Frustrated Nonphases as Catalysts for Phases - How Graph Theory Calculates Optimal Balance in Linguistic Fibonacci Trees and Gr

Volume 21
Issue 2
Koji Arikawa
This article examines a graph-theoretical analysis of an externally merged VP as a derivational Fibonacci (F) tree, i.e., a structure written in a Lindenmayer grammar without a linearization condition. We aim to show that the computational procedures of a natural human language (CHL), which is a complex system, solve a dynamically frustrated equation (introduced by Philippe Binder, and proposed as a third factor by Juan Uriagereka) in the human brain. The relevant equation is Kirchhoff’s (electric) current law (KCL) AT y = f, which calculates the equilibrium (balance) in any network.
Applying the standard graph-theoretical method, we calculate the hidden optimal balance among VP, vP, and CP as an F network. We calculate the potential (cumulative relative quantity of features) in each node and then the current (relative force of feature flow) along each edge.
The graph theory shows that an externally merged antisymmetric VP with binary-node connections conceals an extremely symmetric balance (a KCL solution of zero), wherein almost all the edges disappear and the nodes are almost completely disconnected. In this scenario, KCL qualifies as a dynamically frustrated equation caught in a dilemma between two contradictory dynamics: symmetry breaking and symmetry generation.
Our challenge is threefold. First, we aim to calculate the optimal balance hidden in sentential F trees. Second, we reassess Richard Kayne’s insightful “connectivity” proposal. Third, we propose a new hypothesis of phase existence. A frustrated nonphase catalyzes the creation of phases, forcing loop structures of feature inheritance and internal merges. These loops are nonzero solutions of the KCL.

Keywords: connectivity, Fibonacci (F) tree, optimal balance, feature inheritance, frustrated equation, graph theory, Kirchhoff’s current law (KCL), phase, third factor, symmetry