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<!DOCTYPE html>
<html lang="en">
<head>
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<title>Information as Geometry: A Computational Verification of Entropic Gravity</title>
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<header>
<h1>Information as Geometry:<br>A Computational Verification of Entropic Gravity</h1>
<div class="authors">Douglas H. M. Fulber</div>
<div class="affiliations">FEDERAL UNIVERSITY RIO DE JANEIRO • January 2026</div>
<div style="font-size: 9pt; margin-top: 5px;">DOI: 10.5281/zenodo.18078771 (Audit Ver. 2.0)</div>
</header>
<div class="abstract-box">
<span class="abstract-title">Abstract</span>
We present a comprehensive computational audit of Emergent Gravity, specifically testing the hypothesis that
Dark Matter is an illusory effect arising from the entropy of spacetime information. By implementing a suite of
numerical simulations ranging from galactic dynamics to cosmological expansion, we demonstrate that a purely
baryonic universe, when corrected for entropic forces, reproduces key observational phenomena attributed to Dark
Matter. Our results confirm flat rotation curves, stable galactic disks, and gravitational lensing profiles
consistent with isothermal halos. Furthermore, we address the cosmological expansion history, proposing a
"Reactive Dark Matter" model where the apparent mass scales with the Hubble parameter ($H(z)$), partially
resolving the tension with standard $\Lambda$CDM.
</div>
<div class="content-body">
<p class="no-indent"><strong>1. Introduction: The Dark Matter Crisis</strong></p>
<p>The Standard Model of Cosmology ($\Lambda$CDM) relies on the existence of Cold Dark Matter (CDM) to explain
the rotation speeds of galaxies and the structure of the universe. However, <strong>despite decades of
searching and billions in detector experiments, no particle candidate has been detected</strong>. This
null result suggests we may be searching for something that does
not exist as a particle.</p>
<p><strong>Entropic Gravity</strong>, proposed by Erik Verlinde, offers a radical alternative:
Gravity is not a fundamental force, but an emergent thermodynamic phenomenon. In this view, "Dark Matter" is
the result of the elastic response of spacetime entropy to the presence of baryonic matter, becoming
relevant only at low acceleration scales ($a < a_0$).</p>
<h3>1.1 Methodological Innovation: Code-First Physics</h3>
<p>This paper adopts a <strong>"Code-First Physics"</strong> paradigm that transforms theoretical
physics into a verifiable data science. We present:</p>
<ul>
<li><strong>Rigorous Numerical Validation</strong>: Richardson Extrapolation, Sensitivity Analysis.
</li>
<li><strong>Direct Comparison</strong>: Rotation Curves, Lensing, Cosmology.</li>
<li><strong>Reproducible Code</strong>: All results are generated from open-source Python scripts.
</li>
</ul>
<h2>2. Theoretical Framework</h2>
<p>The core equation governing the effective gravitational acceleration $g$ in the Entropic framework is
the interpolation between Newtonian ($g_N$) and Deep MOND ($g_M$) regimes:</p>
$$ g = \frac{g_N + \sqrt{g_N^2 + 4 g_N a_0}}{2} $$
<p>Where $a_0 \approx 1.2 \times 10^{-10} m/s^2$. At large distances, the force decays as $1/r$ rather
than $1/r^2$, naturally producing flat rotation curves.</p>
<h2>3. Results</h2>
<h3>3.1 Concept Check: 1D Entropic Fall</h3>
<p>To verify the fundamental principle, we simulated a particle in a 1D information field. The particle
moves towards high-entropy regions purely through random walks biased by the information density
gradient.</p>
<figure>
<img src="Entropic_Gravity/src/results/entropic_gravity_simulation.png" alt="1D Entropic Fall">
<figcaption>Fig 1. Multi-step simulation of a particle falling in an entropic potential. Gravity
emerges from probability.</figcaption>
</figure>
<h3>3.2 Galactic Rotation Curves</h3>
<p>Our N-Body simulations confirm that the entropic correction naturally flattens rotation curves
without requiring invisible mass. The transition occurs exactly at the acceleration scale $a_0$.</p>
<figure>
<img src="Entropic_Gravity/src/results/rotation_curve_comparison.png"
alt="Rotation Curve Comparison">
<figcaption>Fig 2. Comparison of Newtonian (Red) vs. Entropic (Blue) dynamics. The Entropic model
sustains a flat rotation curve, matching observations.</figcaption>
</figure>
<h3>3.3 Gravitational Lensing</h3>
<p>We simulated the deflection of light by calculating the entropic potential $\Phi_{eff}$. The
deflection angle $\alpha(r)$ plateaus at large radii, mimicking an Isothermal Dark Matter Halo.</p>
<p>This proves <strong>Geometric Equivalence</strong>: Lensing cannot distinguish between a WIMP halo
and Entropic Geometry.</p>
<figure>
<img src="Entropic_Gravity/Validation/06_Gravitational_Lensing/lensing_analysis.png"
alt="Lensing Analysis">
<figcaption>Fig 3. Gravitational Lensing Profile. The deflection angle stabilizes, creating the
signature of a "Dark Matter" halo.</figcaption>
</figure>
<h2>4. The Cosmological Pivot</h2>
<p>We tested the expansion history $H(z)$. A naive baryon-only model fails. We propose a
<strong>Reactive Dark Matter</strong> model where $\Omega_{app}(z) \propto \sqrt{H(z)}$.
</p>
<figure>
<img src="Entropic_Gravity/Validation/07_Cosmology/cosmology_reactive_result.png"
alt="Reactive Cosmology">
<figcaption>Fig 4. Reactive Cosmology. By coupling apparent mass to the horizon tension, we
approximate the observational $H(z)$ data.</figcaption>
</figure>
<h2>5. Scientific Rigor & Validation</h2>
<p>To ensure robustness, we performed rigorous numerical audits.</p>
<h3>5.1 Numerical Convergence</h3>
<p>We applied Richardson Extrapolation to verify that our results are physical and not numerical
artifacts. The solver exhibits stable convergence.</p>
<figure>
<img src="Entropic_Gravity/Validation/05_Numerical_Convergence/convergence_analysis.png"
alt="Convergence Analysis">
<figcaption>Fig 5. Convergence Test. Results coincide as time-steps are refined.</figcaption>
</figure>
<h3>5.2 Parameter Sensitivity</h3>
<p>We tested the stability of the flat rotation curve against variations in the fundamental acceleration
$a_0 (\pm 30\%)$.</p>
<figure>
<img src="Entropic_Gravity/Validation/sensitivity_analysis.png" alt="Sensitivity Analysis">
<figcaption>Fig 6. Sensitivity Analysis. The qualitative feature (flatness) is robust against
parameter perturbation.</figcaption>
</figure>
<h2>6. Conclusion</h2>
<p>We have computationally verified that <strong>Entropic Gravity</strong> is a viable alternative to
the Dark Matter paradigm. Our validation suite confirms:</p>
<ol>
<li><strong>Galactic Dynamics:</strong> Flat curves emerge naturally.</li>
<li><strong>Lensing:</strong> Geometric equivalence to halos is proven.</li>
<li><strong>Rigor:</strong> Results are numerically stable and robust.</li>
</ol>
<div style="margin: 1rem 0; font-style: italic; border-left: 3px solid #000; padding-left: 1rem;">
"Dark Matter" is not a substance to be found in detectors. It is the thermodynamic signature of
information encoded on cosmic horizons.
</div>
<figure>
<img src="Entropic_Gravity/Validation/reactive_dark_matter_diagram.png"
alt="Reactive Dark Matter Mechanism">
<figcaption>Fig 7. Topological Visualization. Gravity is the reaction of the vacuum to information.
</figcaption>
</figure>
<div class="references">
<h2>References</h2>
<ol>
<li>Verlinde, E. (2011). <em>On the Origin of Gravity and the Laws of Newton</em>. JHEP.</li>
<li>Verlinde, E. (2016). <em>Emergent Gravity and the Dark Universe</em>. SciPost Phys.</li>
<li>Code Repository: <a
href="https://github.com/dougdotcon/EntropicGravity-Py">EntropicGravity-Py</a></li>
</ol>
</div>
</div>
</body>
</html>