Multi-Planar Hierarchy in Pan-American Seismic Fractality: A Bayesian Resolution to the Projection Paradox

Facundo Firmenich , Pau Firmenich, León Firmenich

Earthquake catalogs worldwide exhibit an enduring paradox: seismicity appears confined to planar faults (correlation dimension D2≈ 2.0), yet operates within volumetric lithospheric deformation. We resolve this paradox through Bayesian fractal analysis of 50,010 earthquakes across seven Pan-American tectonic regimes (2010–2025), revealing that apparent planarity reflects instrumental projection bias—not physical confinement. Correlation dimension D2 ranges 2.08–2.57, with hierarchical Rényi spectrum analysis identifying multi-planar architecture (hierarchical index H = D1 − D0 > 0) in four of six regions. Topological graph structure confirms 29–125 discrete seismogenic communities, with graph fractal dimension (Dgraph = 1.40–1.83) systematically lower than Euclidean D2. Bayesian scale transformation reveals intrinsic volumetric structure (D3= 3.00) in five of six regions, establishing a projectiondominated observation regime where USGS catalog location uncertainty (∼5–10 km) systematically reduces apparent dimensionality by ∆D ≈ 0.4–0.9 despite true 3D seismogenic volumes. This finding exhibits remarkable mathematical isomorphism (not physical equivalence) with the cosmic web paradox in large-scale structure formation, where intrinsically volumetric matter distribution appears filamentary due to observational projection—suggesting universal principles of the projection operator under incomplete sampling across disparate physical systems. Methodological sensitivity analysis demonstrates core conclusions persist across magnitude completeness strategies, scaling region detection algorithms, and coordinate systems. Results challenge pure planar confinement hypotheses, supporting a multi-planar “deck of cards” model where seismicity organizes along discrete reactivated planes embedded in volumetric deformation fields.