CCEGA Paper 17: The Hard Threshold as an Intrinsic Property of the Sen Tachyon Vacuum — Exact Derivation of κ(ρ) (v1.1)

Derives the exact density-dependent coupling κ (ρ) from the Sen tachyon field equation for the power-law form Gₑff (ρ) = G (ρc/ρ) ^2/3. Shows that the divergence at ρc is intrinsic to the non-analytic structure of L ∝ φ^1/2 at its vacuum — not a pathology but a critical point analogous to a second-order phase transition. v1. 1 adds Section 6: Reconciliation with Exponential Coupling. The exponential form Gₑff = G exp (−ρ/ρc), adopted in subsequent CCEGA papers (P25, P34, P43, P44), is more consistent with observational constraints. The key insight — intrinsic hard threshold from tachyon tension divergence — applies to both forms. P17 is not retracted; the power-law derivation remains valid and pedagogically important. Related: P25 v1. 1 (doi: 10. 5281/zenodo. 19284768), P34 v2. 0 (doi: 10. 5281/zenodo. 19766662), P43 v1. 1 (doi: 10. 5281/zenodo. 19394258).