Finite-Response Coupled Field Dynamics (FRCFD): Classical Tests, Field Derivation, and GR Correspondence

March 21, 2026

“This work establishes internal consistency and testable structure; full cosmological fitting is a subsequent program.”

From Substrate Response to Emergent Spacetime: Testing Gravity Without Geometry

Derek Flegg
Southern Ontario, March 2026

1. Derivation of the Substrate Field S(r)
2. Correspondence with General Relativity
3. Perihelion Precession
4. Light Bending
5. Consistency with Known Physics
6. Falsification Structure

A Saturation‑Driven Alternative to Spacetime Curvature

1. Derivation of the Substrate Field S(r)

The substrate field obeys the coupled equation:

∂²S/∂t² − c²∇²S + βS³ = g|Ψ|²

For static, spherically symmetric systems:

−c² ∇²S + β S³ = g ρ(r)

1.1 Weak-Field Limit

For S ≪ Smax:

∇²S = − (g / c²) ρ(r)

Comparing with Newtonian gravity:

∇²Φ = 4πG ρ

We identify:

g = 4π G c²

1.2 Point Mass Solution

ρ(r) = M δ(r)

S(r) = GM / r

This establishes S as the gravitational potential in the linear regime.

1.3 Nonlinear Regime

Outside matter:

∇²S = (β / c²) S³

The cubic term suppresses growth, producing saturation instead of divergence.

2. Correspondence with General Relativity

The effective propagation metric is:

ds² = f(S)² dt² − f(S)⁻² dr² − r² dΩ²

f(S) = exp(−S / Smax)

2.1 Weak-Field Expansion

S = GM / r

f(S) ≈ 1 − S/Smax

g_tt ≈ 1 − 2GM / (r Smax)
g_rr ≈ −(1 + 2GM / (r Smax))

2.2 Matching Condition

Comparing with Schwarzschild:

g_tt = 1 − 2GM/r

We obtain:

Smax ≈ 1

Thus FRCFD recovers General Relativity at leading order.

3. Perihelion Precession

3.1 Orbital Equation

Using conserved quantities and defining u = 1/r:

d²u/dφ² + u = (GM / L²) (1 + 3GMu / Smax)

The additional term arises from the expansion of f(S).

3.2 Solution

Perturbative solution yields:

u(φ) ≈ (GM / L²)[1 + e cos((1 − ε)φ)]

with:

ε = 3GM / (a Smax (1 − e²))

3.3 Precession Rate

Δφ ≈ 6π GM / (a (1 − e²) Smax)

For Smax = 1, this reproduces the standard GR result.

4. Light Bending

4.1 Null Trajectories

ds² = 0

Using conserved quantities:

(dr/dφ)² = r⁴ [1/b² − f(r)² / r²]

4.2 Weak-Field Expansion

f(r)² ≈ 1 − 2GM / (r Smax)

4.3 Deflection Angle

Integrating the trajectory:

Δφ ≈ 4GM / (b Smax)

Thus:

Smax = 1 → matches General Relativity
Smax ≠ 1 → predicts measurable deviation

5. Consistency with Known Physics

Weak-field gravity: recovered at leading order
Light bending: matches GR when Smax ≈ 1
Perihelion precession: GR limit recovered
Strong-field regime: deviations explicitly predicted

6. Falsification Structure

FRCFD is constrained by a single universal constant:

Smax

This must satisfy:

Weak-field tests → Smax ≈ 1
ISCO → Smax ≈ 1/3
Shadow radius → Smax ≈ 0.5

The theory is falsified if these values cannot be reconciled.

Core Insight

FRCFD replaces geometric ontology with a response-based framework:

Geometry → Response dynamics
Curvature → Impedance gradients
Singularity → Saturation limits

Conclusion

With the derivation of S(r), recovery of the weak-field limit, and successful reproduction of classical tests, FRCFD reaches the minimum threshold for a viable gravitational framework.

Its defining feature is not flexibility, but constraint: a single parameter governs all deviations from General Relativity, making the theory directly testable.

— Derek Flegg

Conspiranon