拓扑能隙指数 d + η 的谱起源：机制、核、分解与适用范围
Spectral origin of the topological gap exponent d + η: mechanism, kernel, decomposition, and scope
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中文总结 AI 辅助
本文解析推导了拓扑能隙 Δ 的谱起源，通过谱积分分解为体积与反常维度因子，证明红外主导条件限制维度，并用 Potts 模型数值验证，揭示了磁化相关性与协方差修正机制。
中文摘要 AI 辅助
拓扑能隙 Δ——临界点云相对于密度匹配零点的超额 H1 总持续同调——按 Δ ~ L^{d+η} 标度。我们解析地推导了这一结果：谱积分 I(α) = ∑_{k≠0} S_conn(k)|k|^α 在红外主导时按 L^{2-α-η} 标度，从而给出 I(-2η) ~ L^{d+η}。分解 I(-2η) = I_0 · I_shape 将体积因子（I_0 ∝ N(1-m^2) ~ L^d）与反常维度（I_shape ~ L^η）分离；体积因子解释了磁化驱动的每构型 Δ 方差。我们证明该机制要求 d < 2 + η（红外主导），从而将物理系统限制在 d = 2；在 d = 3 时谱积分为紫外主导，这解释了为何需要密度归一化。对 Potts q = 4 在 L = 32–256 上的 α 扫描发现 α_opt 位于 [-0.75, -0.5]，与 -2η_Ising 一致，而与 -2η_{q=4} = -1 不一致；我们将其标记为暂定结果，有待 L ≥ 1024 的确认。⟨m^2 · I(-2η)⟩ 超标度乘积由通过共享 I_0 振幅的相关系数 r(m^2, I) ≈ -0.98 主导，因此我们将其报告为协方差修正分析。在将 Divol–Polonik 推广到非齐次泊松强度的启发式论证下，裸 PH 核是平坦的；有效核仅在临界点获得 k 依赖性。Δ 与 I(-0.5) 之间的每构型一致性主要是磁化相关性：在 L = 256 时 R^2 = 0.91，一旦剔除 |M| 的影响，R^2 降至约 0。每构型证据证实了 I_0 的 Parseval 恒等式，但不支持 |k|^{-2η} 形状因子；后者由系综 L 标度确立。
英文摘要
The topological gap $Δ$ -- the excess $H_1$ total persistence of a critical point cloud over a density-matched null -- scales as $Δ\sim L^{d+η}$. We derive this analytically: the spectral integral $I(α) = \sum_{k\neq 0} S_{\mathrm{conn}}(k)\,|k|^α$ scales as $L^{2-α-η}$ when IR-dominated, giving $I(-2η) \sim L^{d+η}$. The decomposition $I(-2η) = I_0 \cdot I_{\mathrm{shape}}$ separates volume ($I_0 \propto N(1-m^2) \sim L^d$) from anomalous dimension ($I_{\mathrm{shape}} \sim L^η$); 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 $α_{\mathrm{opt}}$ in $[-0.75, -0.5]$, consistent with $-2η_{\mathrm{Ising}}$ and inconsistent with $-2η_{q=4} = -1$; we flag this as tentative pending $L \geq 1024$ confirmation. The $\langle m^2 \cdot I(-2η)\rangle$ hyperscaling product is dominated by the correlation $r(m^2, I) \approx -0.98$ 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 $R^2 \approx 0$ 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.
