CCEGA Paper 17: The Hard Threshold as an Intrinsic Property of the Sen Tachyon Vacuum — Exact Derivation of κ(ρ)

Derives exactly the density-dependent coupling κ(ρ) deferred in Paper 16. The result diverges at ρ_c and vanishes at infinity. Shows this divergence is intrinsic to L∝φ1/2: the tachyon tension T(φ)=T₀φ-1/2 diverges at the vacuum φ=0, making ρ_c a critical point analogous to a second-order phase transition. The hard threshold is not imposed — it emerges from the non-analytic structure of the Lagrangian. Verified with SymPy. All predictions of Paper 16 unchanged.