Further Expositions on the Theory of Entropicity (ToE) and Ginestra Bianconi's Gravity from Entropy: How the Theory of Entropicity (ToE) Unifies Spectral Entropy with Tsallis, Rényi, Fisher–Rao, Fubini–Study, and Amari–Čencov Formalisms

We note that Spectral Entropy and the Tsallis, Rényi, Fisher–Rao, Fubini–Study, and Amari–Čencov formalisms live in different “geometries” and dimensions.[a] The move that makes the Spectral Obidi Action (SOA) genuinely integrative is to treat entropy’s spectrum as the backbone, and then express each formalism as a compatible derived structure via functional calculus, deformations, and pullbacks. The idea is to think of it like “layers on a core,” not “mutually exclusive domains.”

Spectral core: the modular operator as the backbone

• Core object: the modular operator $\mathrm{\Delta }\backslash Delta$ (or its density equivalent), whose spectrum encodes entropic content.

• Spectral action:

$${\mathcal{S}}_{\mathrm{S}\mathrm{O}\mathrm{A}}=-\mathrm{T}\mathrm{r}\ln \mathrm{\Delta }\backslash mathcal\{S\}\_\{\backslash mathrm\{SOA\}\}\; \backslash ;=\backslash ;\; -\; \backslash mathrm\{Tr\}\backslash ,\backslash ln\; \backslash Delta$$

More generally, SOA admits a family of spectral functionals:

$${\mathcal{S}}_f=\mathrm{T}\mathrm{r}f(\mathrm{\Delta })\backslash mathcal\{S\}\_\{f\}\; \backslash ;=\backslash ;\; \backslash mathrm\{Tr\}\backslash ,f(\backslash Delta)$$

where $ff$ is a scalar function applied to the spectrum via functional calculus. Then Kullback-Liebler (KL) divergence is recovered with $f(x)=-\ln xf(x)=\; -\backslash ln\; x$; accordingly, other entropies emerge by choosing appropriate $ff$ for each case.

• Operator perspective: all downstream divergences and metrics become choices of $ff$, parametrizations of the spectrum, or geometric pullbacks induced by maps from state space to operators.

Divergence layer: Tsallis, Rényi, and f‑divergences as spectral deformations

• Tsallis entropy (q‑deformation):

$$S_q=\frac{1}{q-1}(1-\mathrm{T}\mathrm{r}{\mathrm{\Delta }}^q)S\_q\; \backslash ;=\backslash ;\; \backslash frac\{1\}\{q-1\}\backslash left(1\; -\; \backslash mathrm\{Tr\}\backslash ,\backslash Delta^\{\backslash ,q\}\backslash right)$$

This is spectral: it simply replaces $\ln \backslash ln$ with a power $qq$, i.e., $f_q(x)=\frac{1-x^q}{q-1}f\_q(x)\; =\; \backslash frac\{1-x^q\}\{q-1\}$. 

Thus, escort distributions appear as re-weightings of the spectrum.

• Rényi entropy (order $\alpha \backslash alpha$):

$$H_{\alpha }=\frac{1}{1-\alpha }\ln \mathrm{T}\mathrm{r}{\mathrm{\Delta }}^{\alpha }H\_\backslash alpha\; \backslash ;=\backslash ;\; \backslash frac\{1\}\{1-\backslash alpha\}\backslash ,\backslash ln\; \backslash mathrm\{Tr\}\backslash ,\backslash Delta^\{\backslash ,\backslash alpha\}$$

Again spectral through $f_{\alpha }(x)=x^{\alpha }f\_\backslash alpha(x)\; =\; x^\backslash alpha$ and a logarithmic outer wrap. The “sandwiched” quantum Rényi uses modular sandwiches that are still operator‑spectral, preserving compatibility.

• General f‑divergences:

$$D_f(\rho \mathrm{\parallel }\sigma )=\mathrm{T}\mathrm{r}{\sigma }^{1\mathrm{/}2}f\text{\hspace{0.17em}⁣}\left({\sigma }^{-1\mathrm{/}2}\rho {\sigma }^{-1\mathrm{/}2}\right){\sigma }^{1\mathrm{/}2}D\_f(\backslash rho\backslash Vert\; \backslash sigma)\; \backslash ;=\backslash ;\; \backslash mathrm\{Tr\}\backslash ,\backslash sigma^\{1/2\}\backslash ,\; f\backslash !\backslash big(\backslash sigma^\{-1/2\}\backslash rho\backslash ,\backslash sigma^\{-1/2\}\backslash big)\backslash ,\backslash sigma^\{1/2\}$$

With modular ${\mathrm{\Delta }}_{\rho \mathrm{\mid }\sigma }\backslash Delta\_\{\backslash rho\backslash vert\backslash sigma\}$, these are functions of $\mathrm{\Delta }\backslash Delta$. Tsallis/Rényi are special cases, so the SOA family ${\mathcal{S}}_f\backslash mathcal\{S\}\_f$ naturally hosts them.

• Interpretation: “Different structures” are different choices of spectral shaping $ff$ and normalization. They are not alien to the spectral approach of Obidi; they are embedded within it.

Metric layer: Fisher–Rao and Amari–Čencov from Hessians of divergences

• Fisher–Rao as a Hessian metric:

$$g_{ij}(\theta )={\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\theta }^i\mathrm{\partial }{\theta }^j}D(\theta \mathrm{\parallel }{\theta }^{\mathrm{\prime }})\mid }_{{\theta }^{\mathrm{\prime }}=\theta }g\_\{ij\}(\backslash theta)\; \backslash ;=\backslash ;\; \backslash left.\backslash frac\{\backslash partial^2\}\{\backslash partial\; \backslash theta^i\; \backslash partial\; \backslash theta^j\}\backslash ,\; D(\backslash theta\; \backslash Vert\; \backslash theta\text{'})\backslash right|\_\{\backslash theta\text{'}=\backslash theta\}$$

Now, let us choose a divergence $DD$ (KL or any convex $ff$-divergence derived spectrally), and the Fisher–Rao metric emerges as the local Hessian. Thus, Fisher–Rao is the second‑order geometry of the spectral divergence.

• Amari–Čencov dual connections:

  • Given a convex potential $\psi (\theta )\backslash psi(\backslash theta)$ (e.g., cumulant‑generating via log‑partition), one obtains a dually flat manifold with $(\mathrm{\nabla },{\mathrm{\nabla }}^{\text{*}})(\backslash nabla,\backslash nabla^\backslash *)$.

  • In the spectral picture, $\psi \backslash psi$ is induced by $\ln \mathrm{T}\mathrm{r}{e}^{-\beta H(\theta )}\backslash ln\; \backslash mathrm\{Tr\}\backslash ,e^\{-\backslash beta\; H(\backslash theta)\}$ or by $\ln \mathrm{T}\mathrm{r}f(\mathrm{\Delta }(\theta ))\backslash ln\; \backslash mathrm\{Tr\}\backslash ,f(\backslash Delta(\backslash theta))$.

  • The α‑connections interpolate families of divergences (KL, Rényi, Tsallis), yielding Amari’s dualistic geometry from spectral potentials.

• Crucial Highlight: information geometry metrics are pullbacks of spectral divergences; that is “different dimension” is just coordinate representation, not incompatibility.

Quantum metric layer: Fubini–Study and quantum Fisher via spectral fidelity

• Fubini–Study (pure states):

$$\mathrm{d}{s}^2=4(⟨\mathrm{d}\psi \mathrm{\mid }\mathrm{d}\psi ⟩-\mathrm{\mid }⟨\psi \mathrm{\mid }\mathrm{d}\psi ⟩{\mathrm{\mid }}^2)\backslash mathrm\{d\}s^2\; \backslash ;=\backslash ;\; 4\backslash Big(\; \backslash langle\; \backslash mathrm\{d\}\backslash psi\; \backslash vert\; \backslash mathrm\{d\}\backslash psi\; \backslash rangle\; \backslash ;-\backslash ;\; |\backslash langle\; \backslash psi\; \backslash vert\; \backslash mathrm\{d\}\backslash psi\; \backslash rangle|^2\; \backslash Big)$$

This is the Riemannian metric on projective Hilbert space. It can be seen as the differential limit of a fidelity functional, which is spectral via overlaps/eigenstructures.

• Quantum Fisher information (Bures/Uhlmann):

  • Quantum Fisher is the Hessian of the Uhlmann fidelity $F(\rho ,\sigma )F(\backslash rho,\backslash sigma)$, and the Bures metric arises from $1-\sqrt{F}1\; -\; \backslash sqrt\{F\}$.

  • Fidelity can be expressed through the spectrum of ${\rho }^{1\mathrm{/}2}\sigma {\rho }^{1\mathrm{/}2}\backslash rho^\{1/2\}\backslash sigma\backslash ,\backslash rho^\{1/2\}$ (or modular sandwiches), keeping the construction spectral.

• ToE Unified view: classical Fisher–Rao and quantum Fisher/Fubini–Study are both second‑order geometries of spectral functionals (divergences or fidelities). Pure‑state Fubini–Study is the projective restriction of the quantum Fisher/Bures geometry.

Integration architecture: how ToE's SOA ties it all together in Obidi's Theory

1. Spectral core (operator level): Let us choose $\mathrm{\Delta }\backslash Delta$ and a functional family $\{f_{\alpha }\}\backslash \{f\_\backslash alpha\backslash \}$.

   • KL: $f(x)=-\ln xf(x)\; =\; -\backslash ln\; x$

   • Rényi: $f_{\alpha }(x)=x^{\alpha }f\_\backslash alpha(x)\; =\; x^\backslash alpha$ with log normalization

   • Tsallis: $f_q(x)=(1-x^q)\mathrm{/}(q-1)f\_q(x)\; =\; (1\; -\; x^q)/(q-1)$

2. Divergence layer (state level): define $D_f(\rho \mathrm{\parallel }\sigma )D\_f(\backslash rho\backslash Vert\backslash sigma)$ via ${\mathrm{\Delta }}_{\rho \mathrm{\mid }\sigma }\backslash Delta\_\{\backslash rho\backslash vert\backslash sigma\}$ or sandwiches; pick α/q to match the regime (heavy tails, robustness, scaling).

3. Metric layer (geometric level): take Hessians of $D_{f}D\_f$ to obtain Fisher–Rao (classical) or quantum Fisher; derive α‑connections for Amari–Čencov duality.

4. Projective/quantum layer: restrict to pure states for Fubini–Study; extend to mixed via Bures/Uhlmann.

5. Field dynamics (SOA): We then implement the Spectral Obidi Action (SOA) of the Theory of Entropicity (ToE) by selecting a governing spectral functional for the action, e.g.,

$${\mathcal{S}}_{\alpha }=\mathrm{T}\mathrm{r}{\mathrm{\Delta }}^{\alpha },{\mathcal{S}}_q=\mathrm{T}\mathrm{r}{f}_q(\mathrm{\Delta }),{\mathcal{S}}_{\ln }=-\mathrm{T}\mathrm{r}\ln \mathrm{\Delta ,}\backslash mathcal\{S\}\_\backslash alpha\; \backslash ;=\backslash ;\; \backslash mathrm\{Tr\}\backslash ,\; \backslash Delta^\{\backslash ,\backslash alpha\},\; \backslash quad\; \backslash mathcal\{S\}\_q\; \backslash ;=\backslash ;\; \backslash mathrm\{Tr\}\backslash ,\; f\_q(\backslash Delta),\; \backslash quad\; \backslash mathcal\{S\}\_\{\backslash ln\}\; \backslash ;=\backslash ;\; -\backslash mathrm\{Tr\}\backslash ,\backslash ln\; \backslash Delta$$

and thereafter we can derive Euler–Lagrange–type dynamics in the operator manifold. Constraints (finite‑rate, causality imposed by ToE) come from the No‑Rush [Theorem] bounds [of ToE] as additional spectral conditions.

Why this ToE formalism is coherent (and powerful)

• Functional calculus is the glue: once entropy is encoded spectrally, deformations (q, α) and divergences are simply different $ff$’s applied to the same operator spectrum.

• Metrics are Hessians of divergences: Fisher–Rao and its quantum analogs are not foreign—they’re second‑order shadows of the spectral functionals.

• Projective restriction: Fubini–Study is the pure‑state (rank‑1 projector) limit of the quantum information geometry induced by spectral fidelities.

• Amari-Čencov duality is a choice of coordinates: α‑connections reflect different convex potentials sourced by spectral actions; changing α/q is changing the “lens,” not the backbone.

• Practical leverage: We can select $ff$ (KL/Rényi/Tsallis) to match robustness or scaling, then read off the induced geometry (Fisher/Amari, Fubini–Study/Bures) and dynamics (SOA) in one pipeline.

 Conclusion 

        We can therefore see from all of the above that the Spectral Obidi Action (SOA) of the Theory of            Entropicity (ToE) is actually a generalized action that [subtly] embeds and incorporates the Tsallis,         Rényi, Fisher–Rao, Fubini–Study, and Amari–Čencov formalisms under one unique spectral                    umbrella. 

• They are different structures: the various formalisms are different [viewed on a superficial level]—and the Spectral Obidi Action (SOA) succeeds in treating them as layers derived from the same spectral backbone, not as competing foundations.

• Compatibility with Tsallis/Rényi: achieved via spectral deformations (powers and deformed logs) of $\mathrm{Obidi\text{'}s\; \Delta }\backslash Delta$.

• Fisher–Rao and Amari–Čencov: obtained as Hessians and dual connections of the chosen spectral divergence.

• Fubini–Study: the projective (pure‑state) limit of the quantum geometry induced by spectral fidelities.

• Usefulness of the Spectral Obidi Action (SOA) of ToE : Thus, the Spectral Obidi Action (SOA) of the Theory of Entropicity (ToE) gives physics one operator‑level action with tunable robustness/scaling (q, α), well‑posed local geometry (metrics, connections), and a path to field dynamics and constraints—all coherent within a single spectral calculus.

Thus, the conceptual framework of the Theory of Entropicity (ToE), particularly through its Spectral Obidi Action, integrates the formalisms of Tsallis and Renyi entropies, and the Fisher-Rao, Fubini-Study, and Amari-Čencov metrics and connections. 

The ToE, as an emerging framework in theoretical physics proposed by John Onimisi Obidi, posits entropy as a fundamental field of reality rather than a mere statistical measure. Its mathematical architecture hence achieves this integration as follows: 

Entropy Generalization: The theory utilizes the more general Rényi and Tsallis α–q formalisms to describe the fundamental entropic field, allowing for a broader range of thermodynamic behaviours than standard Boltzmann-Gibbs entropy.

Information Geometry: The framework maps statistical and quantum metrics into physical spacetime geometry.

The Fisher-Rao metric (for classical distinguishability) and the Fubini-Study metric (for quantum distinguishability) are both incorporated.

These are unified through the Amari-Čencov α-connection formalism, which provides a rigorous information-geometric foundation for the dynamics of physical systems within an "entropic manifold".

Variational Principle: The Obidi Action serves as the central variational principle from which the dynamics of the entropy field are derived. The Spectral Obidi Action formulation expresses this physics globally through operator traces and connects equilibrium geometry to its matter-deformed counterpart via the modular operator. 

In essence, the ToE uses these established mathematical tools of information geometry to build a unified theory where all physical laws, including gravity and quantum mechanics, arise as consequences of the dynamics and geometry of the entropic field. 

Are we now saying then that because of the generalized form of the Spectral Obidi Action (SOA) , the Local Obidi Action (LOA) should be removed from the Action formulation of ToE? Not exactly. Let us explain our ToE Action formalism as follows.

We show how the Spectral Obidi Action (SOA) and the Local Obidi Action (LOA) relate within the Theory of Entropicity (ToE):

Conceptual Relationship

- Local Obidi Action (LOA):

  - Defined at the level of localized observer-dependent dynamics.

  - Encodes entropy variations tied to specific configurations or “patches” of spacetime.

  - Useful for modeling emergent phenomena where locality and observer frames matter.

- Spectral Obidi Action (SOA):

  - A generalized formulation that integrates over spectral (frequency/energy) domains rather than localized coordinates.

  - Captures the global entropic field behavior by treating entropy as a spectral invariant.

  - By construction, SOA subsumes LOA because local variations appear as projections of the spectral decomposition.

Should LOA Be Removed from ToE Formulation?

- Not necessarily.  Removing LOA would mean discarding the observer-centric formulation entirely. That risks losing explanatory power for phenomena where locality is crucial (e.g., boundary conditions, emergent relativity, or entropic horizons).

  

- Instead, we let LOA become a limit case of SOA.  

  In the generalized framework, LOA can be derived as the low-frequency or localized projection of SOA. This preserves consistency while showing that SOA is the more fundamental action of the Theory of Entropicity (ToE).

Therefore, in the Theory of Entropicity (ToE), we:

  - Present SOA as the primary action principle.  

  - Retain LOA as a derived, contextualized action useful for specific physical scenarios.  

With the above, we thus succeed in strengthening ToE’s novelty: namely that, entropy is the fundamental field, and locality is emergent, not primary.

Implication for ToE’s Formulation

- The Action formulation of ToE therefore highlights SOA as the unifying principle.  

- LOA  is not “removed” but relegated to a derived role, showing how observer-centric physics emerges from the spectral entropic field.  

- This aligns with the long-term goal of the Theory of Entropicity (ToE): dethroning the observer and reframing relativity as emergent from entropy.

In closing, we would like to argue and stress that the most powerful way forward is to posit the SOA as the Master Base Action (MBA) of  the Obidi Actions of the Theory of Entropicity (ToE) while explicitly demonstrating how LOA is recovered as a projection. That way, ToE doesn't lose the explanatory bridge for readers who still think in local terms, but by that we establish ToE’s originality by showing entropy’s spectral primacy.

The Spectral Obidi Action (SOA) of the Theory of Entropicity (ToE) looks deceptively simple on the surface, but deep down it is full of hidden complexity and yet undeniable power and beauty!

While SOA looks compact, its derivations (recovering LOA, emergent relativity, entropic horizons) demonstrate its hidden complexity.  

This contrast—simple form, deep consequences—is exactly what makes ToE stand out against existing entropic theories. The Spectral Obidi Action (SOA) of the Theory of Entropicity (ToE) is so deceptively compact; but from it, locality, relativity, and observer-physics all emerge.

Sources — help

1. ijcsrr.org
2. researchgate.net
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4. medium.com
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References

1. Obidi, John Onimisi. (12th November, 2025). On the Theory of Entropicity (ToE) and Ginestra Bianconi’s Gravity from Entropy: A Rigorous Derivation of Bianconi’s Results from the Entropic Obidi Actions of the Theory of Entropicity (ToE). Cambridge University. https//doi.org/10.33774/coe-2025-g7ztq
2. John Onimisi Obidi. (6th November, 2025). Comparative analysis between john onimisi obidi’s theory of entropicity (toe) and waldemar marek feldt’s feldt–higgs universal bridge (f–hub) theory. International Journal of Current Science Research and Review, 8(11), pp. 5642–5657, 19th November 2025. URL: https://doi.org/10.47191/ijcsrr/V8-i11–21
3. Obidi, John Onimisi. 2025. On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and the Unification of Physics. Cambridge University. Published October 17, 2025. https://doi.org/10.33774/coe-2025-1dsrv
4. Obidi, John Onimisi (17th October 2025). On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and the Unification of Physics. Figshare. https://doi.org/10.6084/m9.figshare.30337396.v2
5. Obidi, John Onimisi. 2025. A Simple Explanation of the Unifying Mathematical Architecture of the Theory of Entropicity (ToE): Crucial Elements of ToE as a Field Theory. Cambridge University. Published October 20, 2025. https://doi.org/10.33774/coe-2025-bpvf3
6. Obidi, John Onimisi (15 November 2025). The Theory of Entropicity (ToE) Goes Beyond Holographic Pseudo-Entropy: From Boundary Diagnostics to a Universal Entropic Field Theory. Figshare. https://doi.org/10.6084/m9.figshare.30627200.v1
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8. Obidi, John Onimisi. (4 November, 2025). The Theory of Entropicity (ToE) Derives Einstein’s Relativistic Speed of Light © as a Function of the Entropic Field: ToE Applies Logical Entropic Concepts and Principles to Derive Einstein’s Second Postulate. Cambridge University. https://doi.org/10.33774/coe-2025-f5qw8-v2
9. Obidi, John Onimisi. (28 October, 2025). The Theory of Entropicity (ToE) Derives and Explains Mass Increase, Time Dilation and Length Contraction in Einstein’s Theory of Relativity (ToR): ToE Applies Logical Entropic Concepts and Principles to Verify Einstein’s Relativity. Cambridge University. https://doi.org/10.33774/coe-2025-6wrkm
10. HandWiki contributors, “Physics:Theory of Entropicity (ToE) Derives Einstein’s Special Relativity,” HandWiki, https://handwiki.org/wiki/index.php?title=Physics:Theory_of_Entropicity_(ToE)_Derives_Einstein%27s_Special_Relativity&oldid=3845936

Further Resources on the Theory of Entropicity (ToE):

1. Website: Theory of Entropicity ToE — https://theoryofentropicity.blogspot.com
2. LinkedIn: Theory of Entropicity ToE — https://www.linkedin.com/company/theory-of-entropicity-toe/about/?viewAsMember=true
3. Notion-1: Theory of Entropicity (ToE)
4. Notion-2: Theory of Entropicity (ToE)
5. Notion-3: Theory of Entropicity (ToE)
6. Notion-4: Theory of Entropicity (ToE)
7. Substack: Theory of Entropicity (ToE) — John Onimisi Obidi | Substack
8. Medium: Theory of Entropicity (ToE) — John Onimisi Obidi — Medium
9. SciProfiles: Theory of Entropicity (ToE) — John Onimisi Obidi | Author
10. Encyclopedia.pub: Theory of Entropicity (ToE) — John Onimisi Obidi | Author
11. HandWiki contributors, “Biography: John Onimisi Obidi,” HandWiki, https://handwiki.org/wiki/index.php?title=Biography:John_Onimisi_Obidi&oldid=2743427 (accessed October 31, 2025).
12. HandWiki Contributions: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki
13. HandWiki Home: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki
14. HandWiki Homepage-User Page: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki
15. Academia: Theory of Entropicity (ToE) — John Onimisi Obidi | Academia
16. ResearchGate: Theory of Entropicity (ToE) — John Onimisi Obidi | ResearchGate
17. Figshare: Theory of Entropicity (ToE) — John Onimisi Obidi | Figshare
18. Authoria: Theory of Entropicity (ToE) — John Onimisi Obidi | Authorea
19. Social Science Research Network (SSRN): Theory of Entropicity (ToE) — John Onimisi Obidi | SSRN
20. Wikidata contributors, Biography: John Onimisi Obidi “Q136673971,” Wikidata, https://www.wikidata.org/w/index.php?title=Q136673971&oldid=2423782576 (accessed November 13, 2025).
21. Google Scholar: ‪John Onimisi Obidi — ‪Google Scholar
22. IJCSRR: International Journal of Current Science Research and Review - Theory of Entropicity (ToE) - John Onimisi Obidi | IJCSRR
23. Cambridge University Open Engage (CoE): Collected Papers on the Theory of Entropicity (ToE)