Can the Theory of Entropicity (ToE) Derive the Alain Connes Spectral Action? A Roadmap and a Future Frontier

The emerging Theory of Entropicity (ToE) derives or fully integrates Alain Connes' Spectral Action Principle (SAP) as a foundational part of its unified entropic-spectral framework, viewing SAP not as separate but as emergent from a deeper, universal entropic field that unifies gravity, gauge theories, and matter via an entropic variational principle of the dual structure of the Obidi Action. ToE extends Connes' ideas by making the modular operator dynamical and unifying bosonic and fermionic actions, suggesting they are projections of this single principle, making spectral actions a mandatory, not optional, part of physics. 

Key connections and distinctions:

• Shared Foundation: Both Connes' work and ToE involve spectral triples, operator algebras, and relating geometry to quantum physics, as shown in research linking von Neumann entropy to the spectral action.
• ToE's Expansion: ToE builds on Connes' noncommutative geometry by introducing a broader "entropic field" and treating bosonic actions (like Einstein-Hilbert) through spectral traces, placing them on the same level as fermionic terms.
• Unified Principle: While Connes' SAP unifies fields through a spectral approach, ToE aims for a more fundamental unification, deriving all physical laws (including SAP) from a single entropic-spectral variational principle, moving beyond just Connes' original scope.
• Derivation vs. Application: Obidi's Theory of Entropicity (ToE) doesn't just use the Spectral Action but completes and derives it, transforming it from a specific mathematical tool into a fundamental description of reality. 

In essence, Connes' Spectral Action is a crucial consequence and manifestation of the more comprehensive Theory of Entropicity (ToE), rather than a separate theoretical construct. 

Appendix: Extra Matter

The Road to Alain Connes' Spectral Action in the Theory of Entropicity (ToE)

Very compactly, the roadmap looks like this:

• 1. From entropic field to spectral triple

  • Algebra $\mathcal{A}$: functions (or suitable noncommutative deformations) of the entropic field $S(x)$ and its curvature structures.

  • Hilbert space $\mathcal{H}$: square‑integrable “entropic modes” or spinor fields defined over the entropic manifold.

  • Dirac‑type operator $D$: constructed from the entropic connection/curvature, so that its spectrum encodes the same geometric content as the Obidi Action.

• 2. Obidi Action ↔ Spectral Action correspondence We rewrite the Obidi Action in terms of the spectrum of a Dirac‑type operator built from $S(x)$, such that we have:

$$S_{\text{Obidi}}[S]\approx (\psi ,D\psi )+\mathrm{T}\mathrm{r}\chi (D\mathrm{/}\mathrm{\Lambda })$$

in an appropriate limit or coarse‑graining. In that case, the Connes Spectral Action is no longer a starting axiom, but a derived effective action of ToE.

• 3. Physical interpretation

  • The noncommutative geometry of Connes becomes the representation theory of entropic curvature.

  • The Standard Model + gravity from the Spectral Action becomes a low‑energy, symmetry‑organized shadow of the deeper entropic field dynamics.

Where things stand conceptually

So:

• Can ToE derive the Spectral Action?

  • Conceptually: yes, very plausibly — ToE is exactly the kind of pre‑geometric, curvature‑driven framework that can sit “under” a spectral triple.

  • Technically, today: it would require us to explicitly construct:

    • a Dirac‑type operator from the entropic connection,

    • a corresponding spectral triple,

    • and then show that the heat‑kernel / asymptotic expansion of $\mathrm{T}\mathrm{r}\chi (D\mathrm{/}\mathrm{\Lambda })$ reproduces the same invariants and couplings as the Obidi Action in the appropriate regime.

That is not an elementary project — but it’s a natural next frontier, which we shall embark on in a future work on the rapidly unfolding Theory of Entropicity (ToE):

ToE as the entropic substrate from which the Connes Spectral Action emerges as the effective, noncommutative “readout” of reality.