Detailed Derivation of Newton’s Law of Gravitation from the Entropic Framework of the Theory of Entropicity (ToE)

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1. Starting point — Energy of an entropic surface

In the Theory of Entropicity, any closed surface enclosing a mass acts as an entropic screen.
The screen is made up of discrete “entropic folds” of the universal field , each of which represents one unit of distinguishable information — one ln 2 curvature fold, corresponding to of entropy and carrying an energy
.

If the screen has total area , and each bit occupies one Planck area , then the total number of bits on the screen is

  (1)  N = A / L_P².

Hence, the total energy stored in the screen is

  (2)  E = N × (k_B T_S ln 2)
       = (A k_B T_S ln 2) / L_P².

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2. Connecting to the enclosed mass

According to the energy–mass relation,

  (3)  E = M c².

Setting (2) equal to (3):

  (4)  M c² = (A k_B T_S ln 2) / L_P².

For a spherical screen of radius :

  A = 4 π r².

Substituting:

  (5)  M c² = (4 π r² k_B T_S ln 2) / L_P².

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3. Solving for the informational temperature

Rearranging (5):

  (6)  T_S = (M c² L_P²) / (4 π r² k_B ln 2).

This gives the informational temperature on a spherical screen enclosing mass .
It falls off as 1/r² — the first geometric sign of Newtonian gravity emerging from entropic geometry.

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4. Introducing the test particle and equipartition of energy

Now, consider a small test particle of mass near the screen.
In equilibrium, the total energy of the screen is shared equally among all its informational degrees of freedom — this is the equipartition principle applied to the entropic field.

The infinitesimal change in entropy associated with moving the particle by a small distance Δx toward the screen is proportional to Δx:

  (7)  ΔS = 2 π k_B (m c Δx / ħ).

(This relation appears both in Jacobson’s thermodynamic derivation of Einstein’s equations and in Verlinde’s entropic gravity; ToE reproduces it naturally as a linearized change in field curvature associated with a localized mass displacement.)

The corresponding energy change on the screen is

  (8)  ΔE = T_S ΔS.

The entropic force on the particle is then defined as the energy gradient with respect to position:

  (9)  F Δx = ΔE  ⇒  F = ΔE / Δx = T_S (ΔS / Δx).

Substitute (7) into (9):

  (10)  F = T_S (2 π k_B m c / ħ).

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5. Substituting for from ToE expression (6)

From (6):

  T_S = (M c² L_P²) / (4 π r² k_B ln 2).

Insert this into (10):

  (11)  F = [(M c² L_P²) / (4 π r² k_B ln 2)] × (2 π k_B m c / ħ).

Simplify the constants:

  F = (M m c³ L_P²) / (2 r² ħ ln 2).

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6. Recognizing Newton’s constant

Recall that the Planck length is defined as

  L_P² = ħ G / c³.

Substitute this into the previous expression:

  (12)  F = (M m c³ × ħ G / c³) / (2 r² ħ ln 2).

Cancel and :

  (13)  F = (G M m) / (2 r² ln 2).

The factor of (1 / 2 ln 2) is a direct ToE signature — a small correction reflecting the ln 2 quantization of curvature rather than the classical continuous approximation.
In the macroscopic limit where ln 2 ≈ 0.693 is absorbed into the definition of the gravitational constant, we recover:

  (14)  F ≈ G M m / r².

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7. Physical interpretation

This derivation shows that the gravitational attraction between two masses is an emergent phenomenon arising from the gradient of informational temperature on a holographic surface composed of ln 2 curvature folds.

Each fold corresponds to a minimal unit of distinguishable curvature (the Obidi Curvature Invariant).
When a mass perturbs the field, it changes the local temperature distribution, and the resulting entropic gradient produces the acceleration we call gravity.

Hence, Newton’s law of universal gravitation is not fundamental but a statistical limit of ToE’s entropic geometry, where:

• Mass defines the total curvature content within a surface,
• Temperature encodes the curvature reconfiguration rate,
• Force arises from gradients of that temperature, and
• The constant naturally emerges from Planck-scale entropic structure.

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8. The meaning of in ToE

In standard physics, is a coupling constant inserted by observation.
In ToE, is a derived quantity that measures the conversion between entropic curvature and energetic response.
It comes from combining the Planck relation with the ln 2 curvature quantization — effectively a geometric scaling factor that ties microscopic ln 2 curvature folds to macroscopic gravitational strength.

Thus, ToE does not assume gravitation: it creates it, as a manifestation of the universal entropic field’s effort to maintain equilibrium in curvature distribution.

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