Systematic Survey of Vacuum Energy–Gravity Coupling in Casimir Geometries: Perturbative Mechanisms, Cutoff Prescriptions, and the Holographic Approach to Dark Energy

The cosmological constant problem — the 121-order-of-magnitude discrepancy between quantum field theory predictions and the observed dark energy density — motivates in- vestigation of how vacuum energy couples to gravity. We use a specific Casimir geometry, the Bi: Ni crystal lattice at interplanar spacing d = 2. 035 Å with verified energy density u ≈ 10⁹ J/m³, as a concrete test bed. We systematically evaluate twelve distinct grav- itational coupling mechanisms spanning general relativity, scalar-tensor (Brans-Dicke) theory, f (R) gravity, asymptotic safety, entanglement entropy (Jacobson thermodynam- ics), holographic information theory, and quantum electrodynamic vacuum polarization. All perturbative mechanisms are Planck-suppressed, yielding ΔG/G between 10⁻¹⁶ (Brans- Dicke binding energy) and 10⁻¹³⁷ (non-minimal coupling). We then evaluate six vacuum energy cutoff prescriptions. The Cohen-Kaplan-Nelson holographic bound, applied with spherical geometry and the Hubble radius as infrared cutoff, gives ρCKN = ρcrit, over- shooting the observed dark energy density by a factor of 1/Ω_Λ ≈ 1. 46 with zero free parameters. With the de Sitter event horizon (Li 2004), the match is exact within that model. We show that Standard Model species counts cancel in the CKN bound and that the result is independent of H₀. These results reduce the cosmological constant prob- lem from 121 orders to an O (1) question about the correct infrared scale, and provide a complete map of what works, what fails, and why.