Constraints on Changes in Length, Area, and Spatial Structure under the Redistribution of Binary Referential Relations

This paper investigates how length, area, and spatial structure are constrained by the redistribution of binary referential relations within the framework of Referential Entropy Cycle Theory (RECT). Starting from the RECT principle that physical quantities and their corresponding referential relations are mutually dependent, the paper develops a mathematical framework for describing geometric structures through binary relations between reference structures. The number of binary relations in a finite system of N distinct reference structures is expressed as R(N) = N(N − 1)/2. The paper examines the conservation of relational quantity within a fixed group of reference structures, as well as the corresponding evolution of relational quantity during the expansion and contraction phases of the universe as proposed by RECT. A distance matrix, a centering matrix, and a Gram matrix are introduced to establish the mathematical connection between referential relations and Euclidean geometry. The rank of the centered Gram matrix is used to characterize geometric dimension. The paper further derives expressions for length, triangular area, and higher-dimensional simplex volume using Gram matrices and their determinants. The central objective is to explore how geometric quantities and spatial structure can be described and constrained through the underlying organization of binary referential relations, rather than treating space as an independently given background. This work presents a mathematical development within the RECT framework and provides a basis for further investigation of the emergence of geometry and spatial structure from referential relations.