On the Originality of the Spectral Obidi Action (SOA) in the Full Obidi Action of the Theory of Entropicity (ToE)

The Spectral Obidi Action (SOA) undoubtedly has the full weight of the history of ideas behind it — because it is a deep idea.

We ask: Is the Spectral Obidi Action (SOA) truly original to ToE, or is ToE simply repeating what others have already done?

The short answer is direct:

The form of the SOA is not original.

The role, operator, ontology, and integration into ToE are original — and no one in the literature has done that.

Let us break this down clearly and rigorously.

1. What is not original (and cannot be claimed as original)

The following elements are not original to ToE:

1.1 The mathematical template

$$S=\mathrm{Tr}f(\mathcal{O}\mathrm{/}{\mathrm{\Lambda }}^2)$$

This is the Chamseddine–Connes spectral action (1997).

1.2 The heat-kernel expansion

$$K(t)\sim \sum t^{(n-4)\mathrm{/}2}\int \sqrt{g}{a}_n$$

This is standard Seeley–DeWitt theory (1960s–1980s).

1.3 The identification of

• $a_0$ → cosmological constant term

• $a_2$ → Einstein–Hilbert term

• $a_4$ → quadratic curvature terms

This is textbook spectral geometry.

1.4 Using Mellin transforms to relate spectral action to heat kernel

This is standard in spectral geometry and noncommutative geometry.

So, these parts are not original. They are mathematical infrastructure.

But that is not where ToE’s originality lies.

2. What is original — and this is where ToE is doing something no one else has done

There are four absolutely original contributions in ToE's SOA section that no other researcher has done, because they require the entire ToE framework.

Let’s go through them one by one.

2.1 The operator is new: the Entropic Laplacian 

${\mathrm{\Delta }}_S$

In all previous spectral-action literature:

• the operator is the Dirac operator $D$,

• or a Laplace-type operator on spacetime.

No one has ever defined a Laplacian on an information manifold twisted by an entropic field.

This is a ToE invention.

This alone is enough to justify the SOA of ToE as original.

2.2 The domain is new: the Information Manifold, not spacetime

In Chamseddine–Connes:

• the spectral action is defined on a geometric manifold (or almost-commutative geometry).

In ToE:

• the spectral action is defined on the information manifold $({\mathcal{M}}_I,g_I)$,

• whose metric is the Fisher–Entropic metric,

• whose curvature is information curvature, not spacetime curvature.

This is a completely different ontology. No one has ever applied spectral geometry to Fisher-information geometry in this way.

This is original.

2.3 The purpose is new: SOA generates global spacetime from information geometry

In the spectral-action literature:

• the spectral action is the gravitational action.

In ToE:

• the LOA generates local metric structure from entropic dynamics,

• the SOA generates global and topological structure of emergent spacetime.

This two-tier architecture — local entropic dynamics + global spectral generation — is unique to ToE.

No one has done this.

2.4 The interpretation of constants is new:

Newton’s constant and the cosmological constant become entropic spectral invariants

In standard spectral action:

• $G$ and ${\mathrm{\Lambda }}_{\text{CC}}$ are parameters to be matched.

In ToE:

• $G$ becomes

$$G=\frac{6\mathrm{\pi /}}{f_2{\mathrm{\Lambda ^}}^2}$$

an entropic constant derived from the spectrum of ${\mathrm{\Delta }}_S$.

• The cosmological constant becomes an entropic cosmological function, not a free parameter.

This is a conceptual revolution:

Gravity’s constants are not fundamental — they are informational spectral invariants.

No one in the literature has made this move.

2.5 The SOA is embedded inside a larger entropic theory

In Chamseddine–Connes:

• the spectral action is the theory.

In ToE:

• the SOA is one component of a larger architecture:

  • LOA (local entropic dynamics)

  • SOA (global spectral structure)

  • GEFE (General Entropic Field Equations/[Obidi Field Equations (OFE)])

  • OCI (Obidi Curvature Invariant)

  • ESSM (Entropic Seesaw Model)

  • Entropic substrate Ω

  • Information manifold ${\mathcal{M}}_I$

This is a new theoretical ecosystem.

No one has embedded the spectral action into an information-theoretic, entropic, emergent-spacetime framework.

3. So, is it worth talking about in ToE?

Yes, it is — because ToE is not copying the spectral action.

It is repurposing it in a completely new ontological setting.

The originality of the Spectral Obidi Action (SOA) lies in:

• the operator (entropic Laplacian),

• the domain (information manifold),

• the interpretation (entropic emergence),

• the constants (entropic spectral invariants),

• the role (global generation of spacetime),

• the integration into a larger entropic theory.

This is absolutely worth talking about, because:

No one has ever used spectral geometry to generate spacetime from information geometry.

No one has ever defined a spectral action on the Fisher–Entropic manifold. No one has ever derived Newton’s constant from entropic spectral data. No one has ever built a two-tier entropic + spectral emergence architecture.

This is original.

This is the elegance of the Theory of Entropicity (ToE).