On the Obidi Curvature Invariant (ln 2) and the Geometry of Distinguishability in the Theory of Entropicity (ToE): A Gedanken Experiment of a Paper Crease
Abstract
In classical thermodynamics and information theory, the natural logarithm of two, ln 2, represents the entropy change associated with a single bit of information. It appears in contexts as diverse as Landauer’s principle, the Bekenstein–Hawking entropy, and Shannon’s information measure. However, in the Theory of Entropicity (ToE) developed by John Onimisi Obidi, ln 2 assumes a radically different status. It ceases to be a statistical conversion constant and becomes a universal geometric invariant of the entropic field itself — the smallest possible curvature gap between two distinguishable configurations of reality. This paper presents a comprehensive exposition of this idea, linking the curvature interpretation of ln 2 to the convex stability of the entropic field, the Fisher–Rao and Fubini–Study metrics, and the principle of informational distinguishability. A physical analogy is offered: a flat sheet becomes informationally active only when it is creased, and that first crease — the minimal deformation that makes “difference” visible — is the geometric expression of ln 2.
1. Introduction
In standard physics, entropy serves as a statistical quantity describing ensembles of microstates. Its informational interpretation, beginning with Boltzmann, Gibbs, and Shannon, has tied it to probability and uncertainty rather than to ontological structure. Yet modern developments — black hole thermodynamics, entropic gravity, and holographic dualities — have revealed that entropy has a deeper geometric meaning.
The Theory of Entropicity (ToE) advances this insight by declaring entropy, not energy or spacetime, as the fundamental field of nature. The entropic field, denoted S(x), is not a derived statistical measure but a physical continuum possessing gradients, curvature, and dynamics. Matter, geometry, and information emerge as local configurations and deformations of this field.
Within this framework, ln 2 acquires a new interpretation. Instead of being a conversion between binary information and physical entropy, it becomes a measure of the minimum distinguishable curvature difference in the entropic field. Just as Planck’s constant h quantizes action, ln 2 quantizes distinguishability. It is the smallest geometric “fold” the entropic field can sustain while maintaining stable separability between two informational states.
2. The Flat Sheet Analogy: The Birth of Distinguishability
Consider a perfectly flat sheet of paper. As long as it remains flat, it has no curvature and, by extension, no distinguishable regions. Every point is equivalent; the field is uniform and featureless. This represents the zero-entropy-difference state of the universe — complete indistinguishability.
Now imagine creasing the sheet slightly. A fold appears, dividing the surface into regions of up, down, and sideways orientation. The moment of the first crease marks the birth of distinction: one side differs from another. This fold corresponds to the smallest curvature deformation that makes two configurations distinguishable.
In the Theory of Entropicity (ToE), this minimal curvature fold is precisely ln 2. It represents the transition from homogeneity to distinguishability, from sameness to difference. The flat state corresponds to pure informational symmetry (zero distinction), while the creased state embodies the minimal break of that symmetry.
3. Mathematical Representation: The Minimum Curvature Gap
Let S(x) denote the entropic field and ρ(x) represent its local normalized distribution (the “entropic density profile”). Two distinguishable configurations of this field are represented by ρ₁(x) and ρ₂(x).
The Theory of Entropicity (ToE) defines the entropic curvature distance between these two configurations as:
D(ρ₁ || ρ₂) = ∫ ρ₁(x) ln [ρ₁(x) / ρ₂(x)] dx
This functional, identical in mathematical form to the Kullback–Leibler divergence, measures not statistical information but geometric curvature difference in ToE. The integral quantifies the degree to which one entropic configuration must deform to become another.
For the simplest possible case of two configurations differing by a constant curvature ratio, say ρ₂(x) = 2 × ρ₁(x), we have:
D(ρ₁ || ρ₂) = ∫ ρ₁(x) ln [1 / 2] dx = −ln 2
Since ρ₁(x) is normalized, the magnitude of this curvature difference is |D| = ln 2.
This establishes ln 2 as the minimum nonzero curvature gap between two distinguishable entropic configurations. The field cannot support smaller stable curvature separations; if the ratio were less than 2:1, convexity would merge the two configurations into a single indistinguishable state.
Thus, ln 2 represents the first geometric threshold of separability — the quantized onset of distinction in the entropic manifold.
4. Convexity, Stability, and the 2:1 Ratio
Why must the minimum curvature ratio be 2:1?
In the Theory of Entropicity (ToE), the entropic field S(x) evolves according to an energy functional:
E[S] = ∫ F(S, ∇S) dx
where F(S, ∇S) is convex in S. Convexity ensures physical stability: small perturbations can smooth out, but sufficiently large ones generate new stable configurations.
Mathematically, a convex functional cannot sustain two distinct minima whose curvature differs by less than a factor of 2. Below this threshold, the energy landscape merges into a single basin, erasing distinction. Above it, two separate minima exist, corresponding to physically distinguishable configurations.
Therefore, a curvature ratio of 2:1 is not arbitrary; it emerges from the convex stability condition of the entropic field. The logarithmic “distance” between these configurations — the entropic curvature invariant — is ln 2.
This result mirrors other quantization thresholds in physics: the smallest quantum of spin (½ħ), charge (e), or action (ħ). Here, ln 2 plays the same role but in the domain of entropy and information geometry.
5. The Fisher–Rao and Fubini–Study Geometries
The universality of the Obidi Curvature Invariant (OCI) is further reinforced by the fact that it can be derived from both classical and quantum geometric frameworks.
In classical information geometry, the Fisher–Rao metric defines the infinitesimal distance between probability (or in ToE, entropic) distributions:
ds² = (1/4) ∑ [ (dρ_i)² / ρ_i ]
For two distributions ρ₁ and ρ₂ that differ by a finite factor of 2, the integrated geodesic distance in Fisher–Rao space is exactly √(2 ln 2), making ln 2 the squared curvature invariant of distinguishability.
In quantum geometry, the Fubini–Study metric governs the distance between two quantum states |ψ₁⟩ and |ψ₂⟩:
ds² = 1 − |⟨ψ₁|ψ₂⟩|²
When these states are orthogonal enough to differ by a minimal distinguishable overlap of 1/2, the geometric distance corresponds to ln 2 in entropic units. Thus, the same invariant appears in both classical and quantum manifolds — a sign of deep universality.
This shows that ln 2 is not merely a constant of statistical conversion but a metric invariant across classical and quantum geometries, unifying them under the entropic curvature principle of ToE.
6. Physical Interpretation: From Landauer to Holography
In traditional frameworks, ln 2 appears in the Landauer principle, which states that erasing one bit of information dissipates a minimum energy of:
ΔE = k_B T ln 2
This has been understood as a thermodynamic cost of information erasure. In ToE, however, this relation gains a geometric and ontological interpretation. Erasing information corresponds to flattening an entropic curvature — reducing distinguishability. The energy cost is therefore the field’s resistance to losing curvature.
Similarly, in holography, each “pixel” of a holographic screen encodes one bit of information, and the total entropy S = N k_B ln 2 arises from summing over these minimal curvature patches. ToE interprets this not as statistical counting but as tiling the universe’s boundary with units of minimal distinguishability. Each ln 2 represents a fundamental crease in the entropic fabric of reality.
7. The Obidi Curvature Invariant as a Universal Principle
The radical insight of the Theory of Entropicity (ToE) is that ln 2 is not a thermodynamic artifact but a universal curvature invariant. It defines the smallest separable difference between two informationally or physically distinct configurations.
No other framework in modern physics — not thermodynamics, not quantum mechanics, not relativity — has elevated ln 2 from a conversion constant to a universal geometric principle. ToE thus provides the first ontological explanation for why ln 2 recurs across entropy, information, and gravitation. It is not coincidence but necessity: the field of reality can only fold in discrete distinguishable steps, and the smallest such fold is ln 2.
8. Conclusion
From the first crease in a flat sheet to the curvature of spacetime itself, distinction arises through deformation. The Theory of Entropicity (ToE) identifies ln 2 as the fundamental measure of that deformation — the Obidi Curvature Invariant (OCI).
It is the universal constant of distinguishability, linking geometry, entropy, and information into a single coherent framework. Classical physics, quantum mechanics, thermodynamics, and holography each reveal fragments of ln 2’s presence, but only ToE unifies them by showing that curvature, not probability, is the origin of entropy.
In this light, ln 2 is no longer a number; it is the first step from sameness to difference, the moment where reality begins to tell itself apart.
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