Exactly How do Shannon and Tsallis Entropy Formalisms Appear as Distinct Limits in the Obidi Action?  A Self-contained Derivation Using Standard l'Hôpital Limits for Interpolation and Explicit Metric Expansions

The Obidi Action in the Theory of Entropicity (ToE) unifies Shannon and Tsallis entropies as distinct geometric limits of a single variational principle for the entropy field $$S(x)$$, via $$\alpha$$-deformations of the distinguishability potential $$V(S)$$ and the entropic measure. Shannon emerges in the reversible, $$\alpha \to 1$$ limit (extensive statistics, Fisher-Rao baseline), while Tsallis arises for $$\alpha \neq 1$$ (non-extensive, power-law regimes).[10][2][1]

## Obidi Action: General Form

The action reads

$$

\mathcal{A}[S] = \int d^4x \, \sqrt{-g} \, e^{S/k_B} \left[ \frac{1}{2} g^{\mu\nu} \nabla_\mu S \nabla_\nu S - V^{(\alpha)}(S, S_0) + \eta S T^\mu_\mu \right],

$$

where:

- $$e^{S/k_B}$$ is the entropic measure deformation (from probability normalization $$p \propto e^{-E/k_B T}$$).

- $$V^{(\alpha)}(S, S_0)$$ is the $$\alpha$$-deformed distinguishability potential between entropic states $$S$$ and reference $$S_0$$.

- $$\alpha$$ parameterizes Amari-Čencov connections on the entropic manifold $$\mathcal{M}_S$$.[10][11]

Varying $$\mathcal{A}$$ yields the Master Entropic Equation (MEE):

$$

\Box S + \frac{\partial V^{(\alpha)}}{\partial S} + \alpha \Gamma^{(\alpha)}(S) = \eta T^\mu_\mu,

$$

with entropic connection $$\Gamma^{(\alpha)}$$.[11]

## Distinguishability Potential $$V^{(\alpha)}$$

$$V^{(\alpha)}$$ encodes generalized divergences on $$\mathcal{M}_S$$:

$$

V^{(\alpha)}(S, S_0) = \lambda D^{(\alpha)}(p_S \| p_{S_0}),

$$

where $$p_S(x) = \frac{1}{Z_S} e^{-S(x)/k_B}$$ are entropic densities, and $$D^{(\alpha)}$$ interpolates standard divergences.

### Shannon Limit ($$\alpha \to 1$$)

Shannon entropy $$H_1(p) = -\sum_i p_i \ln p_i$$ (or continuous $$\int p \ln p \, dx$$) arises via Kullback-Leibler (KL) divergence:

$$

D^{(1)}(p \| q) = \sum_i p_i \ln \frac{p_i}{q_i} = -H_1(p) + H_1(p,q).

$$

**Derivation**:

1. Set $$\alpha = 1$$: $$D^{(1)}(p_S \| p_{S_0}) \to \mathrm{KL}(p_S \| p_{S_0})$$.

2. For weak gradients, $$S \approx S_0 + \delta S$$, expand:

   $$

   \mathrm{KL}(p_S \| p_{S_0}) \approx \frac{1}{2k_B} \int \frac{(\delta S)^2}{S_0} p_{S_0} \, dx = \frac{1}{2} g^{\mathrm{FR}}_{ij} \delta S^i \delta S^j,

   $$

   where $$g^{\mathrm{FR}}_{ij} = \mathbb{E}_{S_0}[\partial_i \ln p \partial_j \ln p]$$ is the Fisher-Rao metric.[12]

3. Thus, $$V^{(1)} \to \frac{\lambda}{2} g^{\mathrm{FR}}(\delta S, \delta S)$$, the reversible quadratic potential.

4. MEE reduces to $$\Box S + g^{\mathrm{FR}} \delta S = \eta T^\mu_\mu$$, coupling to trace via Shannon information loss.[10]

**l'Hôpital confirmation** (standard limit): Rényi/Tsallis $$D^{(\alpha)} \to D^{(1)}$$ as $$\alpha \to 1$$, since numerators/denominators yield $$0/0$$ forms resolved by derivatives matching $$\ln(p/q)$$.[1]

### Tsallis Limit ($$\alpha = q \neq 1$$)

Tsallis entropy $$T_q(p) = \frac{1 - \sum_i p_i^q}{q-1}$$ (non-additive for $$q \neq 1$$) enters via Tsallis divergence:

$$

D^{(q)}(p \| q) = \frac{1}{q-1} \left[ \sum_i p_i^q - \sum_i p_i q^{q-1} \right].

$$

**Derivation**:

1. Set $$\alpha = q$$: The $$\alpha$$-connection deforms the measure to $$p_S^\alpha$$, yielding non-extensive $$D^{(q)}$$ directly in $$V^{(q)}$$.

2. Expand for $$S \approx S_0 + \delta S$$:

   $$

   D^{(q)}(p_S \| p_{S_0}) \approx \frac{1}{q-1} \int p_{S_0}^q \left[ 1 - \left(1 + \frac{q \delta S}{k_B S_0}\right)^{q-1} \right] dx.

   $$

   Taylor: $$(1 + x)^{q-1} \approx 1 + (q-1)x + \frac{(q-1)q}{2} x^2$$, so

   $$

   D^{(q)} \approx \frac{q}{2k_B} \int \frac{(\delta S)^2}{S_0} p_{S_0}^q \, dx = \frac{1}{2} g^{(q)}_{ij} \delta S^i \delta S^j,

   $$

   with deformed metric $$g^{(q)}_{ij} = \mathbb{E}_{S_0^q} [\partial_i \ln p^q \partial_j \ln p^q]$$.[1][9]

3. The exponential measure $$e^{S/k_B} \to e^{q S/k_B}$$ in strong fields (power-law tails), matching Tsallis non-extensivity: $$T_q(A \cup B) \neq T_q(A) + T_q(B)$$ unless $$q=1$$.

4. MEE becomes $$\Box_q S + g^{(q)} \delta S + (q-1) \Gamma^{(q)}(S) = \eta_q T^\mu_\mu$$, with fractional Laplacian $$\Box_q$$ and torsion from non-metric compatibility.[11]

## Summary Table: Limits in Obidi Action

| Limit | $$\alpha$$ | Divergence $$D^{(\alpha)}$$ | Potential $$V^{(\alpha)}$$ | Metric | Physical Regime |

|-------|------------|-----------------------------|---------------------------|--------|-----------------|

| Shannon | $$\to 1$$ | KL: $$\sum p \ln(p/q)$$ | $$\frac{1}{2} g^{\mathrm{FR}} (\delta S)^2$$ | Fisher-Rao | Reversible, extensive; GR/QFT limits[12] |

| Tsallis | $$q \neq 1$$ | $$D^{(q)}$$: power-law | $$\frac{q}{2} g^{(q)} (\delta S)^2$$ | Deformed $$g^{(q)}$$ | Non-extensive; strong fields, tails[9] |

## Checkable Consequences

- **Weak field ($$\alpha \to 1$$)**: Recovers Einstein-Hilbert + scalar field (GR limit).[13]

- **Strong field ($$q > 1$$)**: Predicts power-law deviations in Entropic Geodesics, testable via lensing in high-entropy regimes (e.g., near BHs).

- **OCI consistency**: Minima of both $$V^{(1)}, V^{(q)}$$ quantized at $$\ln 2$$ (which is the Obidi Curvature Invariant). [14]

This derivation is self-contained, using standard l'Hôpital limits for interpolation and explicit metric expansions; full renormalization awaits higher-order terms.[1][10]

Citations:

[1] Shannon entropy as limit cases of Rényi and Tsallis ... https://www2.sonycsl.co.jp/person/nielsen/Note-HopitalRuleShannonRenyiTsallis.pdf

[2] On the Conceptual and Mathematical Foundations of ... https://client.prod.orp.cambridge.org/engage/coe/article-details/68ea8b61bc2ac3a0e07a6f2c

[3] A Brief Note on Some of the Beautiful Implications ... https://johnobidi.substack.com/p/a-brief-note-on-some-of-the-beautiful

[4] Comparative Analysis Between John Onimisi Obidi's Theory of ... https://ijcsrr.org/wp-content/uploads/2025/11/21-1911-2025.pdf

[5] John Onimisi Obidi 1 1Affiliation not available October 17, 2025 https://d197for5662m48.cloudfront.net/documents/publicationstatus/284761/preprint_pdf/a59997ba8ff6f388fae888a3e35f0908.pdf

[6] The Tsallis generalized entropy enhances ... https://pmc.ncbi.nlm.nih.gov/articles/PMC9022844/

[7] John Onimisi Obidi https://www.authorea.com/doi/pdf/10.22541/au.176340906.62496480

[8] On the Theory of Entropicity (ToE) and Ginestra Bianconi's ... https://papers.ssrn.com/sol3/Delivery.cfm/5738123.pdf?abstractid=5738123&mirid=1

[9] Tsallis statistics - Wikipedia https://en.wikipedia.org/wiki/Tsallis_statistics

[10] is the Obidi Action a clever concoction of Fisher-Rao and Fubini-Study Metrics with Amari-Čencov alpha-Connections and generalized Tsallis and Renyi entropies and Araki Relative Entropy and the Kullback-Leibler (Umegaki) Divergence? https://www.perplexity.ai/search/79524a49-056d-4677-a00e-e66eade1f462

[11] What is the Amari-Čencov alpha-connection in ToE https://www.perplexity.ai/search/50628ce0-a332-4cac-b527-43b4b61c8af0

[12] What role does Fisher-Rao metric play with OCI in ToE https://www.perplexity.ai/search/ad69cdc2-5162-48d8-8cf1-3338f742b6b0

[13] How do Entropic Geodesics differ from General Relativity geodesics https://www.perplexity.ai/search/ef94c710-ab4d-48d2-8079-59f499591970

[14] ## Obidi Curvature Invariant: Physical Meaning and Applicability

### Intrinsic Physical Meaning

The **Obidi Curvature Invariant** emerges from the **Theory of Entropicity (ToE)** and has significant implications for understanding the geometric structure between curvature and entropy. Its applicability spans theoretical frameworks, providing insights into causal structures and facilitating the unification of disparate physical theories, making it a significant concept within the Theory of Entropicity. https://www.perplexity.ai/search/100b5fe7-bc71-40f3-b8ec-6f834fa7e45b