A Self‑Consistent Finite‑Response Field Theory of Gravitation: Finite‑Response Coupled Field Dynamics

Finite-Response Coupled Field Dynamics (FRCFD): A Minimalist Field-Theoretic Framework for Non-Pathological Gravity

Author: Derek Flegg
Date: March 21, 2026

Abstract

Finite‑Response Coupled Field Dynamics (FRCFD) is formulated as a 3+1‑dimensional field theory in which the vacuum is modeled as a finite‑response substrate with a maximum stress capacity \(S_{max}\). This replaces the infinite‑capacity geometric assumptions of General Relativity (GR) with a saturating response function that constrains curvature and temporal evolution in high‑stress regimes. The substrate is treated as the fundamental field whose state determines the effective metric, characterized by a displacement scalar \(S\), an update‑rate function \(f(S) = \exp(-S/S_{max})\), and a saturation boundary at \(S_{max}\). These elements define an operational hierarchy governing the infrared and ultraviolet behavior of the theory.

The Role of Each Equation in FRCFD

1. Substrate Response Equation

∂ₜ² S − c² ∇² S + β S³ = σ(x,t) F_R(C[Ψ])

Physical Role: This equation governs the dynamics of the displacement scalar \(S\), which represents the local state of the finite‑response substrate. It determines how the substrate responds to matter‑field loading and how curvature‑like behavior emerges from the field dynamics.

Finite‑Capacity Structure: The non‑linear term \(β S^{3}\) provides a saturating response that prevents divergent solutions. This term implements the finite‑capacity constraint associated with the upper bound \(S_{max}\).

Coupling: The source term \(\sigma(x,t) F_R(C[\Psi])\) specifies how the matter‑field configuration \(\Psi\) contributes to substrate stress. This defines the interaction between matter and the finite‑response field.

2. Matter‑Field Equation

∂ₜ² Ψ − v² ∇² Ψ + μ Ψ + λ |Ψ|² Ψ = κ S Ψ

Physical Role: This equation describes the propagation of the matter field \(\Psi\) within the finite‑response substrate. It determines how matter evolves under both its own self‑interaction and the influence of substrate stress.

Interaction Term: The coupling term \(κ S Ψ\) links the matter field to the substrate displacement scalar. This term encodes the dependence of matter‑field dynamics on the local substrate state.

Resulting Behavior: The coupling modifies the effective temporal and spatial evolution of \(\Psi\), producing scale‑dependent effects such as variations in local temporal rate (\(f(S)\)) and stress‑dependent corrections relevant to strong‑field systems.

To maintain consistency with Solar System tests while permitting strong‑field deviations, the framework introduces a running effective capacity \(S_{eff}(\sigma)\), yielding scale‑dependent Parametrized Post‑Newtonian (PPN) parameters \(\gamma(\sigma) = 1/S_{eff}(\sigma)\) and \(\beta(\sigma) = 1/S_{eff}(\sigma)^2\). The formalism recovers GR predictions in low‑stress environments while allowing departures in high‑stress systems such as neutron stars and black‑hole candidates, which can be evaluated against Event Horizon Telescope (EHT) measurements. The theory also examines a non‑expansionary account of cosmological redshift as a path‑integrated substrate impedance.

Overall, FRCFD aims to address the internal consistency of gravitational theory by modifying the response properties of the vacuum itself, exploring whether several auxiliary constructs in contemporary physics—such as dark matter and extra dimensions—may arise from assuming an unbounded geometric substrate.

I. Formal Definition of the Substrate as a Field

In FRCFD, the substrate is defined as a fundamental field whose behavior generates the structure of spacetime. It does not occupy space; its local state, represented by the displacement scalar \(S\), defines the effective geometry. This places the substrate in the same formal category as vacuum fields in quantum field theory, but with a finite response capacity. Introducing a maximum stress \(S_{max}\) provides a natural regulator that prevents curvature divergences and point‑mass singularities.

II. Operational Hierarchy and the Update-Rate Relation

The substrate dynamics follow a structured hierarchy:

  • Ground State: Substrate at rest, corresponding to flat geometry.
  • Substrate Stress \(S\): Displacement induced by mass‑energy.
  • Update‑Rate \(f(S)\): Internal response frequency governing local temporal evolution.
  • Saturation: The limit \(S \to S_{max}\), forming a high‑impedance boundary.

The coupling is defined by:

f(S) = exp(−S / S_max)

III. Phase 4: Scale-Dependent Capacity and PPN Drift

The running effective capacity \(S_{eff}(\sigma)\) ensures an infrared fixed point where \(S_{eff} \approx 1\), recovering the Solar System values \(\gamma = 1\) and \(\beta = 1\). In high‑stress ultraviolet regimes, the substrate undergoes softening. This scale dependence provides a testable trajectory for gravitational behavior, predicting that EHT observations should detect a slight stiffening of the metric near the photon ring, where \(\gamma(\sigma)\) deviates from the GR constant.

IV. Cosmological Redshift as Substrate Impedance

The framework investigates a non‑expansionary account of cosmological redshift. Photon energy loss is modeled as a path‑integrated interaction with the finite‑response substrate, producing redshift through vacuum impedance rather than metric expansion. This mechanism is evaluated for consistency with energy conservation, blackbody spectral preservation, and reciprocity relations, while maintaining a static 3+1‑dimensional background.

V. Conclusion: Theoretical Economy

FRCFD achieves theoretical economy by removing the infinite‑capacity postulate. By identifying singularities, dark matter, and extra dimensions as potential artifacts of a divergent geometric substrate, the theory replaces auxiliary scaffolding with a self‑regulating feedback mechanism. As the displacement scalar \(S\) increases, the update‑rate \(f(S)\) slows and the effective capacity \(S_{eff}\) runs, limiting field dynamics before pathological divergences can form. This yields a finite‑response field formalism capable of reproducing observational phenomena across weak‑field, strong‑field, and cosmological scales.

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