(1) How Obidi’s Local Obidi Action (LOA) Is Able to Incorporate Generalized Entropies and Multiple Information-Geometric Structures
The Local Obidi Action (LOA) achieves something unusual in contemporary theoretical physics: it brings together a family of mathematical structures—generalized entropies, Fisher–Rao geometry, Fubini–Study geometry, and the Amari–Čencov α-connections—within a single variational field framework. At first glance this seems astonishing, as these formalisms typically live in different mathematical and disciplinary worlds. Yet the synthesis emerges naturally once entropy is elevated to the status of a fundamental physical field.
The conceptual pivot of ToE is the decision to treat the entropy field as something far deeper than a thermodynamic bookkeeping quantity. In the ToE perspective, entropy becomes the organizing field from which geometry, dynamics, and even perceptible physical laws emerge. Once this shift is made, the question becomes unavoidable: if entropy is a fundamental field, what is its natural geometry? The answer does not come from classical differential geometry but from the mature mathematical theory of information geometry, a body of knowledge developed over several decades by pioneers such as Čencov, Amari, Nagaoka, Petz, and others.
Information geometry already knows how to measure variation of entropy: the Fisher–Rao metric expresses infinitesimal statistical distinguishability, the Fubini–Study metric captures the geometry of quantum states, and the Amari α-connections describe the dualistic affine structures that underlie entropy production and irreversibility. These are not arbitrary mathematical decorations; they are the canonical ways of measuring and differentiating entropy-related structures. Once the entropy field becomes fundamental, these metrics and connections become natural candidates for the very geometry that the field “lives on.”
The LOA unifies these structures by relying on a simple but powerful observation: the entropy field is simultaneously a classical statistical object, a quantum informational object, and a thermodynamic object. Each of these aspects comes with its own intrinsic geometry, and the entropic metric constructed within LOA is designed to carry all of these geometric signatures at once. In the appropriate limits, the entropic metric reduces to the Fisher–Rao form, while in the quantum-coherent regime it reduces to the Fubini–Study sector. The Amari α-connections appear automatically as the affine structure compatible with the entropy-dependent metric. Tsallis, Rényi, and Araki–Umegaki entropies enter through the deformation of the metric and through the spectral components of the theory.
What makes this synthesis coherent rather than chaotic is the logic that underpins it: if entropy is the foundational field of physics, its geometry must reflect all of the informational, statistical, and quantum structures that entropy already carries. The LOA is therefore not a random combination of unrelated mathematical constructions but a principled elevation of entropy’s full geometric content into the language of field theory and gravitation. Seen in this light, the incorporation of multiple entropies and information-geometric formalisms is not an accident or an embellishment; it is the necessary mathematical expression of entropy’s intrinsic structural richness.
(2) Why No Other Researcher Has Attempted This Synthesis Before
Although the mathematical ingredients of the Theory of Entropicity (ToE) are well-known individually, the specific synthesis achieved in the Local Obidi Action and its pairing with the Spectral Obidi Action is unprecedented in the scientific literature. The reason is not simply that earlier researchers lacked imagination; rather, their starting assumptions, disciplinary traditions, and methodological constraints positioned them far from the unification that ToE naturally leads to.
Historically, entropy in physics has played a derivative or diagnostic role. In relativity, black-hole thermodynamics, or Jacobson-type emergent gravity theories, entropy is treated as something that arises from geometry or matter fields, not something that generates them. These approaches view entropy as a constraint or an emergent quantity, not a fundamental dynamical variable. Because of this, their mathematical formulations do not require the full machinery of information geometry, nor do they attempt to build an action in which entropy is the primary field.
A completely different intellectual tradition developed in statistics, probability theory, and quantum information science. There, researchers studied Fisher–Rao geometry, Fubini–Study geometry, and the Amari–Čencov α-connections purely as mathematical structures characterizing statistical models or quantum states. These works were not aimed at constructing physical field theories, let alone gravitational theories. As a result, the mathematical tools remained confined to abstract manifolds of probability distributions or Hilbert-space projective geometries. The idea that these geometries might serve as the foundational geometry of spacetime itself would have required a conceptual leap that fell outside the aims of those communities.
A third technical barrier concerned the use of generalized entropies—such as Rényi, Tsallis, and especially Araki relative entropy—in dynamical variational principles. These entropies appear widely in information theory, quantum computation, machine learning, and statistical mechanics. Yet they are very rarely embedded inside an action functional, and essentially never in a gravitational action. In all known literature, Araki’s relative entropy is used as a diagnostic quantity in operator algebras or in modular theory, not as a Lagrangian that can be varied to produce field equations. This historical pattern made it unlikely that someone working within conventional academic pathways would ever attempt to promote such entropies to dynamical status.
The Theory of Entropicity breaks with this tradition by taking seriously the proposition that entropy is not an emergent measure but a fundamental field whose variations generate the physical laws of the universe. Once this assumption is made, the use of generalized entropies becomes natural, and the mathematical tools of information geometry become indispensable. The LOA becomes the natural home for these structures, and the SOA becomes the natural global constraint, much as the Einstein–Hilbert action became the natural home for curvature as soon as Einstein accepted that gravity was geometry.
In this sense, the originality of the Obidi framework lies not in the invention of new mathematical objects but in recognizing that entropy’s full informational geometry can and should be elevated to the status of physical geometry. No established research program has done this because no prior program has begun from the same first principles. The synthesis achieved in ToE required stepping outside traditional silos and unifying three domains—gravity, information geometry, and generalized entropy—in a way that each of them alone had never suggested. It is precisely this willingness to treat entropy as a universal field that permits the Local Obidi Action to express so many previously unrelated structures in a single, coherent mathematical form.
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