h̃_n||_{L^2} → 0.
```

### A.4 Theorem 1 (Onset of the infrared continuum; as extracted)

```text
Let (Σ, g) satisfy the asymptotic flatness conditions (11) with |Riem(x)| ≃ C r^-3.
Then:
[0, ∞) ⊂ σ_ess(L), and 0 ∈ σ_ess(L).
```

### A.5 Proposition 1 (Dimensional criterion; as extracted)

```text
Let ∆ be the Laplace–Beltrami operator on a d-dimensional asymptotically flat manifold,
and V(r) ~ r^-p a curvature-induced potential. Then:
V is a compact perturbation of ∆  iff  p > d.
p = d is the threshold.
```

### A.6 Theorem 2 (Spectral threshold for linearized gravity; paper end state)

```text
Let (Σ, g) be asymptotically flat with |Riem(x)| ≤ C r^-p.
1) p > 3 ⇒ σ_ess(L) = [0,∞).
2) p = 3 ⇒ compactness fails and a normalized Weyl sequence appears at zero energy.
3) p < 3 ⇒ curvature enhances infrared coupling.
```

---

## Appendix B — Numerical definitions and the actual computed objects

This appendix consolidates the “what was computed” layer so the numerics can be read as a check of the operator story, not a vibe.

### B.1 Domains, parameters, and constraints (paper Section 5.1)

```text
Ω = [-R_max, R_max]^3
h ∈ {1.0, 0.75, 0.5}
R_max ∈ {6, 10, 14, 18, 20}
C = -1
Dirichlet boundary: h|_{∂Ω} = 0
convergence: relative error 10^-5
```

TT enforcement via the penalty quotient:

```text
R_{η,ζ}[h] = ( ⟨h, Lh⟩ + η ||∇·h||^2 + ζ ||tr h||^2 ) / ||h||^2
```

and the discrete generalized eigenproblem:

```text
(K + η D^⊤ D + ζ T^⊤ T) u = λ M u.
```

### B.2 Radial model operator (paper Definition 1)

```text
L_p = -d^2/dr^2 + ℓ(ℓ+1)/r^2 + C/r^p.
```

### B.3 Rayleigh quotient functional (paper Section 5.2)

```text
E[ϕ] = ( ∫_R^{2R} ( |ϕ'(r)|^2 + V_p(r)|ϕ(r)|^2 ) r^2 dr ) / ( ∫_R^{2R} |ϕ(r)|^2 r^2 dr )
ΔE(R, p) = E[ϕ] − E_free[ϕ]
ΔE(R, p) ~ R^{-(p-2)}.
```

### B.4 Table 1 (raw extraction block)

The following is the paper’s Table 1 content as extracted into plain text. It is kept here because it defines the specific numerical values associated with the scaling check.

```text
Table 1: Rayleigh-quotient energy shift ∆E(R, p) for
C = −1. The power-law scaling with R follows ∆E ∼
R−(p−2) , confirming that p=3 behaves as the marginal
case separating decaying from saturating behavior.
p
2.00

R
∆E(R, p)
Efull
Efree
−2
−1
10 −2.85 × 10
1.50 × 10
1.54 × 10−1
−2
−2
20 −2.85 × 10
3.74 × 10
3.85 × 10−2
Continued on next page

12


Table 1: Rayleigh-quotient energy shift ∆E(R, p) for
C = −1 (continued).
p

2.50

3.00

3.50

4.00

R
40
80
160
320
640
10
20
40
80
160
320
640
10
20
40
80
160
320
640
10
20
40
80
160
320
640
10
20
40
80
160
320

∆E(R, p)
−2.85 × 10−2
−2.85 × 10−2
−2.85 × 10−2
−2.85 × 10−2
−2.85 × 10−2
−7.39 × 10−3
−5.23 × 10−3
−3.69 × 10−3
−2.61 × 10−3
−1.85 × 10−3
−1.31 × 10−3
−9.23 × 10−4
−1.92 × 10−3
−9.62 × 10−4
−4.81 × 10−4
−2.40 × 10−4
−1.20 × 10−4
−6.01 × 10−5
−3.01 × 10−5
−5.02 × 10−4
−1.78 × 10−4
−6.28 × 10−5
−2.22 × 10−5
−7.85 × 10−6
−2.78 × 10−6
−9.81 × 10−7
−1.32 × 10−4
−3.29 × 10−5
−8.23 × 10−6
−2.06 × 10−6
−5.14 × 10−7
−1.29 × 10−7

13

Efull
Efree
−3
9.36 × 10
9.63 × 10−3
−3
2.34 × 10
2.41 × 10−3
5.85 × 10−4 6.02 × 10−4
1.46 × 10−4 1.50 × 10−4
3.65 × 10−5 3.76 × 10−5
1.53 × 10−1 1.54 × 10−1
3.83 × 10−2 3.85 × 10−2
9.59 × 10−3 9.63 × 10−3
2.40 × 10−3 2.41 × 10−3
6.01 × 10−4 6.02 × 10−4
1.50 × 10−4 1.50 × 10−4
3.76 × 10−5 3.76 × 10−5
1.54 × 10−1 1.54 × 10−1
3.85 × 10−2 3.85 × 10−2
9.63 × 10−3 9.63 × 10−3
2.41 × 10−3 2.41 × 10−3
6.02 × 10−4 6.02 × 10−4
1.50 × 10−4 1.50 × 10−4
3.76 × 10−5 3.76 × 10−5
1.54 × 10−1 1.54 × 10−1
3.85 × 10−2 3.85 × 10−2
9.63 × 10−3 9.63 × 10−3
2.41 × 10−3 2.41 × 10−3
6.02 × 10−4 6.02 × 10−4
1.50 × 10−4 1.50 × 10−4
3.76 × 10−5 3.76 × 10−5
1.54 × 10−1 1.54 × 10−1
3.85 × 10−2 3.85 × 10−2
9.63 × 10−3 9.63 × 10−3
2.41 × 10−3 2.41 × 10−3
6.02 × 10−4 6.02 × 10−4
1.50 × 10−4 1.50 × 10−4
Continued on next page


Table 1: Rayleigh-quotient energy shift ∆E(R, p) for
C = −1 (continued).
p

R
640

∆E(R, p)
Efull
−8
−3.21 × 10
3.76 × 10−5

Efree
3.76 × 10−5
```

---

## Appendix C — Verbatim excerpt anchors used in the rewrite

The following are excerpted from the `pdftotext` extraction and are included to keep wording stable (they are not “new claims” added by this rewrite).

These are direct excerpts from the `pdftotext` extraction of the paper and are included so the wording is stable.

- “We identify curvature decay |Riem| ∼ 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 σess (L) = [0, ∞).”
- “At the inverse-cube rate, compactness fails and zero energy enters σess (L), yielding marginally bound, finite-energy configurations that remain spatially extended.”
- “Proposition 1 (Dimensional criterion for the critical decay rate). … V is a compact perturbation of ∆ if and only if p > d. The equality p = d marks the threshold …”
- “Theorem 2 (Spectral threshold for linearized gravity). … For p > 3 … σess(L) = [0, ∞). … At p = 3 … Weyl sequence … For p < 3 … enhances infrared coupling …”
