Finite Tension and Spacetime Singularities: A Regularity Boundary in the Enchan Field

Abstract

This analytic note isolates an exact regularity boundary in a reduced finite-tension Enchan field branch and, separately, derives a curvature-divergence criterion in an exact static planar subsector of the associated Enchan metric. For the constitutive response μ(D) = 1/(1 + D^(3/2)), the longitudinal principal coefficient changes sign at the exact threshold D* = 2^(2/3). The subcritical sector has positive principal response, the threshold is degenerate, and the supercritical sector loses longitudinal hyperbolicity with unbounded frozen high-frequency growth. The corresponding gradient-energy primitive is also non-coercive in the large-gradient limit. Separately, in the canonical static planar Enchan metric, divergence of the first or second spatial derivative of a bounded field forces divergence of the Kretschmann scalar. The result establishes a precise finite-tension regularity boundary. It does not claim inevitable finite-time singularity formation, a resolution of the Hawking–Penrose singularity theorems, or Cosmic Censorship.