The Spectral Triple $(A, H, D)$ Spectral Action Principle of Alain Connes and the Obidi Action Principle of the Theory of Entropicity (ToE): Studies in Non-Commutative Geometry and Entropic Information Geometry

1. What Connes’ spectral action actually is

Connes’ spectral action is built on a spectral triple ((A, H, D)):

(A): an involutive algebra (commutative for manifolds, noncommutative for internal degrees of freedom).
(H): a Hilbert space of fermions.
(D): a Dirac operator encoding geometry.

The action is:

[ S_{\text{Connes}} = \langle \psi, D\psi \rangle + \text{Tr}\left( f\left(\frac{D}{\Lambda}\right) \right), ]

where:

(\langle \psi, D\psi \rangle) gives the fermionic action.
(\text{Tr}(f(D/\Lambda))) gives the bosonic action (gravity + gauge + Higgs), via heat kernel expansion.
(f) is a cutoff function, (\Lambda) is an energy scale.

Key points:

The degrees of freedom are encoded in (A, H, D).
The geometry is encoded in the spectrum of (D).
The action is a functional of the spectrum of (D).
It reproduces: Einstein–Hilbert + Yang–Mills + Higgs + fermions.

This is a geometric field theory action. It tells you how fields on a (possibly noncommutative) space evolve and interact.

It does not:

treat entropy as a field.
derive distinguishability.
introduce ln 2 as a curvature invariant.
impose a No‑Rush Theorem or ETL.
unify classical and quantum information geometry.

It is a spectral geometry action, not an entropic ontology action.

2. What the ToE action actually is

The ToE action is built on the entropic field (S(x)), defined on an entropic manifold of configurations. The fundamental object is not a Dirac operator but the curvature of the entropic field.

The ToE action (schematically) is:

[ \mathcal{A}{\text{Obidi}}[S] = \int \mathcal{L}{\text{entropic}}(S, \nabla S, \text{InfoGeom}) , d\mu, ]

where:

(\mathcal{L}_{\text{entropic}}) is constructed from information‑geometric quantities: Fisher–Rao, Fubini–Study, α‑connections.
The field (S) is entropy itself, not a matter field on spacetime.
The curvature of (S) defines distinguishability, structure, and emergence.

From this action, ToE derives:

the Obidi Curvature Invariant (OCI) = ln 2.
the quantum of distinguishability.
the No‑Rush Theorem.
the Entropic Time Limit (ETL).
the emergence of particles as entropic minima.
the emergence of spacetime as a macroscopic shadow of entropic geometry.

So:

Connes’ action: a functional of a Dirac operator on a spectral triple.
ToE action: a functional of an entropic field on an information‑geometric manifold.

They are not the same kind of object.

3. How they differ at a structural level

Here’s the direct structural contrast:

Connes’ spectral action:

Input: ((A, H, D)).
Geometry: encoded in (D).
Action: (\langle \psi, D\psi \rangle + \text{Tr}(f(D/\Lambda))).
Output: gravity + gauge + Higgs + fermions.
Domain: effective field theory on (possibly noncommutative) spacetime.
Ontology: fields on a space.

ToE (Obidi) action:

Input: entropic field (S(x)) on an entropic manifold.
Geometry: encoded in information metrics and entropic curvature.
Action: (\mathcal{A}_{\text{Obidi}}[S]) built from Fisher–Rao, Fubini–Study, α‑connections.
Output: distinguishability, ln 2, ETL, No‑Rush, emergent spacetime, emergent fields.
Domain: ontological substrate of reality.
Ontology: entropy as the field from which space, fields, and time emerge.

Connes starts with a geometry and encodes physics into it.
ToE starts with entropy and generates geometry and physics from it.

4. How ToE subsumes Connes’ action

This is the interesting part.

ToE can subsume Connes’ spectral action if we view Connes’ framework as a special emergent regime of the entropic manifold.

The idea, in outline, is:

From entropic manifold to emergent spacetime
- The entropic manifold, with its information geometry, has regions where the entropic curvature behaves like a smooth Riemannian manifold.
- In that regime, you can define an emergent spacetime metric (g_{\mu\nu}) as a coarse‑grained object derived from entropic curvature.
From entropic curvature to Dirac operator
- On this emergent spacetime, you can define an effective Dirac operator (D_{\text{eff}}) whose spectrum reflects the underlying entropic structure.
- The spectral triple ((A, H, D_{\text{eff}})) then becomes an effective description of a particular entropic phase.
From Obidi Action to Spectral Action
- In the appropriate limit (large‑scale, low‑curvature, coarse‑grained), the Obidi Action reduces to an effective action that depends only on the spectrum of (D_{\text{eff}}).
- In that regime, the ToE action can be shown to approximate something of the form: [ \mathcal{A}{\text{Obidi}}[S] ;\longrightarrow; \langle \psi, D{\text{eff}}\psi \rangle + \text{Tr}\left( f\left(\frac{D_{\text{eff}}}{\Lambda}\right) \right), ] i.e. a Connes‑like spectral action.
Interpretation
- Connes’ spectral action then appears as the effective field‑theoretic shadow of a deeper entropic dynamics.
- The spectral triple is not fundamental; it is an emergent structure in a particular entropic phase.
- The Standard Model + gravity are one entropic phase of the universe.

In other words:

Connes: “Given a spectral triple, here is the action.”
ToE: “Here is the entropic field whose large‑scale, low‑energy phase looks like a spectral triple with a spectral action.”

Connes’ action is horizontal (within physics).
ToE’s action is vertical (beneath physics).

5. Why ToE is more fundamental in this hierarchy

ToE explains:

why distinguishability is quantized (ln 2).
why transitions require finite entropic time (ETL).
why measurement outcomes are discrete.
why spacetime emerges.
why fields exist at all.
why any spectral triple can exist as a physical structure.

Connes assumes:

a spectral triple exists.
a Dirac operator exists.
a Hilbert space of fermions exists.
an algebra of observables exists.

ToE explains why such structures can exist and be distinguishable in the first place.

So, the hierarchy is:

Level 0 (Ontological): ToE, Obidi Action, entropic field, ln 2, No‑Rush, ETL.
Level 1 (Geometric Effective): emergent spacetime, emergent Dirac operator, emergent spectral triple.
Level 2 (Field Theoretic): Connes’ spectral action, Standard Model + gravity.

Connes lives at Level 2.
ToE lives at Level 0 and generates Levels 1 and 2.

6. Summary

Connes’ action: an action on a given geometry.
ToE action: an action of the entropic substrate from which geometry itself emerges.
Connes: “Given ((A, H, D)), here is the physics.”
ToE: “Given entropy as a field, here is why ((A, H, D)) can exist at all.”

So:

Connes has not already done what ToE is doing.
ToE does not compete with Connes; it explains him.
In principle, the Obidi Action can reduce to a Connes‑type spectral action in the appropriate emergent regime.

Theory of Entropicity (ToE) by John Onimisi Obidi

Wikipedia

Wednesday, 21 January 2026