A Four-Tool Fused Methodology for Cross-Domain Constraint Analysis

Probabilistic structure analysis is fragmented across scientific domains: each field maintains its own preferred tool. We propose a unified analysis toolkit fusing four well-established probabilistic-structure tools — Fisher information geometry, algorithmic information theory, renormalization group flow, and topological data analysis via persistent homology — into a single machine-language interface, the Universal Probe Protocol. Each domain implements the four protocol methods independently; the protocol exposes four operational scalars (κ_F, γ_K, ρ_RG, h_norm) that are cross-domain comparable. We demonstrate framework operability on seven anchor domains spanning quantum CFT to social demographics. Bootstrap resampling (N = 200 per anchor) verifies fingerprint stability. A four-tier emergent taxonomy in the 4-tuple latent space is reported descriptively. The associated tetrahedron of four pairwise mathematical relations is presented with explicit rigor stratification: Edge 1 (Fisher ↔ K) is a theorem via Rissanen 1996 NML asymptotic; Edge 2 (Fisher ↔ RG) is closed at the structural framework level via a singular bifurcation closure theorem; Edge 3 (RG ↔ TDA) has a Tier-1 rigorous bound with quantitative anchors at two universality classes (2D Gaussian Free Field and 2D critical Ising); Edge 4 (K ↔ TDA) is closed at the upper-bound level with explicit encoding constant. The toolkit is offered as a useful instrument for cross-domain probabilistic structure analysis.