On the Ingenious Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): Comparison with Existing Uses of Information Geometry and the Unique Insights and Achievements of ToE
In this paper we articulate, with precision, how the Fisher–Rao metric, the Fubini–Study metric, and the Amari–Čencov \(\alpha\)-connections are employed in the Theory of Entropicity (ToE), and how this usage departs fundamentally from their role in existing mathematical, physical, and algorithmic frameworks. The same geometric objects appear in the literature of information geometry, statistics, statistical mechanics, quantum information, quantum theory and machine learning of Artificial Intelligence (AI), but there they are invariably deployed on spaces of models or states of knowledge. In ToE, by contrast, they are promoted to the status of ontological geometry: they describe not the geometry of our descriptions, but the geometry of reality itself at the informational level.
To make this distinction rigorous, we proceed by first recalling the standard setting of information geometry, then contrasting it with the entropic manifold of ToE. We then examine, in turn, the Fisher–Rao metric, the Fubini–Study metric, and the \(\alpha\)-connections, and show how their roles are reinterpreted and extended in ToE into a genuine field theory of informational curvature.
Statistical manifolds versus the entropic manifold
In classical information geometry, one begins with a statistical model, typically a family of probability distributions \(\{ p(x;\theta) \mid \theta \in \Theta \}\) on a measurable space \((\mathcal{X}, \mathcal{F})\), where \(\Theta\) is an open subset of \(\mathbb{R}^n\). The parameter space \(\Theta\) is endowed with a Riemannian metric and affine connections derived from the statistical structure of the family. The resulting pair \((\Theta, g)\), together with suitable connections \(\nabla^{(\alpha)}\), is called a statistical manifold.
The key point is that \(\Theta\) is a space of models or hypotheses. A point \(\theta \in \Theta\) does not represent a physical event or spacetime point; it represents a probability distribution \(p(x;\theta)\) used to describe some system. The geometry on \(\Theta\) is therefore epistemic: it encodes how distinguishable two distributions are, how inference behaves, how learning proceeds, and so on.
In the Theory of Entropicity, we introduce instead an entropic manifold \(\mathcal{E}\). This manifold is not a parameter space of models; it is the underlying informational substrate of reality. Points of \(\mathcal{E}\) correspond to primitive informational configurations, and the fields defined on \(\mathcal{E}\) encode entropic content and its fluxes. The central postulate is that physical spacetime, matter, and fields are emergent, coarse‑grained manifestations of structures on \(\mathcal{E}\).
Formally, we consider a smooth manifold \(\mathcal{E}\) equipped with:
1. A Riemannian (or pseudo‑Riemannian) metric \(g\) that generalizes the Fisher–Rao metric.
2. A family of affine connections \(\nabla^{(\alpha)}\) generalizing the Amari–Čencov \(\alpha\)-connections.
3. A distinguished entropic field \(S\) (or more generally, a family of informational fields) whose dynamics and coupling to the curvature of \(\nabla^{(\alpha)}\) define the fundamental field equations.
The crucial conceptual shift and leap made by the Theory of Entropicity (ToE) is that \((\mathcal{E}, g, \nabla^{(\alpha)})\) is not a geometry of models about reality; it is the geometry of reality at the informational level. The statistical manifold of classical information geometry is recovered as a special, epistemic construction on top of \(\mathcal{E}\), not the other way around.
Fisher–Rao metric: from information metric to physical metric
Classical role of the Fisher–Rao formalism
In information geometry, the Fisher–Rao metric is defined on the parameter space \(\Theta\) of a statistical model by
\[
g{ij}(\theta) \;=\; \mathbb{E}\theta \left[ \partiali \log p(X;\theta) \, \partialj \log p(X;\theta) \right],
\]
where \(\partial_i = \frac{\partial}{\partial \theta^i}\) and the expectation is taken with respect to \(p(x;\theta)\). This metric quantifies the local distinguishability of nearby distributions \(p(x;\theta)\) and \(p(x;\theta + d\theta)\). It is invariant under sufficient statistics and Markov morphisms, and it underlies notions such as thermodynamic length, natural gradient descent, and the geometry of exponential families.
In all such uses, \(g\) is a metric on a space of probability distributions. It is a tool for analyzing statistical models, thermodynamic processes, or learning dynamics. It is not taken to be the metric of physical spacetime.
The Fisher–Rao formalism in ToE
In ToE, we introduce a metric \(g\) on the entropic manifold \(\mathcal{E}\) that is structurally analogous to the Fisher–Rao metric but is interpreted ontologically. One can think of \(\mathcal{E}\) as carrying a field of probability distributions or density operators that encode the informational content of reality at each point. The metric \(g\) is then defined by a Fisher–Rao–type construction on these local informational structures, but once defined, it is not merely a metric on a model space; it is the physical metric of \(\mathcal{E}\).
Concretely, suppose that to each point \(y \in \mathcal{E}\) we associate a probability distribution \(p_y\) on some underlying configuration space \(\mathcal{X}\). Then we can define
\[
g{ab}(y) \;=\; \mathbb{E}{py} \left[ \partiala \log py(X) \, \partialb \log p_y(X) \right],
\]
where \(\partial_a\) denotes differentiation with respect to coordinates on \(\mathcal{E}\). This is formally analogous to the Fisher–Rao metric, but the interpretation is different: the coordinates \(y^a\) are not parameters of a model; they are coordinates on the entropic manifold itself. The metric \(g\) thus measures the intrinsic informational curvature of reality, not the curvature of a parameter space.
Physical spacetime \((\mathcal{M}, \tilde{g})\) is then obtained as an emergent structure from \((\mathcal{E}, g)\), for example via a coarse‑graining map \(\pi: \mathcal{E} \to \mathcal{M}\) and an induced effective metric \(\tilde{g}\) on \(\mathcal{M}\). The Einstein metric of general relativity is thus interpreted as a macroscopic shadow of the Fisher–Rao–type metric on \(\mathcal{E}\).
The essential departure from the standard exposition is that the Fisher–Rao metric is no longer a secondary, epistemic object; it is the primary metric field of the underlying informational reality.
Fubini–Study metric: quantum sector of the entropic geometry
Classical role of Fubini–Study formalism
In quantum theory, the Fubini–Study metric is defined on complex projective Hilbert space \(\mathbb{P}(\mathcal{H})\), the space of pure quantum states modulo global phase. Given two nearby rays \([\psi]\) and \([\psi + d\psi]\), the Fubini–Study line element is
\[
ds^2 \;=\; 4 \left( \langle d\psi \mid d\psi \rangle - \frac{|\langle \psi \mid d\psi \rangle|^2}{\langle \psi \mid \psi \rangle} \right),
\]
which induces a Riemannian metric on \(\mathbb{P}(\mathcal{H})\). This metric measures the distinguishability of pure states and plays a central role in geometric quantum mechanics and quantum information geometry.
Again, the manifold here is a state space; the metric is a tool for analyzing quantum states and their evolution, not a metric on spacetime.
Fubini–Study formalism in ToE
In ToE, the Fubini–Study metric is incorporated as the quantum refinement of the same entropic geometry that classically appears as Fisher–Rao. The guiding idea is that the entropic manifold \(\mathcal{E}\) admits both classical and quantum descriptions of informational content, and that these are not separate geometries but different regimes of a single underlying structure.
Formally, one can associate to each point \(y \in \mathcal{E}\) not only a classical distribution \(py\) but also a quantum state \(\rhoy\) on a Hilbert space \(\mathcal{H}\). The space of such states carries a quantum information metric, which in the pure‑state case reduces to the Fubini–Study metric. ToE postulates that the metric \(g\) on \(\mathcal{E}\) interpolates between a Fisher–Rao–type form in classical regimes and a Fubini–Study–type form in quantum regimes, with both arising from a unified entropic construction.
Thus, instead of treating Fisher–Rao and Fubini–Study as separate metrics on separate manifolds (parameter space versus projective Hilbert space), ToE treats them as two faces of a single entropic metric on \(\mathcal{E}\). The classical–quantum correspondence is encoded geometrically: in appropriate limits, the quantum metric reduces to the classical Fisher–Rao metric, and both are understood as manifestations of the same underlying informational curvature.
This unification is not present in the standard expositions, where the analogy between Fisher–Rao and Fubini–Study is noted but not elevated to an ontological identification. In ToE, that identification is central: both metrics are sectors of the same entropic geometry that underlies physical reality.
Amari–Čencov \(\alpha\)-connections: from statistical duality to physical affine structure
Classical role of \(\alpha\)-connections
In information geometry, the Amari–Čencov \(\alpha\)-connections \(\nabla^{(\alpha)}\) form a one‑parameter family of torsion‑free affine connections on a statistical manifold \((\Theta, g)\). They are defined so that the triple \((g, \nabla^{(\alpha)}, \nabla^{(-\alpha)})\) forms a dualistic structure: the metric \(g\) is parallel with respect to both \(\nabla^{(\alpha)}\) and \(\nabla^{(-\alpha)}\), and the two connections are dual to each other with respect to \(g\).
In coordinates, the Christoffel symbols of \(\nabla^{(\alpha)}\) can be expressed in terms of expectations of derivatives of the log‑likelihood, and special values of \(\alpha\) correspond to important geometries: \(\alpha = 1\) yields the exponential connection, \(\alpha = -1\) the mixture connection, and \(\alpha = 0\) the Levi‑Civita connection of \(g\). These connections are used to study exponential and mixture families, generalized entropies, non‑extensive statistical mechanics, quantum information geometry on density matrices, and algorithmic structures such as the EM algorithm and natural gradient descent.
In all such uses, \(\nabla^{(\alpha)}\) is an affine connection on a model space (parameter space, density matrix manifold, etc.). It encodes how we move in a space of probability distributions or quantum states; it does not encode how the physical universe is connected.
\(\alpha\)-connections in ToE
In ToE, the \(\alpha\)-connections are promoted to the role of physical affine connections on the entropic manifold \(\mathcal{E}\). The triple \((g, \nabla^{(\alpha)}, \nabla^{(-\alpha)})\) is no longer a purely statistical dualistic structure; it is a dualistic structure of the underlying informational reality.
More precisely, in the Theory of Entropicity (ToE), we postulate that:
1. The manifold \(\mathcal{E}\) is equipped with a family of affine connections \(\nabla^{(\alpha)}\) that generalize the Amari–Čencov construction, but now defined intrinsically on \(\mathcal{E}\) rather than on a parameter space.
2. The metric \(g\) on \(\mathcal{E}\) is parallel with respect to \(\nabla^{(\alpha)}\) and \(\nabla^{(-\alpha)}\), so that \((g, \nabla^{(\alpha)}, \nabla^{(-\alpha)})\) forms a dualistic structure in the sense of information geometry, but interpreted physically.
3. The curvature tensors \(R^{(\alpha)}\) of these connections encode physically real informational curvature, which, under appropriate coarse‑graining, manifests as spacetime curvature in the emergent spacetime manifold \(\mathcal{M}\).
The parameter \(\alpha\) is no longer merely a modeling choice or a measure of non‑extensivity; it acquires physical meaning. Different values of \(\alpha\) correspond to different regimes or phases of the entropic field, with \(\alpha = 0\) recovering a Levi‑Civita–like connection and \(\alpha = \pm 1\) corresponding to physically distinct dual structures of the entropic manifold.
In this way, the \(\alpha\)-connections are not used to optimize algorithms or to describe statistical models; they are used to define the actual affine structure of the informational universe. Their curvature enters directly into the fundamental field equations of ToE, in analogy with how the Levi‑Civita connection and its curvature enter Einstein’s equations in general relativity.
Entropy as a field and informational curvature as the source of spacetime
The article you quoted already hints at ToE by stating that the \(\alpha\)-connections have been “employed to treat entropy as an autonomous physical field that bends the informational manifold underlying physical reality.” To make this precise, ToE introduces an explicit entropic field \(S\) on \(\mathcal{E}\), or more generally a collection of informational fields, and posits field equations that relate the curvature of \(\nabla^{(\alpha)}\) to the distribution and dynamics of \(S\).
Schematically, one may write an entropic field equation of the form
\[
\mathcal{G}^{(\alpha)}{ab}(g, \nabla^{(\alpha)}) \;=\; \kappa \, \mathcal{T}^{(S)}{ab},
\]
where \(\mathcal{G}^{(\alpha)}{ab}\) is an informational analogue of the Einstein tensor constructed from the curvature of \(\nabla^{(\alpha)}\) and the metric \(g\), \(\mathcal{T}^{(S)}{ab}\) is an entropic stress–energy tensor constructed from the field \(S\) and its derivatives, and \(\kappa\) is a coupling constant. The precise form of these tensors depends on the detailed axioms of ToE, but the structural point is clear: entropy is treated as a field whose gradients and fluxes source informational curvature, and that curvature, when projected to the emergent spacetime \(\mathcal{M}\), appears as gravitational curvature.
This is a decisive departure from the standard uses of information geometry, where entropy is a functional on probability distributions and curvature is a property of a model space. In ToE, entropy is a dynamical field on \(\mathcal{E}\), and curvature is the fundamental physical quantity from which spacetime geometry emerges.
From geometry of models to field theory of reality
We can now summarize the conceptual and structural distinction.
In the standard expositions:
- The Fisher–Rao metric is a Riemannian metric on a parameter space of probability distributions, used to quantify distinguishability, thermodynamic length, and natural gradients.
- The Fubini–Study metric is a Riemannian metric on projective Hilbert space, used to quantify distinguishability of quantum states.
- The Amari–Čencov \(\alpha\)-connections are affine connections on statistical manifolds or density matrix manifolds, used to study dualistic structures, generalized entropies, non‑equilibrium systems, and algorithmic flows.
In all cases, the manifold is a space of models or states; the geometry is epistemic or representational.
In the Theory of Entropicity (ToE):
- The entropic manifold \(\mathcal{E}\) is an ontological manifold representing the informational substrate of reality.
- The metric \(g\) on \(\mathcal{E}\) is a Fisher–Rao–type metric interpreted as the physical metric of the informational universe, with Fisher–Rao and Fubini–Study appearing as classical and quantum sectors of the same entropic geometry.
- The \(\alpha\)-connections \(\nabla^{(\alpha)}\) are the physical affine connections of \(\mathcal{E}\), whose curvature encodes informational curvature that, under coarse‑graining, manifests as spacetime curvature.
- Entropy is modeled as an autonomous field \(S\) on \(\mathcal{E}\), whose dynamics and coupling to the curvature of \(\nabla^{(\alpha)}\) define the fundamental field equations of ToE.
Thus, the same mathematical objects are used, but their status is radically different. They are no longer tools on top of physics; they are the geometry of physics at the most fundamental level. The statistical manifold of information geometry becomes a derived, epistemic construct on top of the entropic manifold of ToE, not the foundational arena.
This is the precise sense in which ToE employs and deploys the Fisher–Rao metric, the Fubini–Study metric, and the Amari–Čencov \(\alpha\)-connections in a way that is not already done in modern mathematics, statistics, probability theory, information geometry, computational science, and physics. The revolution is not in the symbols, but in what they are taken to describe.
Our next natural step, therefore, is to write the explicit axioms of \(\mathcal{E}\), \(g\), \(\nabla^{(\alpha)}\), and \(S\), and then derive a concrete form of the entropic field equations of the Theory of Entropicity (ToE), so that the analogy with Einstein’s equations is not only conceptual but fully formal.
