4. Mathematical Formalism: The Atomic Flux Generator

To rigorously justify the Corley Momentum Flux Theory, we must strip away macroscopic approximations and define the fundamental unit of gravity: the Atomic Flux Generator.

The Diagnostic: The "Scalar Choice"

Standard physics treats Mass (m) as a scalar invariant—a static value. However, Energy (E) and Momentum (p) are vectors. This "Data Type Mismatch" forces General Relativity to curve the background geometry to reconcile the equations.

CMFT corrects this by defining the atom not as a scalar mass, but as a source of Rotational Momentum Flux. We replace the geometric curvature with a hydrodynamic interaction.

1. The Fundamental Vector (Jrot)

We replace the static scalar density (ρ) with a dynamic Vorticity Vector (Jrot). The atom is a pump. It intakes vacuum pressure and emits a rotational flux field. This field contains both push-pull (radial) and rotational (solenoidal) components.

Jatom  =  ∇ × (Fluxinternal)

Gravity is not a force assigned to the mass; it is the mechanical result of this rotational flux interacting with the vacuum substrate.

2. The Corley Lagrangian Density (L)

The Lagrangian describes the energy balance of the universe. In CMFT, we balance the Elasticity of the Vacuum against the Coupling of the Atomic Generator.

LCMFT  =  [ -1/(4μ) (∇ × Ψ)2 + 1/(2κ) (∂t Ψ)2 ]  -  [ Ψ ċ ∑ Jatom ]

The Vacuum Term (Grey): Describes the substrate elasticity and inertia (limiting speed to c).
The Interaction Term (Red): This describes the coupling. Crucially, there is no "Quadrupole Tensor" hard-coded here. We do not need to tell the universe to behave like a quadrupole. We simply define the rotational input of the atoms.

3. The Hydrodynamic Wave Equation

By applying the Principle of Least Action (δS = 0), we derive the equation of motion. This is a linear Hydrodynamic Wave Equation:

□ Ψ  =  μ ∑ Jatom

Where □ is the D'Alembertian Operator representing wave propagation. The complex gravitational behaviors we observe—such as the Precession of Mercury—are not inputs. They are Emergent Geometric Consequences. When billions of atomic flux generators interact across a finite delay, the summation of their individual rotational vectors creates a net drift. The 1/r4 effect is simply the statistical result of Shape-to-Shape interaction in a delayed medium.

4. The Bridge to Simulation: Discrete Time as Proxy

Because the analytical summation of billions of delayed vectors is computationally prohibitive, our simulations utilize a specific mathematical proxy: Discrete Time Integration.

In a continuous universe, momentum transfer is delayed by c. This delay causes Phase Accumulation—the flux vector arriving at the target is slightly "old," pointing to where the source was, not where it is. This geometric offset creates the Sagitta (the drift vector).

In our Cavendish simulation, we use discrete time steps. This is not "fake gravity." We strictly apply Newton's Third Law to the rotational vectors of the atoms. The "Integration Error" inherent in discrete steps acts as a high-fidelity statistical proxy for "Phase Accumulation." By allowing this integration drift to persist, we successfully model the Sagitta mechanism. The net attraction is not programmed; it emerges naturally from the geometry.