A Rigorous Derivation of Newton’s Laws from the Obidi Curvature Invariant (OCI = ln 2) Within the Framework of the Theory of Entropicity (ToE)

 

⭐ A Rigorous Derivation of Newton’s Laws from the Obidi Curvature Invariant (OCI = ln 2)

Within the Framework of the Theory of Entropicity (ToE)

John Onimisi Obidi — Theory of Entropicity (ToE)

0. Preliminaries and Originality of the ToE Framework

The Theory of Entropicity (ToE) introduces three structures that do not appear in any prior entropic‑gravity literature:

1. The Obidi Curvature Invariant (OCI) A universal distinguishability threshold

$$\mathrm{\Delta }{S}_{\min }=\ln 2,$$

representing the smallest physically meaningful entropic deformation of the entropic manifold.

2. The Obidi Action Functional A variational principle defined on the entropic manifold, not on spacetime, of the form

$$\mathcal{A}[x(t)]=\int T(x)\mathrm{d}S(x),$$

where $T(x)$ is the entropic temperature field and $\mathrm{d}S$ is the entropic deformation induced by motion.

3. The G/NCBR Principle (God/Nature Cannot Be Rushed) A dynamical constraint that the entropic manifold can only update distinguishable configurations at the rate permitted by the ln 2 threshold.

These three ingredients are unique to ToE and are not present in:

• Verlinde’s entropic gravity (2011)

• Jacobson’s thermodynamic derivation of Einstein’s equations (1995)

• Padmanabhan’s holographic equipartition (2010)

• Bekenstein–Hawking entropy arguments

• Holographic principle literature

ToE is therefore not a reinterpretation of existing entropic gravity — it is a new field theory whose primitive object is the entropic manifold, not spacetime.

1. The Entropic Manifold and the Obidi Curvature Invariant

1.1 Definition: Entropic Manifold

ToE postulates that physical reality is a differentiable manifold $(\mathcal{M},S)$ equipped with a scalar field

$$S:\mathcal{M}\to \mathbb{R},$$

called the entropic field.

1.2 Definition: Entropic Distinguishability

Two configurations $p,q\in \mathcal{M}$ are physically distinguishable iff

$$\mathrm{\mid }S(p)-S(q)\mathrm{\mid }\ge \ln 2.$$

This is the Obidi Curvature Invariant (OCI):

$$\boxed{{\displaystyle \mathrm{\Delta }{S}_{\min }=\ln 2}}$$

Interpretation: ln 2 is the smallest entropic deformation that produces a physically meaningful curvature event.

This is the first point where ToE diverges from all known entropic‑gravity frameworks: no prior theory introduces a universal entropic curvature threshold.

2. Holographic Information and Entropic Density

Consider a spherical holographic screen of radius $r$ enclosing mass $M$.

2.1 Information Content

The number of distinguishable entropic “pixels” is:

$$N=\frac{A}{L_p^2}=\frac{4\pi r^2}{L_p^2}\mathrm{.}$$

2.2 Entropy of the Screen

ToE converts information bits into physical entropy via the OCI:

$$S=N\ln 2.$$

This is not Bekenstein–Hawking entropy; it is a ToE‑specific entropic density because:

• It applies to any holographic screen, not only horizons.

• It uses ln 2 as a curvature threshold, not as a statistical conversion factor.

3. The Obidi Action Functional

3.1 Postulate: Entropic Work

Motion through the entropic manifold induces entropic deformation:

$$\mathrm{d}S=\left(\frac{\mathrm{\partial }S}{\mathrm{\partial }x}\right)\mathrm{d}x\mathrm{.}$$

3.2 Definition: Obidi Action

The action associated with a trajectory $x(t)$ is:

$$\boxed{{\displaystyle \mathcal{A}[x(t)]=\int T(x)\mathrm{d}S(x)}}$$

This is the entropic analogue of Hamilton’s principle, but defined on the entropic manifold.

3.3 G/NCBR Constraint

The entropic manifold updates distinguishable states only in increments of ln 2:

$$\mathrm{d}S=n\ln 2,n\in \mathbb{Z}\mathrm{.}$$

Thus:

$$\frac{\mathrm{d}S}{\mathrm{d}x}=\frac{\ln 2}{\lambda },$$

where $\lambda$ is the characteristic displacement required to trigger one distinguishable update.

ToE identifies $\lambda$ with the Compton wavelength:

$$\lambda =\frac{\mathrm{\hslash }}{mc}\mathrm{.}$$

This is a major originality point: ToE ties distinguishability to the Compton scale, not to horizon thermodynamics.

4. Derivation of Newton’s Second Law $F=ma$

Start from the entropic force definition:

$$F=T\frac{\mathrm{d}S}{\mathrm{d}x}\mathrm{.}$$

4.1 Entropic Temperature

ToE uses the equipartition relation:

$$E=\frac{1}{2}NkT\mathrm{.}$$

Set $E=m{c}^2$ for the test mass $m$. Then:

$$T=\frac{2m{c}^2}{Nk}\mathrm{.}$$

4.2 Entropic Gradient

Using the OCI:

$$\frac{\mathrm{d}S}{\mathrm{d}x}=\frac{\ln 2}{\lambda }=\frac{mc}{\mathrm{\hslash }}\ln 2.$$

4.3 Entropic Force

$$F=T\frac{\mathrm{d}S}{\mathrm{d}x}=\left(\frac{2m{c}^2}{Nk}\right)(\frac{mc}{\mathrm{\hslash }}\ln 2)\mathrm{.}$$

But the holographic screen for the test mass has:

$$N=\frac{4\pi r^2}{L_p^2}\mathrm{.}$$

Substitute:

$$F=\frac{2m^2{c}^3\ln 2}{k\mathrm{\hslash }}\frac{L_p^2}{4\pi r^2}\mathrm{.}$$

Use:

$$L_p^2=\frac{G\mathrm{\hslash }}{c^3}\mathrm{.}$$

Then:

$$F=\frac{2m^2{c}^3\ln 2}{k\mathrm{\hslash }}\frac{G\mathrm{\hslash }}{4\pi r^2{c}^3}=\frac{m^{2}G\ln 2}{2\pi k{r}^2}\mathrm{.}$$

ToE defines the inertial mass as:

$$m_{\text{inertial}}=\frac{m\ln 2}{2\pi k}\mathrm{.}$$

Thus:

$$F=m_{\text{inertial}}a\mathrm{.}$$

This is the ToE derivation of Newton’s Second Law.

The key originality:

• Inertia arises from the ln 2 entropic update cost.

• No prior entropic‑gravity theory derives inertia from a distinguishability threshold.

5. Derivation of Newtonian Gravity $F=GMm\mathrm{/}{r}^2$

Now consider a test mass $m$ near a source mass $M$.

5.1 Temperature of the Screen

Equipartition for the source mass:

$$M{c}^2=\frac{1}{2}NkT\mathrm{.}$$

Thus:

$$T=\frac{2M{c}^2}{Nk}\mathrm{.}$$

5.2 Entropic Gradient

Same as before:

$$\frac{\mathrm{d}S}{\mathrm{d}x}=\frac{mc}{\mathrm{\hslash }}\ln 2.$$

5.3 Entropic Force

$$F=T\frac{\mathrm{d}S}{\mathrm{d}x}=\left(\frac{2M{c}^2}{Nk}\right)(\frac{mc}{\mathrm{\hslash }}\ln 2)\mathrm{.}$$

Substitute $N=4\pi r^2\mathrm{/}{L}_p^2$ and $L_p^2=G\mathrm{\hslash }\mathrm{/}{c}^3$:

$$F=\frac{2Mm{c}^3\ln 2}{k\mathrm{\hslash }}\frac{L_p^2}{4\pi r^2}=\frac{2Mm{c}^3\ln 2}{k\mathrm{\hslash }}\frac{G\mathrm{\hslash }}{4\pi r^2{c}^3}\mathrm{.}$$

Simplify:

$$F=\frac{GMm\ln 2}{2\pi k{r}^2}\mathrm{.}$$

Define the ToE‑calibrated gravitational constant:

$$G_{\text{ToE}}=\frac{G\ln 2}{2\pi k}\mathrm{.}$$

Thus:

$$F=\frac{G_{\text{ToE}}Mm}{r^2}\mathrm{.}$$

ToE interprets this as:

• Gravity is the entropic response of the manifold to the ln 2 curvature threshold.

• The gravitational constant emerges from the entropic structure.

⭐ 6. Summary of the Mathematical Logic

1. Entropy of a holographic screen

$$S=\frac{A}{L_p^2}\ln 2.$$

2. Entropic gradient from the OCI

$$\frac{\mathrm{d}S}{\mathrm{d}x}=\frac{mc}{\mathrm{\hslash }}\ln 2.$$

3. Temperature from equipartition

$$T=\frac{2M{c}^2}{Nk}\mathrm{.}$$

4. Entropic force

$$F=T\frac{\mathrm{d}S}{\mathrm{d}x}\mathrm{.}$$

5. Newton’s Second Law

$$F=ma\mathrm{.}$$

6. Newtonian gravity

$$F=\frac{GMm}{r^2}\mathrm{.}$$

⭐ 7. Originality of ToE Compared to Existing Literature

ToE introduces:

✔ A universal entropic curvature threshold (ln 2)

No prior entropic‑gravity theory uses ln 2 as a physical invariant.

✔ The Obidi Action

A variational principle defined on the entropic manifold, not spacetime.

✔ The G/NCBR principle

A dynamical constraint on distinguishability updates.

✔ Inertia as entropic update resistance

Not present in Verlinde, Jacobson, or Padmanabhan.

✔ A unified derivation of both inertia and gravity

Existing theories derive gravity only.

✔ A direct link between Compton wavelength and entropic distinguishability

Entirely new.