ABSTRACTOverview of the Obidi Action Principle
This monograph develops the full mathematical and conceptual foundations of the Obidi Action Principle (OAP), the central variational structure underlying the Theory of Entropicity (ToE). The OAP unifies three geometric sectors— Fisher–Rao, Fubini–Study, and Amari–Čencov -connections—into a single entropic manifold whose curvature, metric, and affine structure jointly determine the dynamics of spacetime, matter, and gauge interactions.
Unlike traditional physics, where geometry, matter, and gauge fields are introduced as separate entities, ToE derives all three from the information‑geometric structure of the entropic field. The Fisher–Rao metric becomes the spacetime metric; the Fubini–Study metric becomes the matter–energy sector; and the -connections become the gauge sector.
The Obidi Action is the unique diffeomorphism‑invariant scalar functional constructed from these structures, and its variation yields the full field equations of ToE: a generalized Einstein–Obidi field equation, entropic matter equations, and entropic gauge equations. This monograph presents the derivation, mathematical justification, and physical interpretation of these results in a rigorous and unified framework.
§ IIntroduction: The Entropic Manifold and the Ontological Elevation of Information Geometry
The Theory of Entropicity (ToE) begins from a radical but mathematically precise premise: information geometry is not epistemic but ontological. In classical statistics, the Fisher–Rao metric measures distinguishability between probability distributions; in quantum mechanics, the Fubini–Study metric measures distinguishability between quantum states; and in information geometry, the -connections encode dualistic affine structures. Traditionally, these structures quantify knowledge about systems. In ToE, they quantify being itself.
The entropic manifold is defined as the space of physically realized entropic configurations. Each point corresponds to a complete specification of the entropic field , and the geometry of encodes the physical laws governing the universe.
The Fisher–Rao metric defines the spacetime interval; the Fubini–Study metric defines the inertial and energetic structure of matter; and the -connections define the gauge structure. These three components form the entropic trinity of ToE.
The Obidi Action Principle (OAP) is the unified variational principle that governs the dynamics of this entropic manifold. It is constructed from scalar invariants of , , and , and its variation yields the full field equations of ToE. The OAP is therefore the entropic analogue of the Einstein–Hilbert action, the Yang–Mills action, and the matter action, but unified into a single geometric functional.
§ IIThe Entropic Manifold : Coordinates, Structure, and Mathematical Foundations
The entropic manifold is a smooth, finite‑ or infinite‑dimensional differentiable manifold whose points represent entropic configurations of the universe. A coordinate chart labels these configurations. The probability distribution is not epistemic but ontological: it is the local expression of the entropic field .
The manifold is equipped with a measure such that defines the entropic density. The Fisher–Rao metric is defined by:
The Čencov–Morozova theorem ensures that this metric is the unique Riemannian metric invariant under sufficient statistics and Markov morphisms. This uniqueness is the mathematical justification for identifying with the spacetime metric: no other metric on satisfies the required invariance properties.
The entropic manifold also carries an internal sector described by the Fubini–Study metric , defined for pure states by:
This metric defines a Kähler geometry whose curvature encodes the inertial and energetic structure of the entropic field. Finally, the manifold carries a family of -connections , whose curvature defines the gauge field strengths.
Together, these structures define the full geometric content of the entropic manifold. The Obidi Action is constructed from these structures and governs their dynamics.
§ IIIThe Fisher–Rao Metric as the Spacetime Sector of the Entropic Manifold
The Fisher–Rao metric occupies a privileged position within the Theory of Entropicity (ToE). It is not merely a statistical measure of distinguishability; it is the unique Riemannian metric compatible with the transformation laws of information‑preserving mappings. This uniqueness, guaranteed by the Čencov–Morozova theorem, elevates the Fisher–Rao metric from a statistical artifact to a fundamental geometric structure of the entropic manifold. In ToE, this structure is identified with the spacetime metric.
Given a family of entropic configurations , the Fisher–Rao metric is:
This metric measures the infinitesimal distinguishability between neighboring entropic states and . In classical information geometry, this is interpreted epistemically: it quantifies how well an observer can discriminate between two statistical models. In ToE, the interpretation is ontological: the distinguishability is a physical separation between two possible configurations of the entropic field .
The Čencov–Morozova theorem states that the Fisher–Rao metric is the only Riemannian metric invariant under sufficient statistics and Markov morphisms. In the entropic ontology, these invariances correspond to the requirement that physical laws be invariant under coarse‑graining and entropic transformations. Thus, the Fisher–Rao metric is the only metric compatible with the fundamental symmetries of the entropic field.
The geodesics of the Fisher–Rao metric satisfy:
where are the Christoffel symbols of . These geodesics represent the physically realized trajectories of the entropic field. The entropic interval:
is the analogue of the spacetime interval in general relativity. The manifold is therefore the spacetime sector of ToE.
Diagram: Fisher–Rao as Spacetime
Entropic Configurations → Fisher–Rao Metric → Spacetime Geometry
Uniqueness (Čencov–Morozova) ⇒ Physical Necessity
§ IVThe Fubini–Study Metric as the Matter–Energy Sector of the Entropic Manifold
The Fubini–Study metric arises naturally in the geometry of quantum states, but in the Theory of Entropicity it is reinterpreted as the matter–energy sector of the entropic manifold. This reinterpretation is not metaphorical; it is a mathematically rigorous identification of curvature in the internal sector of the entropic manifold with the inertial and energetic properties of physical systems.
For pure states , the Fubini–Study metric is:
This metric defines a Kähler geometry on the projective Hilbert space. Its curvature tensor encodes the “stiffness” of the entropic field along internal directions. In ToE, this stiffness is interpreted as inertial resistance.
Directions in the entropic manifold that are costly in the Fubini–Study metric correspond to directions in which the entropic configuration resists deformation. This resistance is the entropic analogue of mass.
The scalar curvature of the Fubini–Study sector contributes to the Obidi Action via:
where is a coupling constant and denotes an appropriate contraction of internal curvature with the spacetime metric.
Regions where correspond to regions where the entropic field stores energy, mass, or internal degrees of freedom. A flat Fubini–Study geometry corresponds to vacuum‑like behavior. Thus, matter and energy are not external inputs to the theory; they are curvature phenomena of the entropic manifold.
Diagram: Fubini–Study as Matter–Energy
Internal Geometry → Curvature → Matter/Energy Content
Matter = Curvature of the Entropic Field
§ VThe -Connections as the Gauge Sector of the Entropic Manifold
The third geometric component of the entropic manifold is the family of Amari–Čencov -connections. In classical information geometry, these connections encode dualistic affine structures and determine how statistical quantities are parallel transported. In the Theory of Entropicity, they are elevated to the status of gauge fields.
On the statistical manifold with Fisher–Rao metric , the -connections are defined by:
where is the Levi–Civita connection of , and is the skewness tensor:
In a gauge‑theoretic representation, one introduces gauge potentials valued in a Lie algebra with structure constants . The curvature (field strength) of the -connection is:
In ToE, the -connections define how internal degrees of freedom are parallel transported along the entropic manifold. This is precisely the role played by gauge fields in Yang–Mills theory. The curvature is interpreted as the gauge field strength, and the holonomies of correspond to physical charges and phase factors.
The gauge sector contributes to the Obidi Action through:
Thus, gauge fields are not added to the theory as external structures; they are intrinsic affine structures of the entropic manifold. The -connections complete the entropic trinity: Fisher–Rao gives spacetime, Fubini–Study gives matter, and -connections give gauge interactions.
Diagram: -Connections as Gauge Fields
Affine Structure → Curvature → Gauge Interactions
Gauge Fields = Affine Geometry of the Entropic Field
§ VIConstruction of the Obidi Action on the Entropic Manifold
The Obidi Action Principle (OAP) is the central variational structure of the Theory of Entropicity (ToE). It unifies the three geometric sectors of the entropic manifold— Fisher–Rao, Fubini–Study, and Amari–Čencov -connections—into a single diffeomorphism‑invariant scalar functional. The OAP is not an arbitrary construction; it is the unique action compatible with the symmetries, invariances, and ontological commitments of the entropic field.
Let be the entropic manifold with coordinates . The geometric data on consist of:
(1) The Fisher–Rao metric , defining the spacetime sector.
(2) The Fubini–Study metric , defining the matter–energy sector.
(3) The -connections , defining the gauge sector.
The action must satisfy three fundamental requirements:
(i) Diffeomorphism invariance on .
(ii) Construction from scalar contractions of curvature and metric tensors.
(iii) Reduction to known physical actions (Einstein–Hilbert, Yang–Mills, matter actions) in appropriate limits.
Let denote the Ricci scalar of the Fisher–Rao metric. Let denote the scalar curvature of the Fubini–Study metric. Let denote the curvature (field strength) of the -connection. The Obidi Action is:
Here is the dimension of the entropic manifold, and is the entropic gravitational constant. The Fubini–Study (matter–energy) sector contributes:
where is a coupling constant and denotes an appropriate contraction of internal curvature with the spacetime metric.
The gauge sector contributes:
The interaction term contains all non‑minimal couplings between the three sectors, constrained by diffeomorphism invariance, gauge invariance, and entropic symmetry.
Diagram: Structure of the Obidi Action
Spacetime (Fisher–Rao) + Matter (Fubini–Study) + Gauge (-Connections)
→ Unified Variational Principle
§ VIIVariational Calculus on the Entropic Manifold
To derive the field equations of the Theory of Entropicity, one performs a systematic variation of the Obidi Action with respect to its independent geometric variables: the Fisher–Rao metric , the Fubini–Study metric (or equivalently the internal fields ), and the gauge potentials associated with the -connections. Each variation yields a distinct sector of the full ToE field equations.
VII.1 Variation with Respect to the Fisher–Rao Metric
Writing the action as:
with:
the metric variation yields:
Setting for arbitrary gives the generalized Einstein–Obidi field equations:
Here contains non‑minimal couplings, and is the total entropic stress–energy tensor.
VII.2 Variation with Respect to the Fubini–Study Sector
Let parametrize the internal manifold. The internal metric and its curvature become functionals of . The variation yields:
These equations describe the dynamics of the entropic matter sector, including mass generation, self‑interaction, and coupling to gauge and gravitational fields.
VII.3 Variation with Respect to the -Connections
Let be the gauge potentials whose curvature is . The variation yields:
Here is the gauge‑covariant derivative, and is the entropic gauge current arising from the internal sector and interaction terms.
These are the entropic generalizations of the Yang–Mills equations.
Diagram: Variational Structure of ToE
→ Gravitational Sector
→ Matter Sector
→ Gauge Sector
All from a Single Action
§ VIIIThe Full Field Equations of the Theory of Entropicity (ToE)
The field equations of the Theory of Entropicity arise from the simultaneous variation of the Obidi Action with respect to all geometric variables. These equations are not independent; they are three projections of a single variational principle on the entropic manifold. Together, they form a unified system describing the dynamics of spacetime, matter, and gauge fields.
VIII.1 Generalized Einstein–Obidi Equation
This equation generalizes Einstein’s field equations. The Fisher–Rao metric plays the role of the spacetime metric, and the total entropic stress–energy tensor includes contributions from the Fubini–Study sector, the gauge sector, and their interactions.
VIII.2 Entropic Matter Equations
These equations describe the dynamics of the internal entropic fields. They include mass terms, self‑interaction terms, and couplings to gauge and gravitational fields. They are the entropic analogue of the Klein–Gordon, Dirac, or nonlinear sigma‑model equations, depending on the structure of .
VIII.3 Entropic Gauge Equations
These are the entropic generalizations of the Yang–Mills equations. The -connections define the gauge structure, and their curvature defines the gauge field strengths. The currents arise from the internal entropic geometry.
Together, these three sets of equations form the complete dynamical system of the Theory of Entropicity. They describe how the entropic field evolves, how spacetime curves, how matter behaves, and how gauge fields propagate—all as manifestations of a single geometric structure.
Diagram: The Full ToE Field System
Generalized Einstein–Obidi Equation
Entropic Matter Equations
Entropic Gauge Equations
Unified by the Obidi Action Principle
APPENDIX AMathematical Preliminaries for the Entropic Manifold
This appendix gathers the mathematical structures required for the rigorous formulation of the Theory of Entropicity (ToE). The entropic manifold is a differentiable manifold equipped with three geometric sectors: the Fisher–Rao metric , the Fubini–Study metric , and the Amari–Čencov -connections . Each of these structures requires a precise mathematical foundation.
A.1 Differential Geometry of the Entropic Manifold
Let be an -dimensional smooth manifold with coordinates . A metric tensor defines the line element:
The Levi–Civita connection associated with has Christoffel symbols:
The Riemann curvature tensor is:
The Ricci tensor and scalar curvature follow:
A.2 Fisher–Rao Geometry
Given a family of entropic configurations , the Fisher–Rao metric is:
The Čencov–Morozova theorem states that the Fisher–Rao metric is the unique Riemannian metric invariant under Markov morphisms. This uniqueness is the mathematical justification for identifying with the spacetime metric in ToE.
A.3 Fubini–Study Geometry
For pure states , the Fubini–Study metric is:
The Fubini–Study metric defines a Kähler manifold with complex structure , symplectic form , and metric satisfying:
A.4 Amari–Čencov -Connections
The -connections are defined by:
where the skewness tensor is:
The curvature of the -connection is:
APPENDIX BWorked Examples in Entropic Geometry
B.1 Fisher–Rao Geometry of a Two‑State System
Consider a binary distribution . The Fisher–Rao metric is:
The line element is:
The geodesic distance between and is:
B.2 Fubini–Study Curvature of a Qubit
A qubit state can be written as:
The Fubini–Study metric becomes:
which is the metric of a sphere of radius . The scalar curvature is:
B.3 Holonomy of an -Connection
For a closed loop in , the holonomy is:
where denotes path ordering. This holonomy corresponds to a physical gauge phase in ToE.
APPENDIX CPhysical Limits of the Obidi Action
C.1 General Relativity Limit
When the Fubini–Study and gauge sectors vanish, the Obidi Action reduces to:
This is the Einstein–Hilbert action with gravitational constant .
C.2 Yang–Mills Limit
When is fixed and is trivial, the action reduces to:
C.3 Quantum Mechanics as an Emergent Hilbert Chart
When the entropic manifold admits a complex structure compatible with , the internal dynamics reduce to Schrödinger evolution in an emergent Hilbert space.
APPENDIX DTables, Diagrams, and Conceptual Maps
Diagram D.1 — The Entropic Trinity
Fisher–Rao → Spacetime
Fubini–Study → Matter–Energy
-Connections → Gauge Fields
Unified by the Obidi Action Principle
Diagram D.2 — Variational Flow of ToE
→ Gravitational Sector
→ Matter Sector
→ Gauge Sector
PART VBibliography, Notation Index, and Glossary
V.1 Bibliography
The following references provide the mathematical, physical, and conceptual background for the Theory of Entropicity (ToE), the Obidi Action Principle (OAP), and the information‑geometric structures employed throughout this monograph.
[1] S. Amari and H. Nagaoka, Methods of Information Geometry, AMS Translations of Mathematical Monographs, Vol. 191, American Mathematical Society, 2000.
[2] N. N. Čencov, Statistical Decision Rules and Optimal Inference, American Mathematical Society, 1982.
[3] B. O’Neill, Semi‑Riemannian Geometry with Applications to Relativity, Academic Press, 1983.
[4] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vols. I–II, Wiley, 1963–1969.
[5] R. M. Wald, General Relativity, University of Chicago Press, 1984.
[6] C. N. Yang and R. L. Mills, “Conservation of Isotopic Spin and Isotopic Gauge Invariance,” Physical Review, 96 (1954), 191–195.
[7] J. Anandan and Y. Aharonov, “Geometry of Quantum Evolution,” Physical Review Letters, 65 (1990), 1697–1700.
[8] M. Nakahara, Geometry, Topology and Physics, 2nd ed., Taylor & Francis, 2003.
[9] J. O. Obidi, Foundational Letters on the Theory of Entropicity (ToE), internal monograph series (unpublished drafts, 2024–2026).
[10] J. O. Obidi, “The Obidi Action Principle (OAP) and the Entropic Manifold,” working manuscript in entropic field theory, 2025–2026.
V.2 Notation Index
This subsection collects the principal symbols used throughout the monograph. All indices follow the conventions stated below unless explicitly indicated otherwise.
— Entropic manifold (space of entropic configurations).
— Coordinates on ; spacetime‑sector coordinates.
— Underlying micro‑configuration variable.
— Ontological probability distribution (entropic configuration).
— Entropic field.
— Fisher–Rao metric (spacetime sector).
— Fubini–Study metric (matter–energy sector).
— Amari–Čencov -connection (gauge sector).
— Christoffel symbols of .
— Christoffel symbols of the -connection.
— Skewness tensor.
— Riemann curvature, Ricci tensor, Ricci scalar.
— Scalar curvature of the Fubini–Study sector.
— Gauge potential.
— Gauge field strength.
— Structure constants of the gauge algebra.
— Invariant metric on the gauge algebra.
— Einstein tensor.
— Non‑minimal coupling tensor.
— Total entropic stress–energy tensor.
— Internal fields.
— Euler–Lagrange expressions for internal fields.
— Entropic gauge current.
— Entropic gravitational constant.
— Fubini–Study coupling constant.
— Gauge coupling constants.
— Obidi Action functional.
— Lagrangian densities.
V.3 Glossary of Core Concepts
Entropic Manifold () — The differentiable manifold whose points represent physically realized entropic configurations of the universe.
Entropic Field () — A scalar or tensorial field representing local entropic density or configuration.
Fisher–Rao Metric () — The unique information‑geometric metric invariant under Markov morphisms; reinterpreted as the spacetime metric in ToE.
Fubini–Study Metric () — The Kähler metric on projective Hilbert space; reinterpreted as the matter–energy sector.
Amari–Čencov -Connections — A one‑parameter family of affine connections compatible with , reinterpreted as gauge connections.
Obidi Action Principle (OAP) — The unified variational principle governing the dynamics of the entropic manifold.
Generalized Einstein–Obidi Equation — The gravitational field equation of ToE.
Entropic Matter Equations — Internal field equations governing the Fubini–Study sector.
Entropic Gauge Equations — Gauge field equations generalizing Yang–Mills theory.
Ontodynamics — The study of dynamical laws at the level of being itself.
Hilbert Space as Emergent Chart — The ToE view that Hilbert space is not fundamental but arises as a coordinate chart on certain sectors of the entropic manifold.
PART VIFully Linked Table of Contents
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