Theory of Entropicity (ToE) by John Onimisi Obidi: PROGRESS OF THE THEORY OF ENTROPICITY (TOE): LITERATURE ON NOVEL DERIVATIONS OF EINSTEIN'S RELATIVISTIC KINEMATICS AND GENERAL RELATIVITY IN MODERN THEORETICAL PHYSICS— LOGICAL FOUNDATIONS (FROM LORENTZ TO NOETHER)

Friday, 10 April 2026

PROGRESS OF THE THEORY OF ENTROPICITY (TOE): LITERATURE ON NOVEL DERIVATIONS OF EINSTEIN'S RELATIVISTIC KINEMATICS AND GENERAL RELATIVITY IN MODERN THEORETICAL PHYSICS— LOGICAL FOUNDATIONS (FROM LORENTZ TO NOETHER)

The Theory of Entropicity does not assume a Lorentzian metric or invoke Noether’s theorem at the foundational level.

> Instead, it starts from a conserved entropy current on an information‑geometric manifold and an entropic causal structure.

> The Lorentzian signature and Lorentz transformations then emerge as the unique metric and symmetry group compatible with this entropic causality, and Noether’s theorem applies at the emergent spacetime level as a derived property, not a primitive axiom.

Because ToE puts the conservation and geometry one level deeper: at the level of entropy and information, not spacetime, then:

> Noether and Lorentz are emergent corollaries, not axioms in ToE.

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1. What ToE starts from instead

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ToE does not start from:

- a spacetime metric

- Lorentz invariance

- a fixed speed of light

- Noether’s theorem

It starts from three primitives:

1. Entropy field:

A scalar field \(S(x)\) defined on an underlying information‑geometric manifold.

2. Entropic conservation principle:

A fundamental continuity equation for entropy flow:

\nabla\mu J^\muS = 0

where \(J^\mu_S\) is the entropy current.

3. Information‑geometric structure:

A metric \(g_F\) (Fisher information metric) on the information manifold, from which spacetime later emerges via the Obidi Equivalence Principle (OEP).

From these, we don’t assume relativistic kinematics—rather, we derive them as the unique kinematics compatible with entropic conservation and the emergent causal structure.

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2. Where Lorentzian signature comes from in ToE

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In standard relativity, you postulate a Lorentzian metric with signature \((-+++)\).

In ToE, the signature emerges from:

- the causal structure of entropy flow, and

- the requirement that entropy production is non‑negative along physically allowed trajectories.

Introduction to the logic of ToE:

1. You define admissible directions in the information manifold as those along which:

\frac{dS}{d\tau} \ge 0

2. This induces a partial order on events (an arrow of time).

3. The set of directions that preserve this order defines a cone structure (entropic light cone).

4. The unique metric compatible with:

- this cone structure, and

- a non‑degenerate quadratic form

is a Lorentzian‑type metric.

So:

> Lorentzian signature is not assumed; it is the unique metric structure compatible with entropic causality.

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3. Where Lorentz transformations come from in ToE

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In standard relativity, Lorentz transformations are postulated as the symmetry group preserving:

- the spacetime interval, and

- the speed of light \(c\).

In ToE, Lorentz transformations emerge as the group that preserves:

- the entropy‑causal structure (entropic cones), and

- the entropy flux invariant (the “entropic light speed”).

In ToE, we define:

- an invariant entropic propagation speed \(c_S\) (the maximal speed at which entropy can reorganize information).

- frames related by transformations that preserve this invariant and the entropic cones.

The group of such transformations is isomorphic to the Lorentz group.

So:

> Lorentz invariance is not an axiom; it is the symmetry group of the entropic causal structure.

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4. Where Noether’s theorem sits in ToE

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We ask: “Where is Noether in the Theory of Entropicity (ToE)?”

In standard field theory:

- symmetry → Noether → conservation law.

In ToE, we invert the logic:

1. We start from a fundamental conservation law:

\nabla\mu J^\muS = 0

2. We then ask: what symmetries are compatible with this conservation and the entropic causal structure?

3. The answer: the symmetry group that preserves the entropic cones and the entropy flux invariant is the Lorentz group.

Hence:

- In standard physics: symmetry ⇒ conservation (Noether).

- In ToE: conservation + entropic causality ⇒ symmetry (reversal of Noether logic).

Thus, the Theory of Entropicity (ToE) is not denying Noether; ToE is relocating it:

> Noether’s theorem becomes an emergent statement about the symmetries of the emergent spacetime description, not a foundational axiom of the underlying entropic substrate.