On the Ingenuity of Obidi's Derivation of the Curvature Invariant of ln 2 from the Kullback-Leibler (Umegaki) Divergence as a Universal Constant Law of Distinguishability Potential 

In ToE, the distinguishability potential [the Obidi Curvature Invariant (OCI) of ln2] is obtained by taking the classical Kullback–Leibler (KL) divergence and rewriting it in terms of the local entropy field $$S(x)$$ and a reference configuration $$S_0(x)$$, then interpreting that KL density as a **potential energy density** in the Obidi / Spectral Obidi Action.[1][2]

## Step 1: Start from KL divergence

For two probability densities $$p(x)$$ and $$q(x)$$, the KL divergence is  

$$ D_{\mathrm{KL}}(p\Vert q) = \int p(x)\,\ln\!\frac{p(x)}{q(x)}\,dx.$$  

ToE uses this structure as the template for comparing two entropic configurations.[1][2]

## Step 2: Replace probabilities by entropic densities (Obidi's First Ingenuity—OFI)

ToE treats the entropy field $$S(x)$$ (or an associated entropic density) as the fundamental variable and introduces a local reference configuration $$S_0(x)$$.[1][2] The probabilistic KL integrand is lifted to an entropic density level by the substitution  

$$ p(x)\to S(x),\quad q(x)\to S_0(x),$$  

together with the standard convex extension that makes the divergence finite and well‑behaved for fields:  

$$

D(x) = S(x)\,\ln\!\frac{S(x)}{S_0(x)} - S(x) + S_0(x).[]

$$  

This is a pointwise KL‑type density with the same key properties: $$D(x)\ge 0$$ and $$D(x)=0$$ iff $$S(x)=S_0(x)$$.[1][3]

## Step 3: Promote the KL density to a potential (Obidi's Second Ingenuity — OSI)

The field‑level distinguishability functional is then  

$$ \mathcal{D}[S\Vert S_0] = \int_{\mathcal{M}} D(x)\,\sqrt{-g}\,d^4x=\int_{\mathcal{M}}\!\Big(S\ln\!\frac{S}{S_0} - S + S_0\Big)\sqrt{-g}\,d^4x.[][]$$  

ToE inserts $$\mathcal{D}[S\Vert S_0]$$ as the **potential term** in the Obidi / Spectral Obidi Action, schematically  

$$\mathcal{A}_E[S] = \int \big(\mathcal{K}[S,\partial S] - V_{\text{dist}}[S,S_0]\big)\sqrt{-g}\,d^4x,$$  

with  

$$ V_{\text{dist}}[S,S_0] \equiv D(x) = S\ln\!\frac{S}{S_0} - S + S_0.$$  

Thus the same structure that measures statistical distinguishability in information theory becomes a driving potential that pushes the entropy field away from or toward the reference configuration.[1][2]

## Step 4: Why this counts as “derived from” KL: Obidi's Ingenuity in Summary 

- The integrand $$S\ln(S/S_0) - S + S_0$$ is the continuum, field‑theoretic analogue of the KL density, preserving non‑negativity, convexity, and vanishing only at equality, which are exactly the properties used in KL‑based information geometry.[1][3][2]

- ToE’s “distinguishability potential” is therefore just the KL divergence reinterpreted in the entropic field language and reinserted into the action as a potential energy density that encodes how costly it is (in curvature/entropy terms) to deform $$S$$ away from $$S_0$$.[1][2]

Citations:

[1] Theory of Entropicity (ToE)'s Post https://www.linkedin.com/posts/theory-of-entropicity-toe_deriving-the-einstein-field-equations-of-activity-7419929069711982593-XgOF

[2] On the Conceptual and Mathematical Foundations of ... https://papers.ssrn.com/sol3/Delivery.cfm/5632191.pdf?abstractid=5632191&mirid=1

[3] Kullback–Leibler divergence - Wikipedia https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence

[4] Applications of a Kullback-Leibler Divergence for Comparing Non ... https://pmc.ncbi.nlm.nih.gov/articles/PMC4006220/

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

[6] Pictures get clear without 100% certainty? https://www.reddit.com/r/ObraDinn/comments/lf7lr9/pictures_get_clear_without_100_certainty/

[7] John Onimisi Obidi* 20 October 2025 https://flame-challenge.authorea.com/users/896400/articles/1348176/master/file/data/A%20Simple%20Explanation%20of%20the%20Unifying%20Mathematical%20Architecture%20of%20the%20Theory%20of%20Entropicity%20(ToE)_V3_S1/A%20Simple%20Explanation%20of%20the%20Unifying%20Mathematical%20Architecture%20of%20the%20Theory%20of%20Entropicity%20(ToE)_V3_S1.pdf?inline=true

[8] John Obidi’s Blog https://t.me/s/JohnObidiBlog?after=624

[9] Some Dissimilarity Measures of Branching Processes and Optimal Decision Making in the Presence of Potential Pandemics https://pmc.ncbi.nlm.nih.gov/articles/PMC7517477/

[10] Jose Gracia's Post - RealClock Quantum Mechanics https://www.linkedin.com/posts/jose-gracia-0686398_realclock-quantum-mechanics-preface-chapter-activity-7420149947049603072-sURJ