Companion F: Proton as 5D Junction: Variational Foundation

We establish the variational foundation for the proton’s Y-junction topology within 5D Elastic Diffusive Cosmology (EDC). Starting from the postulate that baryons are flux-tube networks in the 5D bulk [P], we derive that equaltension junctions minimize total length when arms meet at 120◦ angles [Der]. This provides the missing foundational derivation for Paper 2’s identification of the proton geometry. The key result is that the Steiner configuration emerges from force balance, not as an external input but as a consequence of variational minimization. Calibration boundary: This companion contains zero calibrated parameters. All results follow from geometric postulates and standard variational calculus. The tension parameter σ enters as an overall scale but cancels in angle determinations. Epistemic status: The junction postulate is [P]. Given this postulate, the 120◦ angle is [Der] via explicit variational argument.