Concise Mathematical Exposition of the Spectral Obidi Action (SOA) and Bianconi's Action in the Theory of Entropicity (ToE)

A Concise Mathematical Exposition of the Spectral Obidi Action (SOA) and Bianconi's Action in the Theory of Entropicity (ToE)

In this exposition, we show how the Spectral Obidi Action (SOA) of the Theory of Entropicity (ToE) differs from the Ginestra Bianconi Action in her Gravity from Entropy, even though they look related at a conceptual level. The Spectral Obidi Equation [the Master Entropic Equation (MEE) that arises from the inclusive variation of the Spectral Obidi Action (SOA)] gives you more physics and more predictive structure than what Bianconi’s original “gravity from entropy” action on its own was designed to do.

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1. We ask: Are the two actions literally the same?

Bianconi’s core idea (in a very compressed fashion) is:

• Take a spacetime metric $g_{\mu\nu}$ and a matter/deformed metric $g^m_{\mu\nu}$.

• Define a relative entropy (Kullback–Leibler–type) between geometric states.

• Build an effective gravitational action from that relative entropy, so that:

  • extremizing the entropy produces something equivalent (or close) to Einstein-type field equations;

  • dark energy / dark matter–like corrections appear as entropic/geometric effects.

In schematic form, her action is of the type:

$$I_{\text{Bianconi}} \sim \int d^4x \sqrt{-g}\, D_{\text{KL}}(g \Vert g^m),$$

plus extra pieces for the $G$-field and cosmological term. It is a relative entropy between two metrics. The entropy lives in the comparison of geometries.

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The Spectral Obidi Action that you defined is:

$$I_{\text{SOA}}[S,g] = - \mathrm{Tr}\bigl(\ln \Delta\bigr), \quad \Delta = G\,g^{-1},$$

with $G$$G_{}$ built from the entropic geometry $G_\alpha(S)$ determined by the entropy field $S(x)$.

Key differences:

1. Primary object

   • Bianconi: the primary object is a relative entropy functional $D_{\text{KL}}(g \Vert g^m)$ between two metrics. There is no independent, dynamical entropy field $S(x)$ treated as a fundamental field.

   • ToE: the primary object is the entropy field $S(x)$, and the geometry operators $G$ and $\Delta$ are derived from it.

     • $I_{\mathrm{SOA}}$ is not just “KL of two metrics”; it’s a spectral functional of an operator that encodes the full information geometry (Fisher–Rao, Fubini–Study, Tsallis/Rényi via $\alpha$, etc.).

2. Status of the action

   • Bianconi’s relative entropy is the gravitational action (plus corrections): you extremize it with respect to the metric(s) and get modified Einstein-type equations.

   • The Spectral Obidi Action is one piece (global/spectral) of a two-part variational principle:

     $$I_{\text{tot}} = I_{\mathrm{LOA}}[S,g] + I_{\mathrm{SOA}}[S,g],$$

     where $I_{\mathrm{LOA}}$ already gives you a proper local field theory for $S(x)$and $g_{\mu\nu}$.

3. Level of generality

   • Bianconi is essentially in the Shannon/Fisher regime (extensive entropy, near-equilibrium, small metric deviations).

   • $I_{\mathrm{SOA}}$ is designed to work:

     • for general entropies (Tsallis, Rényi),

     • for full information geometry (Amari $\alpha$-connections),

     • and in a fully nonlinear entropic field theory (ToE).

So: 

Do SOA and Bianconi Action have conceptual kinship? Yes.
Do SOA and Bianconi Action have Identical mathematical object? No. This is because Bianconi’s action is a special, restricted Shannon/Fisher-type limit of the more general spectral framework of ToE.

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2. How does Bianconi’s action appear inside ToE?

Let us draw this apt analogy: Think of the Spectral Obidi Action (SOA) setup as a big machine, and Bianconi’s Action theory as one very specific “mode” of that machine.

If we impose all of the following conditions on the full Obidi Action:

1. Fix the entropic index to the extensive case:

   $$\alpha \to 1,$$

   so information geometry collapses to standard Shannon/Fisher structure.

2. Linearize the entropy field around a constant background:

   $$S(x) = S_0 + \phi(x), \quad |\nabla S| \ll 1,$$

   and keep only quadratic terms (near-equilibrium expansion) in the Local Obidi Action.

3. Restrict attention to metric perturbations and identify the Fisher metric with the quadratic form of metric fluctuations.

Then:

• The quadratic piece of $I_{\mathrm{LOA}}$ + the leading spectral correction from $I_{\mathrm{SOA}}$ combine into an effective functional that is equivalent to a KL-type relative entropy between a background metric and a matter-modified metric.

In that sense:

Bianconi’s “gravity from entropy” can be read as the $\alpha = 1$, weak-field, Shannon/Fisher limit of the full Obidi framework.

So it’s contained in ToE, but ToE goes far beyond it.

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3. So what does the Spectral Obidi Equation give us that Bianconi’s doesn’t?

The Spectral Obidi Equation (schematically):

$$\underbrace{ \nabla_\mu\!\bigl(e^{S/k_B}\nabla^\mu S\bigr) - \frac{1}{2k_B}\,e^{S/k_B}(\nabla S)^2 + \frac{1}{\chi}\,V'(S) }_{\text{local MEE term}} \;-\; \underbrace{ \mathrm{Tr}\Bigl( \Delta^{-1} \frac{\delta G}{\delta S(x)}\,g^{-1} \Bigr) }_{\text{global spectral back-reaction}} = 0.$$

This is a single field equation where:

• The first chunk is the local PDE for the entropy field (Master Entropic Equation - MEE).

• The second chunk is the global spectral term, telling us how the entire spectrum of the entropic geometry feeds back into local dynamics.

Compared to Bianconi, we write as follows about the Spectral Obidi Action (SOA):

3.1. Entropy is a genuine field, not just a functional

In Bianconi’s setup:

• Entropy sits mostly inside a functional of metrics.

• We don’t get a standalone dynamical field equation for “S(x)” that we could evolve in time like a scalar field.

In ToE with the Spectral Obidi Equation:

• We have a bona fide field $S(x)$ with:

  • a local kinetic term,

  • a potential $V(S)$,

  • and a nonlocal spectral self-coupling via $\Delta$.

• That’s a dynamical, predictive field theory, not just a variational identity.

3.2. Built-in information geometry (α–connections, Tsallis/Rényi)

The spectral term involves $G_\alpha(S)$, $\mathcal{D}_S$, $\Delta$, etc., which carry:

• Amari $\alpha$-connections → intrinsic time asymmetry and irreversibility when $\alpha \neq 0$
  

• Tsallis/Rényi structure → ability to describe non-extensive, long-range, non-equilibrium phenomena in a controlled way.

• Fisher-Rao + Fubini-Study in a single unified metric → classical and quantum coherence in the same geometric object.

Bianconi’s action essentially stops at the Shannon/Fisher layer; it does not try to unify Tsallis, Rényi, Fubini-Study, or full $\alpha$-geometry inside a single universal action.

3.3. Global constraints and the dark sector

The spectral term

$$\mathrm{Tr}(\ln \Delta) = \sum_i \ln \lambda_i$$

and its variations give us:

• A spectral energy density:

  $$E_{\mathrm{spec}} \propto \sum_i (\lambda_i - 1)^2,$$

  behaving like cold dark matter.

• A residual entropic pressure when the spectrum is not exactly at equilibrium, interpreted as:

  $$\Lambda_{\mathrm{ent}} > 0,$$

  i.e. a dynamically generated dark-energy–like term.

Bianconi also talks about emergent $\Lambda$ and a $G$-field, but:

• In ToE that $G$-field is explained as the Lagrange multiplier enforcing the global spectral constraint from $I_{\mathrm{SOA}}$.

• We get a direct spectral interpretation: dark matter and dark energy are non-equilibrated spectral properties of the entropic field, not extra fields or particles.

3.4. Irreversibility and the Entropic Time Limit (ETL)

The Spectral Obidi Equation + $\alpha$-connections + local MEE give:

• A built-in arrow of time (because $\nabla^{(\alpha)} \neq \nabla^{(-\alpha)}$ when $\alpha \neq 0$).

• A universal Entropic Time Limit (ETL), leading to finite entanglement formation time (e.g. ~232 attoseconds in ToE's predictive framework).

Bianconi’s entropic gravity is essentially time-symmetric at the level of the equations; it does not encode:

• an intrinsic, geometric arrow of time,

• or a fundamental entanglement time bound.

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4. Physical usefulness: what can we actually do with the Spectral Obidi Equation?

Here’s where it becomes practically different from Bianconi’s:

1. Predictive field evolution

   • We can, at least in principle, solve for $S(x)$ under different initial and boundary conditions, with or without symmetry assumptions (cosmology, black holes, galaxies, etc.).

   • The spectral term ties local evolution to global mode structure; we can study:

     • entropic waves,

     • spectral instabilities,

     • relaxation to equilibrium,

     • and how these show up as gravitational phenomena.

2. Nonlinear corrections to GR

   • Beyond Bianconi’s near-equilibrium, ToE can derive:

     • corrections to lensing,

     • perihelion precession,

     • cosmological expansion (via $\Lambda_{\mathrm{ent}}(t)$),

     • and potentially strong-field signatures near black holes, all as explicit functionals of $S(x)$ and the spectrum of $\Delta$.

3. Unified treatment of classical + quantum geometry

   • Because $G_\alpha$ mixes Fisher-Rao and Fubini-Study, the same field and action govern:

     • classical thermodynamic gravity,

     • quantum coherence and entanglement structure,

     • and their back-reaction on geometry.

4. Model-building for dark matter/energy without new particles

   • Instead of “let’s add a dark field” or “let’s add a cosmological constant by hand”, in ToE we:

     Choose a physically reasonable spectrum for $\Delta$, compute $E_{\mathrm{spec}}$ and $\Lambda_{\mathrm{ent}}$, and fit to cosmological data.

   • That is a concrete, testable program that is not available in that form in Bianconi’s original setup.

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5. Closing Highlights and Conclusion

• Is the Spectral Obidi Action just Bianconi’s action?
  No. It’s strictly more general. Bianconi’s relative-entropy-based “gravity from entropy” can be extracted as a particular limit (Shannon/Fisher, $\alpha = 1$, weak-field, near-equilibrium) of the combined Obidi framework.

• What is physically new or useful about the Spectral Obidi Equation?
  It gives us:

  • a true dynamical field equation for entropy $S(x)$ with both local and global pieces;

  • a unified handle on Tsallis, Rényi, Fisher-Rao, Fubini-Study, Amari $\alpha$ within one action;

  • a built-in mechanism for dark matter and dark energy as spectral effects;

  • an intrinsic arrow of time and ETL directly tied to information geometry;

  • and a pathway to compute nonlinear, testable corrections beyond what Bianconi’s original action was written to handle.

So, from all of the above, we can conclude as follows:

Bianconi sees gravity as a relational effect encoded in relative entropy between geometries.
ToE’s Spectral Obidi Action (SOA) sees that relational picture as just one low-energy specialization of a deeper reality in which entropy is a fundamental field, and its spectral geometry generates gravity, the dark sector, and time’s arrow all at once.