diff --git a/README.md b/README.md index 5f3b958..3183e1e 100644 --- a/README.md +++ b/README.md @@ -1,3 +1,10 @@ # 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 new file mode 100644 index 0000000..e52b892 --- /dev/null +++ b/arxiv/2511.05345/2026-01-01-gm-v3/ARTIFACTS.sha256 @@ -0,0 +1,6 @@ +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 new file mode 100644 index 0000000..dbf16cc --- /dev/null +++ b/arxiv/2511.05345/2026-01-01-gm-v3/README.md @@ -0,0 +1,19 @@ +# 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 new file mode 100644 index 0000000..ac0e12a --- /dev/null +++ b/arxiv/2511.05345/2026-01-01-gm-v3/generated/IF_GM_ARXIV_2511_05345.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 ^{2(δ-j)} |∇^j h|^2_g dV_g. +``` + +Where the shorthand `` is the standard “one plus radius” weight: + +```text + = (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 ^{2(δ-j)} |∇^j h|^2_g dV_g. +``` + +Where the shorthand `` is the standard “one plus radius” weight: + +```text + = (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/index.html b/arxiv/2511.05345/2026-01-01-gm-v3/rendered/index.html new file mode 100644 index 0000000..64fec5f --- /dev/null +++ b/arxiv/2511.05345/2026-01-01-gm-v3/rendered/index.html @@ -0,0 +1,934 @@ + + + + + + InfraFabric GM v3 — arXiv:2511.05345 + + + +
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+ + + + + + + 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 new file mode 100644 index 0000000..5198a47 Binary files /dev/null and b/arxiv/2511.05345/2026-01-01-gm-v3/source/2511.05345.pdf differ 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 new file mode 100644 index 0000000..e1a4994 --- /dev/null +++ b/arxiv/2511.05345/2026-01-01-gm-v3/source/2511.05345.pdf.sha256 @@ -0,0 +1 @@ +992e50be5f84477fdbcf72ddea26bec88e6e1b756d948161af1b71bf03fc6742 diff --git a/arxiv/2511.05345/2026-01-01-gm-v3/source/2511.05345.txt b/arxiv/2511.05345/2026-01-01-gm-v3/source/2511.05345.txt new file mode 100644 index 0000000..e4e924c --- /dev/null +++ b/arxiv/2511.05345/2026-01-01-gm-v3/source/2511.05345.txt @@ -0,0 +1,1512 @@ +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 + + References +[1] B. 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