<p>This paper presents a systematic study of the fixed point card associated with a quadratic Lotka-Volterra mapping defined on the four-dimensional simplex. The geometric configuration and spectral properties of all fixed points are completely classified, and the orientation rules determining the structure of the card are rigorously established. We further investigate the problem of connections between non-adjacent triples of fixed points and derive necessary and sufficient conditions for the existence of such connections. The obtained results clarify the operators spectral structure and provide new contributions to the structural theory of quadratic stochastic Lotka-Volterra operators.</p>