On the Variational Principle of Information: A Mathematical Derivation of Gravity and Collapse from Geometric Constraints

Building on the frameworks established in On the Physical Consequences of Distinguishability and On the Stability and Collapse of Physical Information, this paper derives quantum state collapse and Einstein field equations from a single variational principle incorporating a holographic information constraint.

Following Jacobson’s thermodynamic approach to gravity, I propose that representational capacity — quantified by the Bekenstein bound — constrains both geometric and quantum dynamics. The constraint is formalized using Kolmogorov complexity: quantum states satisfy K(ψ) ≤ Imax(g), where K(ψ) denotes the minimal algorithmic description length. Varying a total action under this constraint yields Einstein equations with an additional stress-energy contribution from information overflow, and a modified Schrödinger equation with a collapse term proportional to excess complexity. The collapse rate emerges as Γ = (c/R) · max(0, K(ψ) − Imax)/Imax, containing no free parameters.

This framework suggests quantum collapse and gravitational curvature represent complementary responses to holographic saturation: energy excess induces curvature, information excess induces collapse. The approach makes testable predictions for interference visibility decay as a function of entanglement depth, distinguishable from existing objective collapse models.