The Equation of State of the ER String: Linear Confinement from an Entanglement-Fixed Throat Area (XIX v4)

Within the pregeometric FCD-RB (Folded Crunch Dipole – Reversed Big Bang) framework, each massive quark constituent is modeled not as a primary pointlike field, but as the visible-brane endpoint of a closed Einstein-Rosen (ER) bridge that spans the fivedimensional bulk and anchors on a CP-conjugate mirror brane. Confinement is reinterpreted not as a gauge-mediated color force, but as the macroscopic elastic response of this ER bridge under stretching. This paper derives the constitutive equation of state of the ER string the relation between its stored energy and its elongation — from a single, robust holographic premise: the closed bipartite bridge behaves as a maximally entangled Thermofield Double (TFD) state whose throat cross-sectional area is held strictly invariant by the conservation of its entanglement entropy (S ent = A throat /4G N ). This fixed-area constraint forces a constant tension, leading to a strictly linear potential V (r) = σr at long distances. The tension is derived analytically without any free parameters: σ = 2(m const c 2 ) 2 /ℏc ≈ 0.99 GeV/fm, yielding a scale parameter √ σ = √ 2m const c 2 ≈ 442 MeV and a Regge slope α ′ = 1/(2πσ) ≈0 81 GeV −2 , both in outstanding agreement (within 1%−10%) with lattice QCD and lightmeson spectroscopy. At ultra-short distances, the throat s Morris-Thorne metric regularizes the coordinate divergence, giving rise to a smooth harmonic core (V 2 ) rather than a Coulombic singularity. This establishes a highly falsi able departure from standard QCD, with light hadrons living entirely in the linear regime while heavy-quarkonium spectroscopy marks the limits of the model s domain.