Unifying Dark Matter and Dark Energy via the Entropic Dynamics of Holographic Spacetime Stiffness (Preliminary Version - Obsolete)

The standard ΛCDM cosmological model invokes two elusive components—dark matter and dark

energy—to explain galactic rotation curves and cosmic acceleration. We propose a first-principles

modification to General Relativity in which the gravitational coupling is dynamically determined by

a local holographic entanglement density field Φ. Starting from a Jordan-frame scalar–tensor effec-

tive action motivated by entanglement thermodynamics, we derive the modified Einstein equations

and the scalar trace equation, enforcing a well-posed variational principle. To reconcile halo-scale

information dilution with local tests, we introduce an environmental dilution closure in which the

scalar potential is suppressed in low-density regions by a dimensionless matter invariant I∝−T(m)

,

yielding a covariant matter–scalar exchange Qν localized to transition layers around a characteris-

tic density ρ⋆. In the halo dilution regime the effective potential is negligible and the weak-field

scalar obeys a Laplace equation, admitting Φ(r) = Φbg + C/r; a wide radial window then exhibits

Φ ∝r−1 and produces approximately flat rotation curves without particle dark matter. We fur-

ther derive a sharp lensing–dynamics consistency relation in the dilution regime, including the mass

ratio Mlens/Mdyn = (2ω+ 3)/(2ω+ 4), and provide toy-model sanity checks (Hernquist and spher-

icalized exponential-disk baryons) showing that the predicted Einstein angles and transition-layer

thicknesses fall in observationally plausible ranges.