Relational Gradient Flow: A Unified Stochastic Equation for Gravity, Quantum Pressure, and Information Coherence

This work introduces the Relational Gradient Flow — a single unifying equation that connects gravitational self-organization, quantum pressure, and informational coherence within one stochastic framework.

Starting from a discrete relational update

M_{t+1}=M_t+\sum_k \alpha\,\frac{s_k-M_t}{\|s_k-M_t\|^2}

-\sum_k \beta\,\frac{s_k-M_t}{\|s_k-M_t\|^3}+\mathcal N(0,\sigma^2),

\partial_t \rho = \nabla\!\cdot\!\Big(\rho\,\nabla[\beta\,\Phi_{\rm grav} + \alpha\,\Psi_{\rm info} + \gamma\,V_Q(\rho)]\Big) + D\,\nabla^2\rho.

Here,

 represents the classical gravitational potential (Poisson field),

 encodes an informational coupling (logarithmic coherence potential), and

 is the Bohm quantum potential that introduces a quantum-pressure–like regularization.

The equation describes the dynamics of any system — from galaxies and dark-matter halos to quantum fluids or cognitive networks — as a stochastic gradient descent in a unified free-energy landscape.

Each term corresponds to a different scale of order:

gravity → macroscopic coherence,

quantum potential → microscopic stability,

informational field → relational synchronization.

Within the Logic of Existence (LoE) framework, this is expressed as the existential condition

\delta J = 0,

The Relational Gradient Flow therefore functions as a bridge between classical and quantum domains — a candidate for a meta-Hamiltonian connecting astrophysical gravitation, emergent information dynamics, and the physics of consciousness through one variational principle.