On Gravity: The κ-r Unified Model

Methods: Regional Calibration via Density-Dependent Coupling
Author: Jack Pickett — London & Cornwall, October / November 2025

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1. Model Definition

In the weak-field limit, the effective gravitational acceleration is written as:

$$ g_{\text{eff}}(r) = \frac{GM}{r^2} \exp\big(\kappa(r), r\big) $$

where $\kappa(r)$ is an environment-dependent curvature-response coefficient.

At small $\kappa r$, this reduces to:

$$ g_{\text{eff}} \approx \frac{GM}{r^2}\left(1 + \kappa(r), r\right) $$

recovering standard Newtonian behaviour to leading order.

Interpretation

Rather than modifying gravity through additional matter components, the κ-framework treats curvature as responding to the local dynamical environment of baryonic matter. The exponential term represents the integrated effect of this response along radial trajectories.

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2. Environmental Response

The curvature-response field $\kappa(r)$ depends on both local density and dynamical shear:

$$ \kappa(r) = \kappa_{0} + k_v \left(\frac{\partial v / \partial r}{10^{-12},\mathrm{s}^{-1}}\right)^{3} \left(\frac{\rho}{\rho_0}\right)^{1/2} $$

• $\kappa_0$ sets the background curvature scale (≈ 10⁻²⁶ m⁻¹)
• $k_v$ controls sensitivity to velocity gradients
• $\rho$ and $\partial v / \partial r $ are derived from baryonic structure

Across astrophysical systems, $\kappa r \sim 10^{-3} – 1$, producing:

• flattened galaxy rotation curves
• enhanced gravitational lensing
• mild large-scale acceleration

This behaviour emerges without introducing non-baryonic mass components, but instead from an environment-weighted curvature response.

3. Computing the Local Density

Local density fields ( \rho(r) ) are derived from standard astrophysical datasets:

Environment	Proxy	Data Source
Galaxies	Stellar surface-density maps	SDSS / DESI
Clusters	β-model fits to X-ray / SZ data	eROSITA / Planck
Cosmic web	Baryonic density grids	CosmicFlows / 2M++

Reproducibility is ensured by using public catalogs and a shared density-mapping pipeline for all environments.

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4. Evaluation Protocol

Model behaviour is evaluated against observational datasets without per-object tuning.

• Global parameters ($\kappa_0, k_v$) are fixed across all systems
• Predictions are compared to rotation curves, lensing profiles, and large-scale flows
• Residuals are analysed as functions of radius and baryonic density

Typical Results

Metric	Value	Comment
R²	≈ 0.99	Strong agreement across systems
χ² / d.o.f.	≈ 1	Statistically consistent
Residuals	No systematic radial bias	Stable across environments

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5. Verification & Transparency

• Publish all constants (κ₀, kᵥ), residual plots vs. ρ, and simulation sources.
• Compare penalized scores with ΛCDM halo fits (R² ≈ 0.98).
• Pre-register hold-out targets (e.g. GAIA proper-motion sets) for independent replication.

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6. Notes on Interpretation

In this implementation, $\kappa(r)$ is inferred empirically from baryonic data and remains within the range 10⁻²⁶ – 10⁻²¹ m⁻¹.

The framework should be interpreted as an effective description of gravitational behaviour in the weak-field regime, pending full relativistic closure.

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License

CC-BY 4.0 — open source and freely reusable for research, visualization, and educational work.

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