THE EQUATION OF STATE OF THE ER STRING (XIX v6)

In the FCD-RBv6 pregeometric framework, hadronic confinement is derived analytically from the constitutive elastodynamics of trans-brane Einstein–Rosen (ER) bridges embedded in the AdS₅ bulk and regularized at the centrifugal Compton floor R₀ = ℏ/(m_const c) ≈ 0.6308 fm. Five core foundations are formalized: (1) The Constitutive Equation of State Dilemma: evaluating Case A (entanglement-fixed throat area, a = const, producing linear confinement V(r) = σ₀ r) against Case B (classical volume-conserving thinning, a² r = const, yielding sublinear non-confining potentials V(r) ∝ √r); (2) Holographic Freezing via Ryu–Takayanagi: modeled as a closed Thermofield Double (TFD) state connecting Brane A and Brane B, the strict conservation of entanglement entropy (dS_ent = 0) algebraically freezes the cross-sectional area of the extremal bottleneck (ℓ = 0) such that da/dr ≡ 0, ruling out classical plastic necking without imposing unphysical rigidity away from the central throat; (3) Longitudinal Elasticity and Hyperbolic Geometry: outside the minimal bottleneck, the bridge retains its hyperbolic Morris–Thorne profile r²(ℓ) = a₀² + ℓ², enabling linear longitudinal elongation with constant string tension σ₀ = 2(m_const c²)²/(ℏ c) ≈ 0.99 GeV/fm up to the elastic snapping threshold at ΔE ≈ 2 m_const c² ≈ 625 MeV (hadronization); (4) Parameter-Free Regge Trajectories: the string tension yields a characteristic energy scale √(σ₀) ≈ 442 MeV (within 1% of lattice QCD simulations) and a Regge slope α' = 1/(2πσ₀) ≈ 0.81 GeV⁻² matching light hadron spectroscopy; and (5) The Non-Singular Harmonic Core: proper radial geometry regularizes the short-distance potential into an exact harmonic well V(ℓ) = V₀ + (1/2)k ℓ² + O(ℓ⁴), banishing Coulombic singularities at r < 0.4 fm. Internal string tension and macroscopic gravitation are strictly decoupled: the former governs strong confinement, while the latter is mediated by adiabatic bulk phase-space depletion (dS = 0).