Finite‑Response Gravity and the Scale‑Dependent Structure of Post‑Newtonian Dynamics

Finite-Response Coupled Field Dynamics (FRCFD) Phase 4: Running Substrate Capacity, Precision Constraints, and Controlled Cosmological Extension

The visualization below illustrates the extreme optical response of spacetime near a compact object. In FRCFD, similar features emerge not from geometry but from the saturation of a finite‑capacity substrate.

Author: Derek Flegg
Date: March 21, 2026

Table of Contents

1. Theoretical Position and Upgrade

FRCFD replaces geometric gravity with a finite-response scalar substrate S. Earlier formulations assumed a constant capacity S_max, which produces a direct inconsistency:

  • Weak-field: S_max ≈ 1
  • Strong-field: S_max ≈ 0.3–0.5

Phase 4 resolves this via: 

S_max → S_eff(σ)

This defines a classical effective field theory with scale-dependent coupling, not a quantum RG flow.

2. Running Capacity and Invariant Scale

σ = |∇S|^2

S(r) ≈ GM / r
∇S = −GM / r^2
σ = G^2 M^2 / r^4

f(S, σ) = exp( − S / S_eff(σ) )

All dynamics are now local and closed in terms of S and its derivatives.

3. Weak-Field Structure

ε(r) = GM / (r S_eff(r))

f^2 ≈ 1 − 2ε + 2ε^2

f^2 ≈ 1 − 2GM / (r S_eff(r))

This reproduces GR structure with a scale-dependent normalization.

4. Orbital Dynamics

d^2u/dφ^2 + u =
(3GM / S_eff) u^2
+ GM u^2 d/du (1 / S_eff)

σ = G^2 M^2 u^4
d/du = 4 G^2 M^2 u^3 d/dσ

d^2u/dφ^2 + u =
(3GM / S_eff) u^2
+ 4 G^3 M^3 u^5 d/dσ (1 / S_eff)

New physics appears as a suppressed u^5 correction.

5. Perihelion Precession

Δφ = 6πGM / (a (1 − e^2) S_eff(r_orbit))

Higher-order corrections are negligible in weak fields.

6. Mercury Constraint

Δφ ≈ 43 arcseconds / century

S_eff(r_Mercury) = 1 ± 10^-5

σ ≈ G^2 M^2 / r^4 ≈ O(1)

dS_eff/dσ |_(σ ≈ 1) ≈ 0

Conclusion: Running must be frozen in the Solar System.

7. EFT Functional Form

S_eff(σ) =
S_UV + (S_IR − S_UV) exp( − (σ / σ_crit)^n )
  • S_IR = 1 ± 10^-5
  • σ_crit ≫ σ_Mercury
  • n ≥ 2

This defines a nonlinear, finite-response EFT with saturation.

8. Two-Regime Structure

Region I — Infrared

  • S_eff ≈ 1
  • GR fully recovered

Region II — Strong Field

  • S_eff → 0.3–0.5
  • Observable deviations

This structure is derived from observational constraints, not assumed.

9. Cosmology (Constrained / Speculative)

Status: Not yet derived from field equations

S(χ) = γ χ   (ansatz)

ds^2 = f(S)^2 dt^2 − f(S)^(-2) dχ^2 − Σ(χ)^2 dΩ^2

1 + z = 1 / f(S_emit)

This reproduces redshift scaling but lacks:

  • Field-equation derivation
  • Geodesic deviation proof
  • Photon conservation proof

Conclusion: Cosmology remains a working hypothesis.

10. Falsifiability

Valid if:

  • No Solar System deviation
  • Strong-field transition observed

Ruled out if:

  • PPN deviations appear
  • No strong-field modification exists

The theory predicts a finite-response gravitational transition scale.

PPN Structure and Solar‑System Constraints in FRCFD

A crucial requirement for any alternative to General Relativity is consistency with Solar‑System precision tests. In the finite‑response framework, these constraints arise naturally once the weak‑field metric is expanded and matched to the standard Parametrized Post‑Newtonian (PPN) form.

Table of Contents — PPN Section

1. Weak‑Field Metric Expansion

The effective propagation metric in FRCFD is:

ds² = f(S)² dt² − f(S)⁻² dr² − r² dΩ²

with substrate response:

f(S, σ) = exp( − S / S_eff(σ) )

In the weak field:

S ≈ GM / r
ε(r) = GM / (r S_eff(r))

Expanding the response function:

f²   ≈ 1 − 2ε + 2ε²
f⁻²  ≈ 1 + 2ε + 2ε²

2. Matching to the Standard PPN Metric

The PPN metric in isotropic coordinates is:

g_tt = 1 − 2U + 2β U²
g_rr = − (1 + 2γ U)
U = GM / r

Substituting ε = U / S_eff into the FRCFD metric gives:

Radial component:

g_rr = − (1 + 2U / S_eff)
⇒ γ(σ) = 1 / S_eff(σ)

Temporal component:

g_tt = 1 − 2U / S_eff + 2U² / S_eff²
⇒ β(σ) = 1 / S_eff(σ)²

3. PPN Parameters as Running Observables

The finite‑response theory predicts:

γ(σ) = 1 / S_eff(σ)
β(σ) = 1 / S_eff(σ)²

Thus:

γ − 1 = (1 − S_eff) / S_eff
β − 1 = (1 − S_eff²) / S_eff²

For small deviations S_eff = 1 + δ:

γ − 1 ≈ −δ
β − 1 ≈ −2δ

Key result: PPN parameters are not constants but scale‑dependent functions of the substrate stress σ.

4. Solar‑System Precision Bounds

Cassini Shapiro Delay (γ):

|γ − 1| < 2.3 × 10⁻⁵
⇒ |S_eff − 1| < 2 × 10⁻⁵

Lunar Laser Ranging (β):

|β − 1| ≲ 10⁻⁴
⇒ |S_eff − 1| ≲ 5 × 10⁻⁵

The Cassini bound dominates:

S_eff(σ_solar) = 1 ± 10⁻⁵

5. Implications for the Running Function

The running variable is:

σ = G² M² / r⁴

In the Solar System, σ ≈ O(1). Thus the running must be frozen:

(dS_eff / dσ)|_(σ ≈ 1) ≈ 0

For the Phase‑4 response function:

S_eff(σ) = S_UV + (S_IR − S_UV) exp[ − (σ / σ_crit)ⁿ ]

PPN consistency requires:

  • σ_crit ≫ 1
  • S_IR = 1 ± 10⁻⁵

This ensures:

  • Infrared (Solar System): S_eff ≈ 1, γ ≈ 1, β ≈ 1
  • Strong‑field regime: S_eff → 0.3–0.5, γ → 2–3, β → 4–10

6. Physical Interpretation

The finite‑response substrate produces a sharp, testable prediction:

PPN parameters become scale‑dependent observables.

  • Weak fields: substrate unsaturated → GR recovered
  • Strong fields: substrate saturated → large deviations

This is the defining signature of FRCFD.

7. Falsifiability

FRCFD is ruled out if:

  • Cassini or LLR detect γ ≠ 1 or β ≠ 1 at Solar‑System scales
  • No strong‑field deviations are observed near compact objects

FRCFD is supported if:

  • Solar‑System tests remain GR‑like
  • Strong‑field transitions are detected (EHT, XRISM, lensing arcs)

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Core Principles of Finite‑Response Coupled Field Dynamics (FRCFD)

Table of Contents

1. The Saturation Principle (UV Completion)

In General Relativity, the gravitational field admits arbitrarily large curvature, leading to mathematical singularities. FRCFD replaces this with a finite‑capacity vacuum substrate characterized by an invariant limit Smax.

  • As field intensity increases, the substrate enters a saturated regime.
  • The gravitational potential becomes bounded.
  • Physical singularities are forbidden.

Result: Compact objects transition into a Saturated Core State, providing a UV‑complete description of gravity that remains finite at extreme energy densities.

2. Spacetime as an Emergent Metric Response

In FRCFD, curvature is not fundamental. Instead, it emerges from the nonlinear impedance of the substrate when stressed by mass‑energy distributions.

  • Mass‑energy induces a Substrate Stress Scalar S.
  • Metric response arises from:
    • Update‑Rate Suppression (time dilation)
    • Radial Metric Stiffening (length contraction)

Interpretation: Gravity is the observable manifestation of the vacuum’s finite relaxation rate under high energy‑density gradients.

3. The Running Capacity (Phase 4 Dynamical Coupling)

Phase 4 introduces a key refinement: the substrate capacity becomes a running quantity, evolving with the local field stress σ = |∇S|².

  • Infrared (IR) Regime: In weak fields, Seff ≈ 1, recovering Einsteinian geometry to 10⁻⁵ precision.
  • Ultraviolet (UV) Regime: Near horizons or ISCO radii, Seff softens toward ~0.3, producing strong‑field deviations from Schwarzschild predictions.

This running capacity is the central dynamical feature of Phase 4.

4. Scale‑Dependent PPN: The Observable Signature

FRCFD transforms the Parametrized Post‑Newtonian (PPN) framework from a set of constants into scale‑dependent observables γ(σ) and β(σ).

  • In GR: γ = β = 1 everywhere.
  • In FRCFD: γ and β flow with σ.

Signature: As one moves from the IR (Earth) to the UV (Sgr A*), these parameters drift, producing measurable anomalies in:

  • light‑bending
  • photon‑sphere structure
  • gravitational‑wave ringdown frequencies

5. Non‑Expansionary Cosmology

FRCFD offers a static‑background alternative to metric expansion. The observed Hubble redshift is reinterpreted as path‑integrated vacuum impedance.

  • Photons lose energy gradually as they propagate through the finite‑response substrate.
  • This cumulative effect produces a redshift identical in form to expansion.

Result: The theory recovers the Etherington Reciprocity Theorem, dL = dA(1 + z)², showing that SN Ia and BAO data can be explained without requiring the physical expansion of space.

6. The Field‑Theoretic Big Picture

FRCFD is a bridging formalism. It preserves the high‑precision geometric limit of Einstein in the IR while introducing saturated, finite‑response dynamics in the UV where geometric gravity fails.

The result is a unified, UV‑complete, observationally testable framework for gravitational physics across:

  • Solar‑System weak fields
  • Black‑hole strong fields
  • Cosmological photon propagation

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