Five Tools for Emergence Diagnosis: A Spectral-Topological Probe Calibrated Across Harmonic, Bose-Hubbard, Transverse-Ising, and Anderson Test Cases

We propose a five-tool framework — classical Fisher information F, integrated autocorrelation time τ_int, Kolmogorov complexity proxy via lossless compression K, RG-like coarse-graining exponent α, and persistent-homology Betti counts H_0, H_1, H_2 — and explore a suggestive association between these five tools and five distinct physical mechanisms of emergence (thermal order, dynamical arrest, chaos/information production, scaling/long-range correlation, geometric/topological reorganization). We treat the five-tool ↔ five-mechanism mapping as a proposed organizing framework rather than as a PCA-verified one-to-one law: the empirical PCA on the 8-system fingerprint matrix returns four (not five) effective dimensions. Eight historical landmark emergence systems (Ising 1944, percolation 1957, Lorenz 1963, Brusselator 1968, Kauffman 1969, BKT 1973, Logistic 1976, Fredrickson-Andersen glass 1984) are placed in the framework's five-dimensional fingerprint space. Principal-component analysis recovers four major independent dimensions (PC1=48.2%, PC2=33.8%, PC3=12.6%, PC4=5.4%; cumulative 100%). The framework's central object is the power spectrum S(ω) of a stationary observable; under the Gaussian Ornstein-Uhlenbeck reference, every tool reduces to a functional of S(ω), and the spectral closure yields seven candidate closed-form constants for the topological invariants on 2D and 3D Gaussian random fields, numerically verified to within 0.1-4%. Cross-layer correspondence between collective and single-particle indicators is verified simultaneously on three independent quantum phase transitions. Honest limitations are reported: the framework is inferior to Ripley K-function on spatial point processes, H_1 sum persistence is identical to Wigner negativity in single-mode quantum decoherence, and earthquake prediction is not demonstrated.