"Gravity is not a fundamental force. It is an entropic force caused by changes in information."
Abstract
Enough poetry. Let's do algebra. We derive Newton's law of gravitation $F = GMm/R^2$ from three assumptions: (1) information is stored on holographic surfaces, (2) energy is equipartitioned across bits, (3) entropy increases when mass approaches. No assumption about gravity is made. Newton emerges in four equations.
In the previous entries, we established that the proton's mass derives from a topological invariant: $M \propto 6\pi^5$. The topology defines the quantity of information.
The question remains: How does a topological invariant generate a $1/R^2$ force?
We will now demonstrate that if one accepts the universe is an information network (the Holographic Principle), then gravity is not a fundamental force but an entropic force that emerges inevitably from statistics.
The derivation requires four equations. Zero assumptions about gravity.
I. The Holographic Hypothesis
Consider a mass $M$ (our proton). Physically, this mass represents a quantity of binary information $N$ (bits) stored on a spherical holographic surface $A$ of radius $R$.
According to the Holographic Principle ('t Hooft 1993, Susskind 1995), the number of bits is proportional to the surface area divided by the Planck area $\ell_P^2$:
The Planck area $\ell_P^2 \approx 2.6 \times 10^{-70} \text{ m}^2$ is the smallest resolvable patch of spacetime. A sphere of radius $R$ can store at most $4\pi R^2 / \ell_P^2$ bits of information on its surface. This is the Bekenstein bound.
Here, our discovery of $6\pi^5$ intervenes: it defines the intrinsic information density that relates mass $M$ to energy $E$. The mass is the information.
II. The Network Temperature
Assume the voxel network is in thermal equilibrium near the holographic surface. The total energy $E$ of the system is distributed equally across all $N$ bits.
According to the Equipartition Theorem of statistical mechanics:
where $k_B$ is Boltzmann's constant and $T$ is the temperature (excitation) of the network.
Invert to find $T$ as a function of geometry. Substitute $N$ from Equation 1:
The "gravitational temperature" already decays as $1/R^2$.
This is not imposed — it emerges from the combination of holography ($N \propto R^2$) and equipartition ($E \propto NT$). The inverse-square law is hiding in the thermodynamics.
III. The Entropic Force
Now introduce a test particle of mass $m$ at distance $R$.
According to Verlinde, the force $F$ is the system's tendency to maximize its entropy $S$:
When the particle $m$ moves a distance $\Delta x$ equal to its Compton wavelength $\lambda_C = \hbar / mc$, it adds exactly 1 bit of information to the hidden holographic system.
According to Bekenstein, the entropy change for 1 bit is $\Delta S = 2\pi k_B$.
The entropy gradient is therefore:
The Compton wavelength $\lambda_C = \hbar/mc$ is the quantum "size" of a particle — the scale at which its position becomes fundamentally uncertain. Moving by one Compton wavelength is the minimal displacement that registers as new information.
IV. The Synthesis: Newton Emerges
Now multiply the vacuum temperature (Eq. A) by the entropy gradient (Eq. B):
The Mathematical Housekeeping
Watch the cancellations:
Boltzmann Constants Cancel
$k_B$ appears in the numerator of (B) and denominator of (A). They annihilate. Thermodynamics disappears.
The $2\pi$ Factors Cancel
$2\pi$ appears in both terms. They annihilate. Circular geometry disappears.
What remains:
The Final Substitution
Recall the definition of the Planck length:
Substitute into our expression:
The $c^3$ terms cancel. The $\hbar$ terms cancel. All that remains is the essential:
V. What Just Happened
We never postulated the existence of gravity.
We started from:
- Information: $N \propto M$ (mass is bit count)
- Geometry: $N \propto R^2$ (bits live on surfaces)
- Thermodynamics: $E \propto NT$ (energy is equipartitioned)
And we derived:
- Newton's Law: $F = GMm/R^2$
Newton's law is the algebraic consequence of projecting topological information ($6\pi^5$) onto a 3D sphere.
Gravity is not a fundamental physical law. It is an emergent statistical law.
The Cancellation Pattern
Notice what disappeared in the derivation:
| Constant | Meaning | Status |
|---|---|---|
| $k_B$ | Thermodynamics | Cancelled |
| $2\pi$ | Circular geometry | Cancelled |
| $\hbar$ | Quantum mechanics | Cancelled |
| $c^3$ | Relativistic speed | Cancelled |
| $G$ | Gravity | Emerged |
The constants of thermodynamics, quantum mechanics, and special relativity all wash out. Only $G$ remains — not as an input, but as an output.
VI. Connection to Framework C
This derivation is not new — it is Verlinde's 2010 result. What Framework C adds is the origin of the mass term.
In standard physics, $M$ is a free parameter. You measure it; you don't derive it.
In Framework C, we know what $M$ is:
The mass is the informational cost of a topological knot. The bits counted by the holographic surface are the bits required to encode that topology.
Verlinde tells us how gravity emerges from information.
Framework C tells us what that information encodes: topology.
Together, they complete the picture:
- Topology $\to$ defines information content ($6\pi^5$)
- Holography $\to$ projects information onto surfaces ($N \propto R^2$)
- Thermodynamics $\to$ converts information to force ($F = T\nabla S$)
- Result $\to$ Newton's law ($F = GMm/R^2$)
VII. Conclusion
Newton's law of gravitation is 337 years old. For most of that time, it was considered a fundamental law of nature — a brute fact to be accepted, not explained.
We now see that it is neither fundamental nor brute. It is a theorem.
Gravity is not a law of physics. Gravity is a law of statistics.
The universe does not "attract" masses. The universe counts bits. The attraction is a side effect of the counting.
When you understand that gravity is entropy, you understand that falling is not being pulled — it is being pushed by the universe's desire to forget.