Resolving the Hubble Tension through TAU Phase Dynamics: Transient Post-CMB Phantom Dark Energy

The application of the **$\mathbf{\TAU}$ Research Protocol ($\mathbf{PR-TAU}$)** to the **Hubble Tension ($\mathbf{H_0}$)** problem reveals a dynamic solution to the discrepancy between early Universe ($\mathbf{CMB}$, low value) and late Universe (standard candles, high value) measurements. $\mathbf{\TAU}$ posits that Dark Energy, anchored in the **Critical Pressure ($\mathbf{\Pcrit} \approx 4.7 \times 10^{-10} \ \text{Pa}$)**, is not a pure cosmological constant ($\mathbf{w}=-1$) but a fluid whose equation of state ($\mathbf{w_{\text{Warp}}}$) has **transiently exceeded $\mathbf{w}=-1$** (a **Phantom** regime) in the late Universe. By utilizing the $\mathbf{\TAU}$ Cosmic Acceleration Equation (Eq. 1), we show that this dynamic $\mathbf{w_{\text{Warp}}} < -1$ behavior induces the required additional acceleration to match the local $\mathbf{H_0}$ value with Supernovae observations. This prediction is directly falsifiable by mapping the Dark Energy equation of state $\mathbf{w}(\mathbf{z})$.