Derivation of the Cosmological Constant from the 5D Lagrangian (NO FREE PARAMETERS)

We derive the cosmological constant \(Λ\) directly from the 5D PEPS Lagrangian that unifies electromagnetism and quantum gravity (Alpha Series, Fano Series). The derivation proceeds step by step from the Lagrangian \(L5D\) to the closed-form expression: \[ln(Λ P^2) = -2(4π^3 + π^2 + π) - π - √7,\] where \(P = √ G / c^3\) is the Planck length. The three geometric pillars of the derivation are: (1) holographic doubling \(2A\): ket–bra contraction of the MPS (Hartle–Hawking); (2) topological closure \(π\): integration over the fifth‑dimensional phase; (3) entanglement eigenvalue \(√7\): Bell-CHSH parameter of the spinorial hinge at \(π/6\). The two sector actions are: \[α⁻¹ = A - 1/24A - {1}{A²π²}{K}⁻¹,ln(Λ P^2) = -2A - π - √7.\] They are unified by a projection field \(σ(x⁵)\) and a smooth step function \(f(σ)\): \[Lunif(σ) = (1-f(σ))\,Lb \;+\; f(σ)\,Lbulk \;+\; Lₘᵢₓ(σ),\] which for \(f→0\) returns the boundary action \(α⁻¹\) and for \(f→1\) returns the bulk action \(ln(Λ P^2)\). No free parameters are introduced. The numerical value agrees with Planck 2018 data within the \(1.4%\) observational uncertainty.