Description: This paper investigates the Fejér‑centered boundary gap energy G(a), which bridges two equivalent criteria for the Riemann Hypothesis (RH): the Fejér first‑moment condition and the critical logarithmic mean‑square condition. It identifies the gap as the meeting point of three obstructions: weighted zero‑packet energy, damped zero pair‑correlation, and prime residual long‑memory variance. 🔹 Fejér‑Centered Gap Defined as G(a)=a∫∣S(X)−aB(X)∣2e−2aXdX. Spectral form involves Hardy energy and the Laplace‑Fourier transform of H0(s). Weight function Wa(t) peaks at microscopic boundary scale ∣t∣≈a. 🔹 Zero‑Packet Analysis Each critical zero contributes a packet of the form 1/(a+i(t−y)). Diagonal packet weight: harmless, scales like y−2, summable over zeros. Off‑diagonal correlations: main obstruction; require cancellation or Bessel condition. Introduces damped pair‑correlation kernel Pa(u)=2a2/(4a2+u2). 🔹 Pair‑Correlation Model In dyadic blocks of zeros, packet interactions modeled by Pa(u). A no‑excessive‑clustering bound on zero pairs would imply boundedness of G(a). Links to Montgomery’s pair correlation but with damping. 🔹 Prime‑Side Interpretation Residual measure dM(u)=∑A(n)/n dlog⁡n−eu/2du. Kernel Ka(u,v)=e−2amax⁡(u,v)(1−a∣u−v∣)/4. Governs long‑memory correlations of weighted prime powers. Equivalent to variance of multiplicative interval residuals E(X,H). 🔹 Obstruction Identified On the zero side: off‑diagonal packet correlations and off‑line poles. On the prime side: critical cancellation in residual correlations, not raw prime‑pair estimates. Controlling G(a) requires either a zero‑packet correlation bound or a prime residual correlation bound. 🔹 Conclusion V120_2 shows that the Fejér‑centered gap energy is the precise obstruction between Fejér holomorphy and Hardy boundary energy. It unifies three perspectives: Zero‑packet energy Damped pair‑correlation Prime residual variance The paper does not prove RH unconditionally, but it diagnoses the exact analytic obstacle that must be overcome.