Daphne Garrido
September 21, 2026
Relational Transfer Matrices.pdf
In the Universal Relational-Geometric Coherence Law, the interactions among the elements of a system are encoded in a matrix whose entries already incorporate geometric protection. This matrix is called the relational transfer matrix.
The present article defines it, introduces the associated coherence budget, and situates both objects inside the classical theory of transfer matrices used in statistical mechanics and dynamical systems.
Transfer matrices have been a standard tool since the 1940s.
In statistical mechanics, one often studies a chain, or a higher-dimensional lattice sliced into successive layers. The statistical weight of going from one configuration of a layer to the next is recorded in a matrix W.
The partition function of the whole system then becomes a product of these matrices, and the free energy is determined by the largest eigenvalue of W.
The same idea appears in dynamical systems and solid-state physics. When a linear recurrence or a discrete Schrodinger equation is written in first-order form, the evolution of a state vector from site n to site n + 1 is given by a transfer matrix T_n.
The long-term behaviour of the system -- growth or decay of solutions, localization of eigenfunctions, and Lyapunov exponents -- is controlled by the product of many successive transfer matrices and by the eigenvalues, or singular values, of that product.
In both settings, the essential point is the same: local transition rules are packaged into a matrix, and global properties are read off from powers or products of that matrix.
Within the URCL, the local transition between two sites i and j is written as
M_ij = kappa_ij * exp(-beta * P_ij).
Here: