Spectral origin of the topological gap exponent d + η: mechanism, kernel, decomposition, and scope
Abstract
The topological gap Δ -- the excess H1 total persistence of a critical point cloud over a density-matched null -- scales as Δ∼ Ld+η. We derive this analytically: the spectral integral I(α) = ∑k≠ 0 Sconn(k) |k|α scales as L2−α−η when IR-dominated, giving I(−2η) ∼ Ld+η. The decomposition I(−2η) = I0 · Ishape separates volume (I0 ∝ N(1−m2) ∼ Ld) from anomalous dimension (Ishape ∼ Lη); the volume factor accounts for the magnetization-driven per-configuration variance of Δ. We prove the mechanism requires d < 2 + η (IR dominance), confining it to d = 2 for physical systems; in d = 3 the spectral integral is UV-dominated, explaining why density normalization is needed. An α-sweep for Potts q = 4 at L = 32--$256$ finds αopt in $[-0.75, -0.5]$, consistent with −2ηIsing and inconsistent with −2ηq=4 = −1; we flag this as tentative pending L ≥ 1024 confirmation. The ⟨ m2 · I(−2η)⟩ hyperscaling product is dominated by the correlation r(m2, I) ≈ −0.98 via the shared I0 amplitude, so we report it as a covariance-correction analysis. Under a heuristic argument extending Divol--Polonik to inhomogeneous Poisson intensities, the bare PH kernel is flat; the effective kernel acquires k-dependence only at criticality. The per-configuration agreement between Δ and I(−0.5) is primarily a magnetization correlation: R2 = 0.91 at L = 256 collapses to R2 ≈ 0 once |M| is partialed out. Per-configuration evidence corroborates the I0 Parseval identity but not the |k|−2η shape factor; the latter is established by ensemble L-scaling.
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BibTeX
@misc{loftus2026spectral,
title = {{Spectral origin of the topological gap exponent d + η: mechanism, kernel, decomposition, and scope}},
author = {Matthew Loftus},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09159},
eprint = {2609.09159},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09159},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spectral origin of the topological gap exponent d + η: mechanism, kernel, decomposition, and scope
AU - Matthew Loftus
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.09159
UR - https://arxiv.org/abs/2609.09159
AB - The topological gap Δ -- the excess H_1 total persistence of a critical point cloud over a density-matched null -- scales as Δ∼ L^d+η. We derive this analytically: the spectral integral I(α) = ∑_k≠ 0 S_conn(k) |k|^α scales as L^2−α−η when IR-dominated, giving I(−2η) ∼ L^d+η. The decomposition I(−2η) = I_0 · I_shape separates volume (I_0 ∝ N(1−m^2) ∼ L^d) from anomalous dimension (I_shape ∼ L^η); the volume factor accounts for the magnetization-driven per-configuration variance of Δ. We prove the mechanism requires d < 2 + η (IR dominance), confining it to d = 2 for physical systems; in d = 3 the spectral integral is UV-dominated, explaining why density normalization is needed. An α-sweep for Potts q = 4 at L = 32--$256$ finds α_opt in $[-0.75, -0.5]$, consistent with −2η_Ising and inconsistent with −2η_q=4 = −1; we flag this as tentative pending L ≥ 1024 confirmation. The ⟨ m^2 · I(−2η)⟩ hyperscaling product is dominated by the correlation r(m^2, I) ≈ −0.98 via the shared I_0 amplitude, so we report it as a covariance-correction analysis. Under a heuristic argument extending Divol--Polonik to inhomogeneous Poisson intensities, the bare PH kernel is flat; the effective kernel acquires k-dependence only at criticality. The per-configuration agreement between Δ and I(−0.5) is primarily a magnetization correlation: R^2 = 0.91 at L = 256 collapses to R^2 ≈ 0 once |M| is partialed out. Per-configuration evidence corroborates the I_0 Parseval identity but not the |k|^−2η shape factor; the latter is established by ensemble L-scaling.
ER -