6. The Rigorous Geometric Construction

(Reference: CMFT Theory v33)

1. Definition of the System

In the Corley Momentum Flux Theory, gravity is not a field generated by a scalar mass. It is the Linear Average of Rotational Drift generated by the interaction of flux vectors.

To simulate the gravitational efficiency of the Earth, we define the system not as a point, but as a Volumetric Object with a specific density profile matching geophysical data:

• Geometry: Sphere (R = 6371 km)
• Stratification:
  • Inner Core (r < 3480 km): High Flux Density (N ≈ 5.10)
  • Mantle (r > 3480 km): Low Flux Density (N ≈ 2.25)

2. Derivation of the Seed: The Sagitta Mechanism

The fundamental unit of gravity in CMFT is not the particle, but the Pairwise Interaction. The "Seed" efficiency factor arises from the geometry of momentum transfer between two source points (A, B) and a target (T).

The Geometric Construction

1. The Flux Arcs (Phase Accumulation):
Due to the finite speed of light limit (c) governing the medium, the momentum transfer from a source point is subject to Phase Accumulation. The pincer motion of the interaction is not instantaneous; the delay across the volume "fuzzifies" the precise vector origin, manifesting effectively as an Arc of Influence. The angular width of this arc is determined by the geometric relationship (separation angle γ) of the interacting pair.

2. The Centroids (C_A, C_B):
To resolve the momentum transfer, the system calculates the Geometric Centroid of these arcs. The magnitude of the centroid vector is governed by the Inverse Square Law (1/r²), but its effectiveness is scaled by the centroid calculation of the arc.

For an arc of angular width γ, the distance to the centroid is scaled by:

Geometric Justification: The Origin of the Sinc Filter

Why does sinc(γ/2) appear? Because momentum transfer propagates at finite speed c, the instantaneous line-of-action from a source to a target is not a delta-direction; it occupies an arc of influence of angular width γ (phase accumulation).

The resolved momentum-transfer direction is therefore the Geometric Centroid of that arc. The centroid distance along the bisector is exactly scaled by:

This is not a fitted filter; it is the closed-form centroid factor for an arc of angular support γ. The Sagitta Drift arises when two sources produce two such centroid vectors whose Linear Average yields a net inward bias at the target.

3. The Drift Vector (D_ij):
The net gravitational pull on the target is the Linear Average of these two centroids.

Geometrically, this resultant vector points towards the midpoint of the chord connecting the two source points. This geometric offset (the "Sagitta") represents the drift.

2.1 Formalization: The Solenoidal Green's Function

The Green's Function in CMFT operates not merely as a decay filter, but as a Vector Selection Mechanism. It connects the Pure Momentum of the atomic flux to the Tension of the vacuum by isolating the specific geometric component that survives cancellation.

1. The Survival of Rotation:
While the radial "push-pull" influence of the flux effectively cancels out in the pressurized medium, the Solenoidal (Rotational) Influence at the target does not. This is because the flux is relativistic momentum independent of the medium, traveling at finite speed c.

2. Phase Accumulation & The Drift:
This finite speed introduces Phase Accumulation across the volume of the source. The influence arriving at the target is therefore not a vector point, but a phase-delayed Arc of Influence. The Green's Function calculates the Geometric Centroid of this rotating flux.

The Sagitta Drift emerges because the linear average of these centroids lies inside the geometric chord. This offset represents a "ghost vector" or defect in the field. The vacuum, maintaining pressure in balance with the flux, exerts Tension to close this geometric tear, which we perceive as the gravitational pull.

• The Far-Field Limit (Newtonian Recovery): As distance increases, the opening angle γ → 0. Since sinc(0) = 1, the function perfectly recovers the standard Newtonian potential (1/r²).
• The Near-Field Limit (Geometric Deficit): As distance decreases, γ increases. The impedance term drops below 1.0, creating the "Geometric Drag" (1/r⁴) responsible for the orbital precession anomalies observed in General Relativity.

This confirms that the CMFT model is a Constructive Lagrangian Solver—it reproduces the necessary curvature not by bending the coordinate system, but by solving the wave equation for a compressible fluid using kinematic geometry.

3. The General Equation: Unified Flux Integration

While the Geometric Seed (Section 2) defines the efficiency of the connection, we must rigorously define the Input being connected. To capture the full kinematic behavior of the stress tensor (including Frame Dragging and Pressure), we replace scalar mass with a Flux 4-Vector.

3.1 The Flux 4-Vector (Φ)

Every voxel i in the volume is defined as a generator of momentum flux:

• Scalar Flux (Φ⁰): The total "Push Back" pressure. Derived from Layer 1 (Density) and Layer 3 (Tension Map). This replaces simple mass.
• Flux Bias (Φ^vec): The directional asymmetry caused by motion (Rotation/Shear). This vector represents the transverse momentum carried by the flux stream.

3.2 The Unified Drift Calculation

The total drift force is derived by integrating the interaction of these 4-Vectors over every unique pair. The Seed Efficiency (sinc) is applied to the radial term, while the Kinematic Shear is applied to the transverse term.

For a target T and a source voxel i with unit vector r̂:

The Integration Logic:

• Radial Term: This utilizes the Geometric Seed derived in Section 2. The sinc(γ/2) filter modifies the static pressure pull, creating the 1/r⁴ Sagitta Drift (Precession) in the near field.
• Shear Term: This utilizes the Cross Product (×). It captures the "Memory of Motion." If the source is rotating, the flux carries that lateral momentum, creating a torque on the target (Frame Dragging) without curving the vacuum.

4. The Sampling Method: Voxel Grid

Because summing 10⁵⁰ atoms is impossible, we utilize a Monte Carlo Voxel Approximation. The Earth is discretized into Representative Volume Elements (RVEs) forming a 3D Point Cloud.

The Grid Construction:

• Radial Steps (dr): The sphere is sliced into concentric shells.
• Theta/Phi Steps (dθ, dφ): Each shell is sectored into grid points.
• Volume Assignment: Each point represents a physical volume dV = r² sin(θ) dr dθ dφ.
• Density Assignment: Each point is assigned an N value based on its depth (Core vs. Mantle).

This reduces the Earth to a cloud of weighted points. The integration runs in O(N²) time, calculating the precise Sagitta Drift for every voxel pair.

5. Calibration: The Far-Field Anchor

To validate the Corley Momentum Flux profile against standard Newtonian gravity, we must establish a calibration baseline. This initial estimate attempts to trace the binding energy of the system via the distribution of electrons (Flux Generators) defined in Subsection 1 (Core vs. Mantle).

Why the Moon?
We initially select the Moon's distance (60 Earth Radii) as an anchor because the angular width of the Earth is negligible (γ ≈ 0). Since sinc(0) = 1.0, this represents the Newtonian Ideal. By pinning the two theories together at this "Clean" far-field point, we can observe how the CMFT profile diverges as we approach the "Messy" near-field surface.

The Calibration Formula:
We calculate the ratio of efficiency between the Surface and the Orbit.

The Result: ~97%
This initial calculation indicates that while the Far Field (Moon) receives 100% of the Earth's potential flux, the Surface receives approximately 97%.

The missing 3% is not error; it is the Geometric Deficit caused by the large opening angles (γ) of flux pairs near the surface, where the "Arc Centroids" drift inward.

Crucially, the exact magnitude of this deficit depends on the density profile used. To understand the true fit, we must examine the entire curve rather than a single point. Readers should refer to Section 11 Simulation: Electron-Flux Coherency Test, which performs a live pairwise integration to visualize this profile and allows you to interactively test different calibration points (Surface vs. Moon).