From e723b416e621b7565dcca5bb193816921dfba33a Mon Sep 17 00:00:00 2001
From: root <root@mtl-01.infrafabric.io>
Date: Thu, 1 Jan 2026 17:34:30 +0000
Subject: [PATCH] Add GM v3 rewrite + rendered page for arXiv 2511.05345

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diff --git a/README.md b/README.md
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 # Michael-Wilson
 
-IF.TRACE / ShadowRT related artifacts (arXiv dossiers, receipts, renders).
\ No newline at end of file
+Artifacts and rewrites related to the Michael–Wilson arXiv paper track.
+
+## arXiv 2511.05345
+- Plain-English Shadow Dossier (older): `arxiv/2511.05345/2026-01-01/`
+- GM v3 rewrite (definition-preserving English): `arxiv/2511.05345/2026-01-01-gm-v3/`
+
+Public render (GM v3):
+- `https://infrafabric.io/static/hosted/review/arxiv/2511.05345/2026-01-01-gm-v3/index.html`
diff --git a/arxiv/2511.05345/2026-01-01-gm-v3/ARTIFACTS.sha256 b/arxiv/2511.05345/2026-01-01-gm-v3/ARTIFACTS.sha256
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+992e50be5f84477fdbcf72ddea26bec88e6e1b756d948161af1b71bf03fc6742  source/2511.05345.pdf
+62509bd85c81d520d3f3c97c5c753bf6728c711f15519a8dab321eb602088407  source/2511.05345.pdf.sha256
+9b555b0f92bd067e219f47693ef8784668d1fd1171733770a40c163cf466cb52  source/2511.05345.txt
+e7b268122172257d4e2004f7d576b43ca5700e76cdd44560f9ba5f067e1c6022  generated/IF_GM_ARXIV_2511_05345.md
+c200e7d18fd0c21e5bf28603cbb3cf79a638d16a8c00f99d7ffac711ffa0c46d  rendered/index.html
+e7b268122172257d4e2004f7d576b43ca5700e76cdd44560f9ba5f067e1c6022  rendered/dossier.md
diff --git a/arxiv/2511.05345/2026-01-01-gm-v3/README.md b/arxiv/2511.05345/2026-01-01-gm-v3/README.md
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+# arXiv 2511.05345 — GM v3 rewrite (English)
+
+- Goal: definition-preserving English rewrite, using `IF.STYLE-BIBLE-v3.0GM-EN.md` guidance.
+- Scope: keep the math (operators, norms, theorem statements, numerical setup) readable for non-specialists.
+
+## Contents
+- `source/2511.05345.pdf` — source paper (PDF)
+- `source/2511.05345.txt` — extracted text (for diff/grepping)
+- `generated/IF_GM_ARXIV_2511_05345.md` — long-form rewrite (Markdown)
+- `rendered/index.html` + `rendered/dossier.md` — standalone rendered page and its Markdown payload
+- `ARTIFACTS.sha256` — checksums for all artifacts in this folder
+
+## Public rendered page
+- Rendered: `https://infrafabric.io/static/hosted/review/arxiv/2511.05345/2026-01-01-gm-v3/index.html`
+- Markdown: `https://infrafabric.io/static/hosted/review/arxiv/2511.05345/2026-01-01-gm-v3/dossier.md`
+
+## Verification
+- Local PDF SHA-256 is recorded inside the dossier and also in `source/2511.05345.pdf.sha256`.
+- `ARTIFACTS.sha256` provides file-level checksums for the full bundle in this folder.
diff --git a/arxiv/2511.05345/2026-01-01-gm-v3/generated/IF_GM_ARXIV_2511_05345.md b/arxiv/2511.05345/2026-01-01-gm-v3/generated/IF_GM_ARXIV_2511_05345.md
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+# 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 …”
diff --git a/arxiv/2511.05345/2026-01-01-gm-v3/rendered/dossier.md b/arxiv/2511.05345/2026-01-01-gm-v3/rendered/dossier.md
new file mode 100644
index 0000000..ac0e12a
--- /dev/null
+++ b/arxiv/2511.05345/2026-01-01-gm-v3/rendered/dossier.md
@@ -0,0 +1,842 @@
+# 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 …”
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+<!doctype html>
+<html lang="en">
+<head>
+  <meta charset="utf-8" />
+  <meta name="viewport" content="width=device-width, initial-scale=1" />
+  <title>InfraFabric GM v3 — arXiv:2511.05345</title>
+  <style>
+    :root {
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+    pre code{background:transparent;padding:0}
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+      <div class="badge"><b>Source</b><span>arXiv:2511.05345</span></div>
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+      <div class="panelhead">
+        <h1>InfraFabric — GM v3 rewrite (English, definition-preserving)</h1>
+        <div class="links">
+          <a class="btn" href="https://arxiv.org/abs/2511.05345" target="_blank" rel="noreferrer">arXiv abstract</a>
+          <a class="btn" href="https://arxiv.org/pdf/2511.05345.pdf" target="_blank" rel="noreferrer">PDF</a>
+          <a class="btn" href="dossier.md">Download Markdown (GM v3)</a>
+        </div>
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+      <div class="foot" style="padding:0 18px 18px">If Mermaid doesn’t render, refresh once; this page needs CDN access for marked + mermaid.</div>
+    </div>
+  </div>
+
+  <script type="text/markdown" id="md"># 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 …”
+</script>
+  <script src="https://cdn.jsdelivr.net/npm/marked/marked.min.js"></script>
+  <script src="https://cdn.jsdelivr.net/npm/mermaid@10/dist/mermaid.min.js"></script>
+  <script>
+    (function(){
+      const md = document.getElementById('md').textContent;
+      marked.setOptions({ gfm: true, breaks: false });
+      const html = marked.parse(md);
+      const content = document.getElementById('content');
+      content.innerHTML = html;
+
+      const codes = content.querySelectorAll('pre code.language-mermaid');
+      for (const code of codes) {
+        const pre = code.parentElement;
+        const div = document.createElement('div');
+        div.className = 'mermaid';
+        div.textContent = code.textContent;
+        pre.replaceWith(div);
+      }
+
+      mermaid.initialize({ startOnLoad: false, securityLevel: 'strict', theme: 'dark' });
+      mermaid.run({ querySelector: '.mermaid' }).catch(() => {});
+    })();
+  </script>
+</body>
+</html>
diff --git a/arxiv/2511.05345/2026-01-01-gm-v3/source/2511.05345.pdf b/arxiv/2511.05345/2026-01-01-gm-v3/source/2511.05345.pdf
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diff --git a/arxiv/2511.05345/2026-01-01-gm-v3/source/2511.05345.pdf.sha256 b/arxiv/2511.05345/2026-01-01-gm-v3/source/2511.05345.pdf.sha256
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+arXiv:2511.05345v1 [gr-qc] 7 Nov 2025
+
+The r −3 Curvature Decay and the Infrared
+Structure of Linearized Gravity
+Michael Wilson
+University of Arkansas at Little Rock
+Department of Physics and Astronomy
+mkwilson3@ualr.edu
+
+November 10, 2025
+Abstract
+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, ∞), corresponding to freely radiating
+tensor modes. At the inverse-cube rate, compactness fails and zero energy enters σess (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(ℓ+1)
+d2
+Lp = − dr
++ rCp
+2 +
+r2
+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 [18], indicating a common r−3 scaling
+that governs the infrared structure of gauge and gravitational fields.
+
+1
+
+1
+
+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
+L = ∇∗ ∇ + VR ,
+
+(VR h)ij = −R ℓ ijm hℓ m ,
+
+(1)
+
+acting on symmetric, trace-free tensor fields hij on an asymptotically flat
+three-manifold (Σ, g). The operator (1) governs stationary, harmonic-gauge
+perturbations of a vacuum background and determines whether small tensor excitations are radiative or spatially correlated. Its essential spectrum
+σess (L) distinguishes these regimes: a purely continuous spectrum [0, ∞) corresponds to freely propagating modes, whereas inclusion of zero in σess (L)
+signals marginally bound, long-range configurations.
+We show that curvature decay |Riem| ∼ r−3 marks the sharp boundary between these behaviors. For faster decay, VR is a compact perturbation of the
+flat tensor Laplacian ∆T = ∇∗ ∇, giving σess (L) = [0, ∞). 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 ∥Lhn ∥ → 0, describing
+extended, finite-energy tensor modes at zero frequency.
+A complementary numerical analysis of the radial model
+ℓ(ℓ + 1) C
+d2
++ p,
+Lp = − 2 +
+dr
+r2
+r
+confirms this transition. The lowest eigenvalue λ1 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,
+2
+
+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 [18], where |FA | ∼ r−3 separates radiative from infrared-sensitive
+behavior. This parallel suggests a common spectral mechanism linking spin1 and spin-2 fields, rooted in the dimensional scaling of curvature in three
+spatial dimensions.
+The paper is organized as follows. Section 2 introduces the analytic framework and the spectral universality principle linking gauge and gravitational
+fields. Section 3 establishes the inverse-cube decay as the critical regime and
+proves that zero enters the essential spectrum of the Lichnerowicz operator. Section 4 generalizes the argument to arbitrary dimension and derives
+the scaling law pcrit = d. Section 5 presents numerical verification of the
+spectral threshold through Rayleigh-quotient scaling and three-dimensional
+eigenvalue analysis, supported by convergence and stability checks. Section 6
+interprets the transition in physical terms, relating it to gravitational memory, late-time tails, and asymptotic symmetries. Section 7 situates the result
+within prior work in spectral geometry and gauge theory, and Section 8 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 5.
+
+2
+
+Spectral Scaling and Structural Parallels
+Across Spin-1 and Spin-2 Fields
+
+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
+
+3
+
+the two theories.
+
+2.1
+
+Geometric setup and harmonic gauge
+
+Let (Σ, g) be a smooth, oriented, three-dimensional Riemannian manifold
+representing a time-symmetric slice of a vacuum spacetime (M, gµν ) satisfying
+Ric(gµν ) = 0. On the asymptotic end, choose coordinates identifying Σ \ K
+with R3 \ BR (0) such that
+gij = δij + aij ,
+
+aij = O(r−1 ),
+
+∂k aij = O(r−2 ),
+
+∂ℓ ∂k aij = O(r−3 ),
+
+where r = |x| and ⟨r⟩ = (1 + r2 )1/2 . These conditions imply Γkij = O(r−2 ),
+|Riem| = O(r−3 ), and ∇Riem = O(r−4 ).
+Expanding the Einstein-Hilbert action
+Z
+√
+1
+R −g d4 x
+(2)
+SEH =
+16πG
+to quadratic order in a perturbation gµν 7→ gµν + ℏ1/2 hµν yields the spatial
+quadratic form [20, 21]
+Z
+√
+(2)
+1
+S [h] = 2 hij Lij kl hkl g d3 x,
+Σ
+
+where
+Lh = ∇∗ ∇h + VR h,
+
+(VR h)ij = −Ri ℓ j m hℓm .
+
+(3)
+
+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.
+
+2.2
+
+Shared scaling structure
+
+An analogous Laplace-type operator appears for spin-1 fields. For a YangMills connection A, the covariant Laplacian on the adjoint bundle is
+∆A = −(∇A )∗ ∇A = −∇∗ ∇ + ad(FA ),
+
+(4)
+
+where ad(FA ) denotes the adjoint action of the curvature. Both (3) and
+(4) therefore consist of a Laplace term plus a curvature potential that decays with distance. In the scalar Schrödinger case, potentials faster than r−3
+4
+
+yield compact perturbations of −∆, 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 |FA | and |Riem| become marginal when p = 3, the
+precise rate at which the potential ceases to be compact.1
+
+2.3
+
+Weighted Sobolev spaces and integration identity
+
+For δ ∈ R, define
+Hδk (Σ; S 2 T ∗ Σ) =
+
+n
+o
+k
+h ∈ Hloc (Σ) : ∥h∥Hδk < ∞ ,
+
+∥h∥2H k =
+δ
+
+k Z
+X
+
+⟨r⟩2(δ−j) |∇j h|2g dVg .
+
+j=0 Σ
+
+Integration by parts using the asymptotic decay of g gives
+⟨Lh, k⟩L2 = ⟨∇h, ∇k⟩L2 + ⟨VR h, k⟩L2 ,
+
+(5)
+
+valid for compactly supported smooth tensors h, k, showing that L is symmetric on Cc∞ (Σ; S 2 T ∗ Σ).
+
+2.4
+
+Self-adjoint realization and spectral framework
+
+Weighted elliptic estimates imply
+L : Hδ2 (Σ; S 2 T ∗ Σ) −→ L2δ−2 (Σ; S 2 T ∗ Σ)
+
+(6)
+
+is bounded for −1 < δ < 0. Essential self-adjointness of ∇∗ ∇ on complete
+Riemannian manifolds follows from Chernoff’s theorem, and VR is symmetric
+and relatively bounded since |VR (x)| = O(r−p ). By the Kato-Rellich theorem,
+L = ∇∗ ∇ + VR is self-adjoint on the same domain as ∇∗ ∇.
+By Lemma C.4 2 , if |Riem(x)| ≤ C⟨r⟩−p with p > 3, then VR is compact
+as a map Hδ2 → L2δ−2 for −1 < δ < 0, and therefore L = ∇∗ ∇ + VR is a
+compact perturbation of ∇∗ ∇. Weyl’s theorem then implies that
+σess (L) = [0, ∞).
+
+(7)
+
+In particular, sufficiently rapid curvature decay leaves the essential spectrum
+of the linearized gravitational field identical to that of flat space, so all finiteenergy tensor excitations are asymptotically radiative.
+1
+2
+
+The analogous threshold for the non-Abelian Laplacian was derived in Ref. [18].
+See supplementary material. Compactness of the curvature potential for p > 3.
+
+5
+
+2.5
+
+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 VR ∼ r−p competes with it when p = 3. Thus
+pcrit = 3,
+
+(8)
+
+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.
+
+3
+
+Infrared Spectrum and Marginal Modes
+
+When the background curvature decays as r−3 , the Lichnerowicz operator
+reaches the scaling threshold identified in Section 2. 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.
+
+3.1
+
+Fredholm and gauge framework
+
+Let (Σ, g) be a smooth, asymptotically flat three-manifold with a single end
+1
+diffeomorphic to R3 outside a compact set, and assume HdR
+(Σ) = 0. The
+vector Laplacian
+∆V = ∇∗ ∇ + Ric,
+(9)
+acts on one-forms and enforces the harmonic gauge constraint. Standard
+elliptic theory gives:
+Lemma 1 (Fredholm property). For −1 < δ < 0, the operator
+∆V : Hδ2 (Σ; T ∗ Σ) −→ L2δ−2 (Σ; T ∗ Σ)
+6
+
+(10)
+
+is Fredholm with bounded inverse. Hence, for each f ∈ L2δ−2 there exists a
+unique X ∈ Hδ2 satisfying ∆V X = f and ∥X∥Hδ2 ≤ C∥f ∥L2δ−2 .
+Proof. This is the weighted elliptic isomorphism theorem of Lockhart and
+1
+(Σ) = 0, and surjectivity from
+McOwen [12]. Injectivity follows from HdR
+asymptotic flatness.
+Under these hypotheses, ∆V defines a gauge correction that enforces
+∇ hij = 0. All subsequent constructions assume this analytic and topological framework.
+j
+
+3.2
+
+Weyl sequence at the critical decay
+
+Assume the metric satisfies
+gij = δij + O(r−1 ),
+
+∂gij = O(r−2 ),
+
+∂ 2 gij = O(r−3 ),
+
+(11)
+
+and that the curvature obeys |Riem(x)| ≃ C r−3 as r → ∞. On L2 (Σ; S 2 T ∗ Σ),
+consider the spatial Lichnerowicz operator
+Lh = ∇∗ ∇h + VR h,
+
+(VR h)ij = −Ri ℓ jm hℓ m ,
+
+(12)
+
+with domain Hδ2 (Σ; S 2 T ∗ Σ), −1 < δ < 0.
+Lemma 2 (Approximate zero modes). Let Hij (ω) be a symmetric, trace-free,
+divergence-free tensor harmonic on S 2 , and define
+hn (r, ω) = An ϕn (r) r−1 Hij (ω),
+
+(13)
+
+where ϕn is a smooth cutoff equal to 1 on [n, 3n/2] and vanishing outside
+[n/2, 2n]. The normalization ∥hn ∥L2 = 1 gives An ≃ n−1/2 . After divergence
+correction by a vector field Xn satisfying ∆V Xn = ∇· hn and setting h̃n =
+hn − LXn g, one obtains
+∥h̃n ∥L2 = 1,
+
+∇j h̃n,ij = 0,
+
+∥Lh̃n ∥L2 → 0.
+
+Proof. The tensor hn is supported on the annulus {n/2 < r < 2n} and normalized so that ∥hn ∥L2 = 1. Derivatives of the cutoff introduce factors of
+n−1 , yielding ∥∇· hn ∥L2 ≲ n−1 and ∥Lhn ∥L2 ≲ n−2 . The gauge correction
+Xn obeys ∥Xn ∥Hδ2 ≲ n−1 and therefore ∥LXn g∥L2 ≲ n−1 . Consequently, h̃n
+is divergence-free and satisfies the bounds above. A complete derivation, including control of curvature remainders, commutators [∇∗ ∇, ϕn ], and explicit
+estimates on L(LXn g), is provided in the Supplementary Material.
+7
+
+Theorem 1 (Onset of the infrared continuum). Let (Σ, g) satisfy (11) with
+|Riem(x)| ≃ C r−3 as r → ∞. Then the Lichnerowicz operator L = ∇∗ ∇+VR
+is self-adjoint on L2 (Σ; S 2 T ∗ Σ) and satisfies
+[0, ∞) ⊂ σess (L),
+
+0 ∈ σess (L).
+
+(14)
+
+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.
+
+3.3
+
+Physical interpretation
+
+The inclusion 0 ∈ σess (L) signifies that the spatial Lichnerowicz operator
+admits extended tensor modes with vanishing energy yet finite L2 norm. In
+the full time-dependent linearized Einstein equation,
+□g hµν + 2Rµρ νσ hρσ = 0,
+these correspond to the ω → 0 limit of radiative solutions.
+This result does not imply confinement or instability but reveals a continuous infrared sector: correlations decay slowly, and certain low-frequency
+components of gravitational radiation remain weakly coupled at large distances. The phenomenon parallels the infrared sensitivity of gauge fields
+at the same scaling threshold. In Yang-Mills theory, slow curvature decay
+localizes color flux; in gravity, it generates persistent correlations and memory effects. Thus the critical decay |Riem| ∼ r−3 delineates the geometric
+transition from purely radiative behavior to an infrared-sensitive, marginally
+bound regime, a structural, though not dynamical, analogue of the nonAbelian case.
+
+4
+
+Dimensional Scaling and Spectral Phase Structure
+
+The preceding analysis showed that when the curvature of an asymptotically flat three-manifold decays as |Riem| ∼ r−3 , the Lichnerowicz operator
+L = ∇∗ ∇ + VR acquires a continuous spectrum extending to zero. This
+identifies the boundary between spectrally transparent geometries and those
+8
+
+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.
+
+4.1
+
+Dimensional analysis of curvature potentials
+
+Let Σd be an asymptotically flat Riemannian manifold of dimension d ≥ 2,
+and let V (r) ∼ r−p denote a curvature-induced potential acting on tensor
+fields through
+Ld = −∇2 + V (r),
+where ∇2 is the Laplace-Beltrami operator on Σd . The potential is shortrange if it defines a compact perturbation of the Laplacian, and long-range
+otherwise. A scaling argument identifies the decay rate separating these
+regimes.
+Proposition 1 (Dimensional criterion for the critical decay rate). Let ∆
+denote the Laplace-Beltrami operator on a d-dimensional asymptotically flat
+manifold, and let V (r) ∼ r−p be a curvature-induced potential. Then V is a
+compact perturbation of ∆ if and only if p > d. The equality p = d marks
+the threshold between short- and long-range behavior.
+R
+Proof. The naive condition obtained by requiring rd−1 |V |2 dr < ∞ tests
+whether V defines a Hilbert-Schmidt perturbation of ∆, which is much
+stronger than compactness. On unweighted spaces H 2 (Rd ) → L2 (Rd ), any
+potential V (x) → 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, Ld acts between weighted Sobolev spaces
+2
+Hδ (Σd ) → L2δ−2 (Σd ) with −1 < δ < 0, appropriate to asymptotically flat
+ends. Compactness can fail at infinity if the decay of V (r) ∼ r−p is too
+slow to suppress contributions from large volumes. For transverse-traceless
+tensor modes, which behave asymptotically as h(r) ∼ r−(d−2)/2 due to the
+asymptotically flat falloff conditions, the weighted L2δ−2 norm of V h on {r >
+R} scales as
+Z ∞
+Z ∞
+2(δ−2)
+2
+2 d−1
+r
+|V (r)| |h(r)| r dr ∼
+r d−5+2δ−2p dr.
+R
+
+R
+
+9
+
+Because −1 < δ < 0, this integral diverges when p ≤ d, showing that for p ≤ 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 Hδ2 → L2δ−2 by the
+weighted Rellich lemma [3, 12]. Hence p = d marks the boundary between
+compact and noncompact behavior.
+Remark 1 (On sharpness). The borderline p = d is therefore not merely
+sufficient but sharp: for p ≤ d one can construct a normalized, transversetraceless Weyl sequence supported at large r with ∥Ld hn ∥ → 0, placing 0 in
+σess (Ld ).
+This reproduces the inverse-cube decay in three dimensions and extends
+it to arbitrary d. The resulting scaling law,
+pcrit = d,
+
+(15)
+
+expresses a dimensional balance between kinetic dispersion ∇2 ∼ r−2 and
+curvature coupling V (r) ∼ r−p . When p > d, curvature effects are integrable
+and the spectrum remains purely continuous; when p ≤ d, the potential
+becomes marginal or long-range, introducing infrared correlations.
+
+4.2
+
+Interpretation and structural implications
+
+Equation (15) 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
+pcrit = 3 in three dimensions is thus one instance of a general relation between
+dimensionality and the asymptotic behavior of curvature-coupled Laplacetype 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.
+
+10
+
+5
+
+Numerical Verification of the Spectral Threshold
+
+The analytic results of Sections 2–4 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.
+
+5.1
+
+Numerical setup and nondimensionalization
+
+The tensor operator L = ∇∗ ∇+VR is discretized on a uniform Cartesian grid
+Ω = [−Rmax , Rmax ]3 with spacing h = ∆x/L0 after nondimensionalizing by a
+fixed asymptotic length scale L0 . Centered finite differences approximate ∇
+and ∇∗ ∇. Dirichlet boundaries h|∂Ω = 0 define a finite–volume eigenproblem;
+the approach to the continuum spectrum is monitored by extrapolation in
+Rmax .
+To enforce the transverse–traceless constraint we use the penalty functional
+⟨h, Lh⟩ + η ∥∇· h∥2L2 (Ω) + ζ ∥tr h∥2L2 (Ω)
+,
+(16)
+Rη,ζ [h] =
+∥h∥2L2 (Ω)
+whose stationary points satisfy (K + ηD⊤ D + ζT ⊤ T )u = λM u, where K
+and M are the stiffness and mass matrices and D, T the discrete divergence
+and trace operators. Varying the penalties η, ζ by factors of 2–4 changes λ1
+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 ∈ {1.0, 0.75, 0.5}, Rmax ∈
+{6, 10, 14, 18, 20}, and C = −1. Each grid contains N 3 points (N = 21–41, up
+to 3.6 × 105 degrees of freedom). All runs use double precision and converge
+within relative error 10−5 .
+Dirichlet boundaries discretize the near–threshold continuum into box
+−2
+modes with λ1 (Rmax ) ∝ Rmax
+; the observed scaling and small residuals confirm that the computed eigenvalues track the physical continuum edge rather
+11
+
+than artificial confinement.
+Definition 1 (Numerical operators). The asymptotic radial model operator
+is
+d2
+ℓ(ℓ + 1) C
++ p,
+Lp = − 2 +
+dr
+r2
+r
+representing a single angular channel of the tensor operator. The three–dimensional
+discretized L = ∇∗ ∇ + VR includes the full curvature coupling and constraint
+enforcement.
+Remark 2 (Radial correspondence). Projecting L onto tensor harmonics of
+order ℓ yields an effective potential Veff (r) = ℓ(ℓ + 1)/r2 + C/rp + O(r−p−1 ),
+so the radial model reproduces the asymptotic channel structure of the full
+operator. For the Schwarzschild tail Ri k j ℓ hkℓ , one has C < 0, corresponding
+to an attractive potential in the far field.
+
+5.2
+
+Rayleigh–quotient scaling test
+
+The Rayleigh–quotient method directly probes the predicted scaling ∆E(R, p) ∼
+R−(p−2) , expressing the competition between the Laplacian and the curvature
+potential. For Vp (r) = ℓ(ℓ + 1)/r2 + C/rp , define
+R 2R
+E[ϕ] =
+
+R
+
+

+|ϕ′ (r)|2 + Vp (r)|ϕ(r)|2 r2 dr
+,
+R 2R
+|ϕ(r)|2 r2 dr
+R
+
+∆E(R, p) = E[ϕ] − Efree [ϕ],
+
+using a normalized bump function ϕ(r) on [R, 2R]. The flat case C = 0
+defines Efree . Table 1 lists the results; the fitted slopes α(p) ≈ −(p − 2)
+confirm the analytic scaling.
+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
+
+Figure 1: Log–log scaling of |∆E(R, p)| for representative p. Measured slopes
+α(p) ≈ −(p − 2) agree with the analytic prediction.
+
+5.3
+
+Three–dimensional eigenvalue analysis
+
+To test the full tensor operator, we compute the lowest eigenvalues λ1 of
+the discretized model for several p and Rmax . All runs use C = −1 and
+the TT–penalty enforcement of Eq. (16). The results are shown in Table 2
+and Fig. 2. The systematic decrease of λ1 with increasing Rmax confirms
+convergence toward the continuum limit. A least–squares fit λ1 (Rmax ) =
+−2
+−3
+aRmax
++ bRmax
+yields small residuals, consistent with the finite–volume interpretation.
+
+14
+
+Table 2: Lowest eigenvalue λ1 of the discretized tensor
+operator for several decay exponents p and outer radii
+Rmax . Decreasing λ1 with larger Rmax confirms convergence toward the infinite-volume limit.
+p
+flat
+
+2.0
+
+2.5
+
+3.0
+
+3.5
+
+4.0
+
+Rmax
+6
+10
+14
+18
+20
+6
+10
+14
+18
+20
+6
+10
+14
+18
+20
+6
+10
+14
+18
+20
+6
+10
+14
+18
+20
+6
+10
+14
+18
+20
+
+λ1
+0.2044
+0.0739
+0.0377
+0.0228
+0.0185
+0.1466
+0.0435
+0.0192
+0.0102
+0.0078
+0.1828
+0.0651
+0.0334
+0.0203
+0.0166
+0.1965
+0.0711
+0.0365
+0.0222
+0.0180
+0.2016
+0.0730
+0.0374
+0.0227
+0.0184
+0.2035
+0.0736
+0.0376
+0.0228
+0.0185
+
+The systematic decrease of λ1 with increasing Rmax confirms convergence to15
+
+ward the continuum limit. Intermediate values at Rmax = 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. 3
+
+Figure 2: Convergence of λ1 with domain size Rmax for several decay exponents p. The flattening of λ1 (Rmax ) for p ≥ 3 shows that curvature becomes
+spectrally negligible beyond the inverse–cube rate, while slower decay (p < 3)
+yields progressively deeper infrared shifts.
+Remark 3 (Interpretation). The continuous approach of λ1 (p) to its flat–space
+value as p increases demonstrates a smooth transition between confining and
+radiative regimes. For p ≤ 2.5, curvature remains spectrally significant; at
+p = 3, curvature and dispersion balance; and for p > 3, the spectrum be3
+
+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 λ1
+between p = 3.0, 2.5, and 2.0 suggests that even slower decay (p ≲ 2) 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.
+
+16
+
+comes 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).
+Lemma 3 (Asymptotic behavior of the lowest eigenvalue). For p ≥ 3,
+−2
+λ1 (Lp ) ∼ Rmax
+and approaches 0+ as Rmax → ∞, 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 λ1
+but no discrete negative modes.
+Remark 4 (Numerical consistency with the analytic threshold). Both diagnostics reproduce the qualitative behavior predicted by the analytic theory:
+power–law Rayleigh scaling with slope α(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.
+
+6
+
+Physical Interpretation and Implications
+
+The analytic and numerical analyses of Sections 3-5 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.
+
+6.1
+
+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
+
+17
+
+quantized linearized gravitational field. In canonical quantization, the equaltime two-point function in harmonic gauge is the inverse of L, so the largedistance 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 [16, 17].
+Remark 5 (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.
+
+6.2
+
+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.
+Remark 6 (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
+18
+
+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 I + , which we leave for future work.
+
+6.3
+
+Relation to Late-Time Tails
+
+Price’s law for black-hole perturbations, ψ ∼ t−2ℓ−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.
+Remark 7 (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.
+
+6.4
+
+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.
+Remark 8 (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.
+
+19
+
+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 R3 . 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.
+
+7
+
+Relation to Previous Work and Threshold
+Phenomena
+
+The spectral threshold established in Sections 3-5 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.
+
+7.1
+
+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 [12]. They proved Fredholm and isomorphism properties for elliptic operators acting between weighted
+Sobolev spaces Hδ2 → L2δ−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.
+Remark 9 (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 |Riem(x)| ≤ C r−p with p >
+3, the curvature potential VR is relatively compact with respect to ∆T and
+20
+
+σess (L) = [0, ∞); 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.
+
+7.2
+
+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 [1]
+proved that potentials decaying faster than r−3 yield purely absolutely continuous spectrum on [0, ∞), 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.
+Remark 10 (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 inversecube 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.
+
+7.3
+
+Relation to late-time tails in black-hole spacetimes
+
+Price [13] and Ching et al. [11] showed that perturbations of the Schwarzschild
+spacetime decay as t−2ℓ−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.
+Remark 11 (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.
+
+21
+
+7.4
+
+Infrared structure and memory
+
+In the context of asymptotic symmetry and gravitational memory, zerofrequency perturbations encode residual deformations between radiative vacua
+at null infinity. The existence of an L2 -normalized Weyl sequence at zero energy provides a spatial realization of these soft configurations.
+Remark 12 (Spectral characterization of the infrared sector). At curvature
+decay |Riem| ∼ 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.
+
+7.5
+
+Parallel thresholds across gauge and gravitational
+systems
+
+A closely related threshold occurs in non-Abelian gauge theory. For the
+covariant Laplacian
+∆A = −(∇A )∗ ∇A = −∇∗ ∇ + ad(FA ),
+acting on adjoint-valued fields with curvature FA , the decay |FA | ∼ r−3 separates spectrally radiative behavior from the onset of infrared sensitivity.
+Proposition 2 (Parallel inverse-cube threshold). For Laplace–type operators on bundles over R3 , 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.
+
+7.6
+
+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.
+22
+
+Remark 13 (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.
+
+8
+
+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.
+Theorem 2 (Spectral threshold for linearized gravity). Let (Σ, g) be a smooth
+asymptotically flat three–manifold with curvature decay |Riem(x)| ≤ C r−p .
+Then:
+1. For p > 3, the curvature potential VR is relatively compact with respect
+to ∇∗ ∇, and the essential spectrum is purely continuous:
+σess (L) = [0, ∞).
+2. 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.
+3. For p < 3, curvature acts as a long-range potential that enhances infrared coupling, but without producing isolated bound states in the tensorial sector.
+Remark 14 (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.
+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
+23
+
+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 3-5.
+Remark 15 (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.
+
+Declarations
+Funding The author received no external funding.
+Conflict of interest The author declares no conflict of interest.
+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.
+
+A
+
+Gauge Correction and Elliptic Estimates
+
+This appendix justifies the harmonic–gauge correction used in Section 3.
+Under the hypotheses of Section 5, namely, (Σ, g) smooth, asymptotically
+flat, simply connected, and with Ric = O(r−3 ), the vector Laplacian
+∆V X = ∇∗ ∇X + Ric(X)
+is uniformly elliptic and symmetric on L2 (Σ; T ∗ Σ).
+Lemma 4 (Isomorphism property). For weights −1 < δ < 0, the mapping
+∆V : Hδ2 (Σ; T ∗ Σ) → L2δ−2 (Σ; T ∗ Σ)
+1
+is Fredholm of index zero and an isomorphism whenever HdR
+(Σ) = 0.
+
+24
+
+Sketch. A direct consequence of Lockhart-McOwen theory (Comm. Pure
+Appl. Math. 38, 603 (1985)), since the Euclidean indicial roots are {0, −2}.
+Lemma 5 (Gauge correction). For each h ∈ Hδ2 (Σ; S 2 T ∗ Σ) with −1 < δ < 0,
+there exists a unique X ∈ Hδ2 (Σ; T ∗ Σ) satisfying ∆V X = ∇· h and ∥X∥Hδ2 ≤
+C∥∇· h∥L2δ−2 .
+Proposition 3 (Corrected Weyl sequence). Let {hn } be the approximate
+sequence of Section 3. Defining h̃n = hn − LXn g with Xn = ∆−1
+V (∇ · hn )
+yields
+∥h̃n ∥L2 = 1,
+h̃n ⇀ 0,
+∥Lh̃n ∥L2 → 0.
+Thus 0 ∈ σess (L) in harmonic gauge.
+
+B
+
+Curvature Structure and the Schwarzschild
+Example
+
+The Schwarzschild metric provides a physical realization of the critical inverse–cube curvature decay analyzed in Section 3.
+Definition 2 (Spatial metric). In isotropic coordinates (t, r, ω), the Schwarzschild
+line element is
+ 1 − M 2
+
+M 4 2
+2
+2r
+ds2 = −
+dt
++
+1
++
+(dr + r2 dω 2 ).
+2r
+1+ M
+2r
+On a time-symmetric slice t = const., the spatial metric is gij = ψ 4 δij with
+.
+ψ(r) = 1 + M
+2r
+Lemma 6 (Asymptotic curvature). For this metric,
+|Riem(x)| ≃ C M r−3
+
+(r → ∞),
+
+so the curvature saturates the inverse–cube decay assumed in Theorem 2.
+Proposition 4 (Effective potential). The spatial Lichnerowicz operator on
+the Schwarzschild background satisfies
+Lh = ∆0 h − (CM )r−3 h + O(r−4 )h,
+25
+
+CM > 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. 4
+
+C
+
+Analytical Framework and Weighted Sobolev
+Spaces
+
+We summarize the analytic conventions and functional-analytic tools used
+throughout.
+Definition 3 (Weighted Sobolev spaces). For a smooth radius function r on
+an asymptotically flat three–manifold (Σ, g) and δ ∈ R,
+XZ
+2
+∥u∥H k =
+(1 + r2 )δ−|α| |∇α u|2 dVg .
+δ
+
+|α|≤k
+
+Σ
+
+Then Hδk (Σ; E) is the completion of Cc∞ (Σ; E) under this norm.
+Lemma 7 (Fredholm property). If P is a uniformly elliptic operator approaching constant coefficients at infinity, then
+P : Hδ2 → L2δ−2
+is Fredholm for all δ not equal to an indicial root [12].
+Proposition 5 (Self-adjointness and essential spectrum). For δ ∈ (−1, 0)
+and V = O(r−p ) with p > 2, operators of the form L = ∇∗ ∇ + V are
+self–adjoint on L2 (Σ; E). The essential spectrum σess (L) is determined by
+the existence of Weyl sequences as in Weyl’s criterion.
+Lemma 8 (Compactness of the curvature potential for p > 3). Let (Σ, g) be
+a smooth asymptotically flat three-manifold with a single Euclidean end, and
+assume
+gij = δij + O(r−1 ),
+
+∂gij = O(r−2 ),
+
+∂ 2 gij = O(r−3 ),
+
+so that |Riem(x)| ≤ C ⟨r⟩−p for some p > 3. Fix a weight −1 < δ < 0, and
+let
+L = ∇∗ ∇ + VR ,
+(VR h)ij = −Ri ℓ j m hℓm .
+4
+
+The overall minus sign arises from the definition (VR h)ij = −Ri k j ℓ hkℓ , which makes
+the curvature coupling attractive for positive mass M > 0.
+
+26
+
+Then the curvature term defines a compact operator
+VR : Hδ2 (Σ; S 2 T ∗ Σ) −→ L2δ−2 (Σ; S 2 T ∗ Σ),
+and therefore L is a compact perturbation of ∇∗ ∇ on L2 (Σ; S 2 T ∗ Σ). In
+particular,
+σess (L) = σess (∇∗ ∇) = [0, ∞).
+Proof. The curvature bound |Riem(x)| ≤ C⟨r⟩−p implies |VR (x)| ≤ C⟨r⟩−p ,
+so that for any h ∈ Hδ2 (Σ; S 2 T ∗ Σ),
+Z
+Z
+2
+2(δ−2)
+2
+|VR (x)h(x)| dVg ≲ ⟨r⟩2(δ−2)−2p |h(x)|2 dVg .
+∥VR h∥L2 = ⟨r⟩
+δ−2
+
+Σ
+
+Σ
+
+Since −1 < δ < 0 and p > 3, the exponent 2(δ − 2) − 2p is less than
+−6, making the weight ⟨r⟩2(δ−2)−2p integrable at infinity in three dimensions.
+Hence VR : Hδ2 → L2δ−2 is bounded.
+To verify compactness, let χR be a smooth cutoff equal to 1 on {r ≤ R}
+and supported in {r ≤ 2R}. Decompose
+(comp)
+
+VR = χR VR + (1 − χR )VR =: VR
+
+(tail)
+
++ VR
+
+.
+
+On the bounded region {r ≤ 2R}, 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 Hδ2 ({r ≤ 2R}) ,→
+L2δ−2 ({r ≤ 2R}) is compact; since VR is smooth there, the multiplication
+(comp)
+operator VR
+is compact.
+(tail)
+The remainder VR
+is supported in {r > R}, where the curvature decay
+dominates. For h with ∥h∥Hδ2 = 1,
+Z
+(tail)
+2
+∥VR h∥L2 ≲
+⟨r⟩2(δ−2)−2p |h(x)|2 dVg .
+δ−2
+
+r>R
+
+Because 2(δ − 2) − 2p < −6, the weight decays faster than r−6 ; combined
+with the uniform bound |h|2 ∈ L2δ , dominated convergence implies that
+(tail)
+
+sup ∥VR
+
+∥h∥H 2 =1
+
+h∥L2δ−2 −→ 0 as R → ∞.
+
+δ
+
+Thus VR is the limit of compact operators with vanishing tails, and hence
+compact.
+27
+
+Finally, since L = ∇∗ ∇+VR is a compact perturbation of the nonnegative,
+self-adjoint operator ∇∗ ∇, Weyl’s theorem ensures that the essential spectra
+coincide:
+σess (L) = σess (∇∗ ∇) = [0, ∞).
+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.
+Remark 16. These weighted spaces and mapping properties justify the estimates and compactness arguments used in Sections 2–3.
+
+D
+
+Numerical Validation and Stability Tests
+
+This appendix documents three numerical consistency checks supporting the
+eigenvalue results reported in Section 5: (i) grid-spacing convergence, (ii)
+finite-volume convergence in Rmax , 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.
+
+D.1
+
+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 ≈ 1.0 (N = 21 points per axis, ∼ 4.1 × 104
+degrees of freedom), and the refined run used h ≈ 0.5 (N = 41, ∼ 3.6 × 105
+degrees of freedom). Converged eigenvalues at both resolutions are listed
+below. The near-invariance of λ1 and λ2 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.
+Coarse resolution (h ≈ 1.0, N = 21 points/axis, ∼ 4.1 × 104 DOFs).
+flat:
+p = 2.5 :
+p = 3.0 :
+p = 3.5 :
+
+λ1 = 0.07386996,
+λ1 = 0.06505331,
+λ1 = 0.07114971,
+λ1 = 0.07298806,
+
+28
+
+λ2 = 0.14713361,
+λ2 = 0.07023744,
+λ2 = 0.07276003,
+λ2 = 0.07351824.
+
+Refined resolution (h ≈ 0.5, N = 41 points/axis, ∼ 3.6 × 105 DOFs).
+flat:
+p = 2.5 :
+p = 3.0 :
+p = 3.5 :
+
+λ1 = 0.07398399,
+λ1 = 0.06505635,
+λ1 = 0.07109678,
+λ1 = 0.07295142,
+
+λ2 = 0.14781594,
+λ2 = 0.07029545,
+λ2 = 0.07279109,
+λ2 = 0.07355686.
+
+Halving the grid spacing changes λ1 by less than 0.2% across all p, demonstrating numerical convergence in h. Crucially, the ordering
+λ1 (p=2.5) < λ1 (p=3.0) < λ1 (p=3.5) ≃ λ1 (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.
+
+D.2
+
+Finite-volume convergence in Rmax
+
+We also varied the outer box size Rmax ∈ {6, 10, 14, 18, 20} at fixed grid
+spacing h ≈ 1.0 (so that increasing Rmax increases the number of unknowns)
+and observed that λ1 decreases monotonically with Rmax for every p tested.
+For example, at p = 3.0 we find
+λ1 (Rmax =6) = 0.1965,
+
+λ1 (Rmax =10) = 0.0711,
+
+λ1 (Rmax =20) = 0.0180.
+
+−2
+agrees with the interpretation of λ1 as
+The approximate scaling λ1 ∼ Rmax
+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.
+
+Penalty strength studies and TT projection cross–check. We sweep
+η, ζ over two decades (e.g. η = ζ ∈ {10, 40, 160, 640} in nondimensional
+units); for p ∈ {2.5, 3.0, 3.5} the relative shifts in λ1,2 between the two largest
+penalties are < 10−3 . We also compute the lowest eigenpair using an explicit TT projection: given any u, let uTT = ΠTT u via Helmholtz–Hodge
+decomposition (solve ∆V X = ∇ · u, then set uTT = u − LX g − 13 (tr u)g).
+29
+
+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.)
+Remark 17 (Conclusion). The convergence in grid spacing, the Rmax 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.
+Code availability. The complete Python script reproducing Table 2 is
+provided as 3d tensor operator.py in the Supplementary Reproducibility
+Package (ResearchGate upload). It implements the finite–difference Laplacian, the regularized tidal curvature term Eij ∼ r−p (ni nj − δij /3), and the
+trace–penalty enforcement of the TT constraint.
+
+E
+
+Weyl Sequence Construction and Verification of the Critical Spectrum
+
+This appendix provides quantitative estimates completing the proof of Lemma 2
+and Theorem 1. Throughout, (Σ, g) satisfies the asymptotic flatness conditions of eq. 11 with |Riem(x)| ≤ Cr−3 .
+
+Normalization and scaling
+Let Hij (ω) be a trace-free, divergence-free harmonic on S 2 and define hn =
+An ϕn (r)r−1 Hij (ω) with ϕn supported on An = {n/2 < r < 2n}. The metric
+volume element satisfies dVg = (1 + O(r−1 ))r2 dr dω, so that
+Z 2n
+
+∥hn ∥2L2 ≃ A2n
+
+r−2 r2 dr ∼ A2n n.
+
+n/2
+
+Choosing ∥hn ∥L2 = 1 gives An ≃ n−1/2 . Derivatives of ϕn are supported on
+the thin shells Sn = {n/2 < r < n} ∪ {3n/2 < r < 2n} where |∇ϕn | ≲ n−1 .
+
+30
+
+Commutator and curvature estimates
+Expanding
+∇∗ ∇(ϕn r−1 H) = ϕn ∇∗ ∇(r−1 H) + 2∇ϕn ·∇(r−1 H) + (∆ϕn )r−1 H,
+the first term vanishes identically in flat space since ∇∗ ∇(r−1 H) = 0 for
+tensor harmonics of order ℓ = 2. The commutator terms are supported on
+Sn and satisfy
+∥[∇∗ ∇, ϕn ]hn ∥L2 ≲ n−2 .
+The curvature potential obeys |VR | ≤ Cr−3 , so that
+Z 2n
+2
+2
+∥VR hn ∥L2 ≲ An
+r−6 r2 dr ∼ n−4 ,
+∥VR hn ∥L2 ≲ n−2 .
+n/2
+
+Together these give ∥Lhn ∥L2 ≲ n−2 .
+
+Gauge correction
+Solving ∆V Xn = ∇· hn with the Fredholm estimate of Lemma 1 yields
+∥Xn ∥Hδ2 ≲ ∥∇· hn ∥L2 ≲ n−1 ,
+
+−1 < δ < 0.
+
+Hence ∥LXn g∥L2 ≲ n−1 and the corrected fields h̃n = hn − LXn g satisfy
+∇j h̃n,ij = 0. Since L(LXn g) involves at most two derivatives of Xn ,
+∥L(LXn g)∥L2 ≲ n−1 ,
+which is subleading relative to ∥Lhn ∥L2 ≲ n−2 . Thus
+∥Lh̃n ∥L2 → 0,
+
+∥h̃n ∥L2 = 1.
+
+Conclusion
+The sequence {h̃n } is normalized, divergence-free, and supported on disjoint
+annuli escaping to infinity, implying h̃n ⇀ 0 weakly in L2 . By Weyl’s criterion, 0 ∈ σess (L), establishing Theorem 1.
+Author contributions The author is solely responsible for all aspects
+of this work.
+31
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