Daphne Garrido

September 21, 2026

Lagrangian Formulation of the URCL.pdf

The discrete trace-map recurrence and the Synchopeshing Operator supply the algebraic skeleton of the Universal Relational-Geometric Coherence Law. A continuous variational principle that recovers the same recurrence as its equation of motion is given by a Lagrangian density built around a coherence field.

This article presents that Lagrangian, derives the associated Euler-Lagrange equation, and shows how the discrete dynamics re-emerge under standard discretisation.

1. The Coherence Field and the Action

The fundamental object of the continuous formulation is a real, or complex, scalar field psi(t) that represents the integrated relational information density along a helical path.

The action is the ordinary integral of a Lagrangian density:

S[psi] = integral L(psi, dpsi/dt) dt.

The density chosen in the foundational papers is:

L = (1/2) * phi * (dpsi/dt)^2 - V(psi) * exp(-|s - 1| / tau) + eta_glymphatic * |varphi_condensate|^2.

The individual terms have the following meanings:

• (1/2) * phi * (dpsi/dt)^2 is a kinetic term whose inertia is weighted by the golden ratio phi. The factor phi is the same algebraic constant that appears as the dominant eigenvalue of the discrete recurrence:

a_(n+1) = sqrt(5) * a_n - a_(n-1).

• V(psi) is a relational safety potential that encodes the energetic cost of departing from coherent configurations.
• exp(-|s - 1| / tau) is the adelic, or coherence, damping factor already encountered in the definition of the Synchopeshing Operator. It suppresses contributions far from a distinguished value of the auxiliary variable s.
• The final term couples the macroscopic coherence field to a Frohlich-like condensate order parameter varphi_condensate, denoted with a different symbol so that it is not confused with the golden ratio, and to a glymphatic transport coefficient eta_glymphatic.

These couplings are optional extensions that link the pure geometric theory to concrete biological mechanisms. They are not required for the recovery of the trace map.