Universality of Low-Band Spectral Dimension in Bounded-Degree Graphs
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2026
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| author | SCHORR, RICHARD |
| author_facet | SCHORR, RICHARD |
| contents | <p>We prove that the low-band spectrum of the normalized Laplacian is governed by a single organizing principle — the volume-growth exponent — for any sequence of bounded-degree graphs satisfying a uniform Poincaré inequality. The paper establishes four theorems and one corollary, constituting a rigorous discrete Weyl law and a complete universality classification for bounded-degree graph sequences.</p> <p><strong>Main Results</strong></p> <p>Theorem 1 — Discrete Weyl Law. For bounded-degree graphs with polynomial volume growth exponent d, the k-th normalized Laplacian eigenvalue satisfies c₁(k/N)^{2/d} ≤ λ_k ≤ c₂(k/N)^{2/d} for constants independent of graph topology.</p> <p>Corollary — Eigenvalue Counting. The integrated eigenvalue density satisfies N(λ) ~ N·λ^{d/2} for small λ.</p> <p>Theorem 2 — Spectral Gap Scaling. The algebraic connectivity scales as λ₂(N) ~ N^{−2/d}.</p> <p>Theorem 3 — Effective Dimension Universality. The low-band effective spectral dimension converges to the volume growth exponent: d_eff(k) → d as N → ∞, independent of graph topology.</p> <p>Theorem 4 — Expander Class. If the spectral gap is bounded away from zero, then d_eff(k) → 0 for any fixed low-band window.</p> <p>All results depend solely on volume growth and the spectral gap; no geometric embedding, translational symmetry, or continuum assumption is required.</p> <p><strong>Two Universality Classes</strong></p> <p>Polynomial-growth class: vanishing spectral gap, λ_k ~ (k/N)^{2/d}, effective dimension d* = d. Representative families: d-dimensional tori, nilpotent Cayley graphs, product graphs.</p> <p>Expander class: non-vanishing spectral gap, flat low-band spectrum, effective dimension d* = 0. Representative families: random-regular graphs, Ramanujan graphs.</p> <p><strong>Connection to PMIR</strong></p> <p>This paper provides the rigorous spectral-geometric foundation for empirical results established across the Phase-Modulated Information Rivalry (PMIR) preprint series. The Discrete Weyl Law guarantees that all eigenvalues within a fixed fractional band scale identically, explaining intra-band collapse of rivalry observables. The Spectral Gap theorem explains why λ₂-rescaling is the minimal rescaling achieving cross-topology collapse. The Expander theorem explains why random-regular graphs show qualitatively different PMIR scaling than lattice graphs.</p> <p><strong>Numerical Illustrations</strong></p> <p>Each theorem is illustrated with figures generated from PMIR numerical experiments on 2D-lattice and random-regular graph families (N ∈ {256, 1024, 4096}, degree 4). Empirical scaling exponents are taken from PMIR Preprint 2 (doi:10.5281/zenodo.18226938), Table 3.1.3: β ≈ −1.99 for 2D lattice (d ≈ 2), β ≈ −0.24 for random-regular (expander-like). Figure-generation code is available at github.com/richardschorriii/PMIR_verification.</p> <p><strong>Related Works (PMIR Preprint Series)</strong></p> <p>Preprint 1 — Topology-Dependent Rivalry Dynamics: doi:10.5281/zenodo.18210474 Preprint 2 — Continuum Scaling Limits: doi:10.5281/zenodo.18226938 Preprint 3 — Anomalous Low-Mode Transport / Emergent Medium: doi:10.5281/zenodo.18275923 Preprint 4 — Universality of Emergent Medium Response: doi:10.5281/zenodo.18282356 Preprint 5 — Spectral-Band Universality and Predictive Closure: doi:10.5281/zenodo.18293869 Preprint 6 — Topology-Dependent Spectral Coupling in Phase-Space Networks: doi:10.5281/zenodo.18509187 Preprint 7 — Chaos Onset as Spectral Regime Transition (N-Body / Hypergraph): doi:10.5281/zenodo.18652630 Preprint 8 — Observation-Induced Frame Dependence and the Hubble Tension: doi:10.5281/zenodo.18652539</p> <p>Analysis assistance provided by Claude (Anthropic, 2026). This work was conducted independently without institutional affiliation or external funding.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18653051 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | Universality of Low-Band Spectral Dimension in Bounded-Degree Graphs SCHORR, RICHARD spectral graph theory Laplacian eigenvalues discrete Weyl law effective spectral dimension volume growth expander graphs universality classes PMIR Phase-Modulated Information Rivalry bounded-degree graphs algebraic connectivity <p>We prove that the low-band spectrum of the normalized Laplacian is governed by a single organizing principle — the volume-growth exponent — for any sequence of bounded-degree graphs satisfying a uniform Poincaré inequality. The paper establishes four theorems and one corollary, constituting a rigorous discrete Weyl law and a complete universality classification for bounded-degree graph sequences.</p> <p><strong>Main Results</strong></p> <p>Theorem 1 — Discrete Weyl Law. For bounded-degree graphs with polynomial volume growth exponent d, the k-th normalized Laplacian eigenvalue satisfies c₁(k/N)^{2/d} ≤ λ_k ≤ c₂(k/N)^{2/d} for constants independent of graph topology.</p> <p>Corollary — Eigenvalue Counting. The integrated eigenvalue density satisfies N(λ) ~ N·λ^{d/2} for small λ.</p> <p>Theorem 2 — Spectral Gap Scaling. The algebraic connectivity scales as λ₂(N) ~ N^{−2/d}.</p> <p>Theorem 3 — Effective Dimension Universality. The low-band effective spectral dimension converges to the volume growth exponent: d_eff(k) → d as N → ∞, independent of graph topology.</p> <p>Theorem 4 — Expander Class. If the spectral gap is bounded away from zero, then d_eff(k) → 0 for any fixed low-band window.</p> <p>All results depend solely on volume growth and the spectral gap; no geometric embedding, translational symmetry, or continuum assumption is required.</p> <p><strong>Two Universality Classes</strong></p> <p>Polynomial-growth class: vanishing spectral gap, λ_k ~ (k/N)^{2/d}, effective dimension d* = d. Representative families: d-dimensional tori, nilpotent Cayley graphs, product graphs.</p> <p>Expander class: non-vanishing spectral gap, flat low-band spectrum, effective dimension d* = 0. Representative families: random-regular graphs, Ramanujan graphs.</p> <p><strong>Connection to PMIR</strong></p> <p>This paper provides the rigorous spectral-geometric foundation for empirical results established across the Phase-Modulated Information Rivalry (PMIR) preprint series. The Discrete Weyl Law guarantees that all eigenvalues within a fixed fractional band scale identically, explaining intra-band collapse of rivalry observables. The Spectral Gap theorem explains why λ₂-rescaling is the minimal rescaling achieving cross-topology collapse. The Expander theorem explains why random-regular graphs show qualitatively different PMIR scaling than lattice graphs.</p> <p><strong>Numerical Illustrations</strong></p> <p>Each theorem is illustrated with figures generated from PMIR numerical experiments on 2D-lattice and random-regular graph families (N ∈ {256, 1024, 4096}, degree 4). Empirical scaling exponents are taken from PMIR Preprint 2 (doi:10.5281/zenodo.18226938), Table 3.1.3: β ≈ −1.99 for 2D lattice (d ≈ 2), β ≈ −0.24 for random-regular (expander-like). Figure-generation code is available at github.com/richardschorriii/PMIR_verification.</p> <p><strong>Related Works (PMIR Preprint Series)</strong></p> <p>Preprint 1 — Topology-Dependent Rivalry Dynamics: doi:10.5281/zenodo.18210474 Preprint 2 — Continuum Scaling Limits: doi:10.5281/zenodo.18226938 Preprint 3 — Anomalous Low-Mode Transport / Emergent Medium: doi:10.5281/zenodo.18275923 Preprint 4 — Universality of Emergent Medium Response: doi:10.5281/zenodo.18282356 Preprint 5 — Spectral-Band Universality and Predictive Closure: doi:10.5281/zenodo.18293869 Preprint 6 — Topology-Dependent Spectral Coupling in Phase-Space Networks: doi:10.5281/zenodo.18509187 Preprint 7 — Chaos Onset as Spectral Regime Transition (N-Body / Hypergraph): doi:10.5281/zenodo.18652630 Preprint 8 — Observation-Induced Frame Dependence and the Hubble Tension: doi:10.5281/zenodo.18652539</p> <p>Analysis assistance provided by Claude (Anthropic, 2026). This work was conducted independently without institutional affiliation or external funding.</p> |
| title | Universality of Low-Band Spectral Dimension in Bounded-Degree Graphs |
| topic | spectral graph theory Laplacian eigenvalues discrete Weyl law effective spectral dimension volume growth expander graphs universality classes PMIR Phase-Modulated Information Rivalry bounded-degree graphs algebraic connectivity |
| url | https://doi.org/10.5281/zenodo.18653051 |