Propagation of Relational Inconsistency in a Pregeometric Connectivity Structure

This paper defines “relational inconsistency” in a pregeometric connectivity structure as a deviation of a local connection state from a self-consistent reference state, and investigates its discrete propagation properties. As the basic structure, we consider a composite cell formed by symmetrically connecting two regular tetrahedra to a regular octahedron, all with the same edge length. The height from a vertex of a regular tetrahedron to the center of its opposite face and the distance between the centers of opposite triangular faces of a regular octahedron are both a√(2/3). Consequently, the composite cell possesses a common straight axis consisting of three equal-length segments. The triangular face of each tetrahedron is perpendicular to this axis, geometrically separating the propagation direction from the transverse degrees of freedom. If an inconsistent state is conservatively transferred between neighboring cells once per fundamental update, the resulting mode propagates along the straight path at a constant speed c0 = ℓ0/τ0. Taking the continuum limit of the right- and left-propagating modes and reconstructing two components in the transverse plane yields a pair of coupled equations isomorphic to the one-dimensional vacuum Maxwell equations. A unidirectional mode also satisfies a transverse-wave condition, possesses two transverse degrees of freedom, and obeys the linear dispersion relation ω = c0k. During these derivations, the mode is not identified as light or an electromagnetic wave. Likewise, the nonnegative quantity associated with relational inconsistency is not called energy at the derivational stage. Only in the final discussion are the derived properties compared with known physical quantities, and possible correspondences with light and energy are considered.