Topological Origin of the Hedgepeth Stress Concentration Factor: $K = 1 + 1/e^{\ln(3)} = 4/3$ from Bidirectional Coordination Geometry

Hedgepeth's (1961) exact shear-lag result for the stress concentration factor (SCF) in a 1D infinite fiber array, $K=4/3$, has lacked a geometric interpretation for over sixty years. In this work, we demonstrate that this fundamental constant follows directly from the bidirectional coordination threshold $\lambda \ell = \ln(3)$ derived in [1]. Key theoretical contributions include: Geometric Interpretation: At the percolation threshold, the fractional load excess on each intact nearest-neighbor fiber equals $1/e^{\ln(3)} = 1/3$, as $e^{\ln(3)} = 3$ represents the minimum cluster size required for bilateral load redistribution. Zero-Parameter Prediction: We predict a critical fiber-break density $\lambda_c = \ln(3)/\ell_c$, where $\ell_c$ is the ineffective length measurable by single-filament fragmentation tests (SFFT). Bridge to 1D Poisson Theory: This provides the first rigorous link between composite failure mechanics and 1D Poisson proximity networks. All results are analytical. This document serves as a pre-registration of the topological derivation of $K=4/3$ and the associated critical density predictions.