Pfaffian Structure and Spectral Dynamics of a Discrete Lotka–Volterra Operator: A Framework for Geological Evidence Integration

Abstract

This paper investigates the connection between Pfaffian invariants, fixed-point geometry, and local spectral dynamics of a discrete Lotka–Volterra operator on the fourdimensional simplex. For an interaction tournament containing four cyclic triples, the unique interior fixed points of the corresponding invariant two-dimensional faces form the vertices of a fixed-point card. Using an established orientation criterion and a specified sign convention, the cyclic configuration of this card is characterized by fourth-order principal Pfaffian invariants. Under positivity of the full Pfaffian vector, an interior fixed point is constructed explicitly through normalized Pfaffian ratios. The characteristic polynomial of the Jacobian at this point is expressed in terms of the interaction coefficients and the same invariants. The simplex tangent space is proved to be invariant under the Jacobian, and the restricted spectrum consists of two complex conjugate pairs with moduli greater than one, establishing locally repelling dynamics with oscillatory components. These results place fixed-point geometry and local spectral behavior within a common algebraic framework, providing explicit formulas suitable for computational analysis. A mathematical interpretation for integrating five groups of geological evidence is also proposed. Distances to fixed-point configurations and interaction-response measures are introduced as descriptors of evidence composition and interaction. Their potential use in gold prospectivity assessment requires calibration against exploration observations and independent empirical validation.