From
Shannon Entropy to Spacetime: A Rigorous Derivation of the Obidi Action from
Shannon Entropy via Information Geometry in the Theory of Entropicity (ToE)
ToE Living
Review Letters — Letter IIF
John Onimisi Obidi
Theory of Entropicity (ToE)
(Dated: Sunday, May 24, 2026)
Abstract
The
Theory of Entropicity (ToE) elevates Shannon entropy from an epistemic measure
to an ontological field whose dynamics underlie both quantum matter and
classical spacetime. In this Letter we provide a rigorous, step‑by‑step
derivation of the Obidi Action—the universal action of
ToE—beginning with the Shannon entropy functional and the information‑geometric
structure it induces. Starting from the continuous Shannon entropy of a
probability density on a differentiable manifold, we promote the density to a
dynamical entropic field 
via the identification 
. The resulting scalar functional serves as the potential
term of an action. The kinetic term is furnished by the Fisher information
metric, the natural Riemannian metric of information geometry, expressed
through the gradient of . Varying the resulting proto‑Obidi action yields the
simplest form of the Master Entropic Equation, a nonlinear
covariant field equation governing the pre‑geometric dynamics of the entropic
field. We then show how the inclusion of curvature invariants built from the
Fisher–Rao metric—through the process of curvature transfer—completes
the proto‑action to the full Obidi Action, from which Einstein’s equations of
general relativity emerge as thermodynamic identities in a coarse‑grained
limit. The derivation establishes a direct logical chain from Shannon’s
original formula to the dynamics of spacetime, rooting gravitation and field
theory in the geometry of distinguishability.
1.
Introduction
A
deep conceptual divide has long separated information theory from fundamental
physics. Shannon’s entropy,
quantifies
uncertainty in a probability distribution, while the Einstein–Hilbert action,
encodes
the dynamics of spacetime curvature. The Theory of Entropicity (ToE) [1, 2, 3]
dissolves this divide by positing that entropy is ontological, not
merely epistemic. In ToE, the probability distribution 
over a configuration manifold is not a state of
knowledge but a real physical field—the entropic field 
—whose variations constitute the fundamental substance of
the universe.
Previous
Letters in this series have established the kinematical framework of ToE
(Letter I), the Master Entropic Equation (Letter IIA), the No‑Rush Theorem
(Letter IIB), and the emergence of quantum mechanics from entropic
distinguishability (Letter IID). The present Letter, IIF, delivers the missing
dynamical cornerstone: a rigorous derivation of the Obidi Action directly
from Shannon entropy, via the tools of information geometry. We demonstrate
that the action functional governing all physical dynamics can be constructed
from two primitive ingredients—Shannon entropy and the Fisher information
metric—without any additional postulates.
The
paper is organized as follows. In §2 we rewrite Shannon’s continuous entropy as
a functional of the entropic field and show that it already possesses the
structural properties of a field‑theory potential term. Section 3 introduces
the Fisher–Rao metric of information geometry and extracts the natural kinetic
scalar for the entropic field. In §4 we assemble the potential and kinetic
terms into the proto‑Obidi Action, the minimal entropic action.
Section 5 varies this action to obtain the Master Entropic Equation in its
simplest covariant form. Section 6 describes the completion of the proto‑action
to the full Obidi Action through curvature transfer, and §7 shows how
Einstein’s equations emerge from the entropic dynamics. We conclude in §8 with
a discussion of the philosophical and physical implications.
Throughout
we work on a 
-dimensional manifold 
with Lorentzian metric 
, signature 
in four dimensions, and use units where the
fundamental entropic length scale 
. Greek indices run from 
to 
; the covariant derivative compatible with is denoted 
, and the d’Alembertian is 
.
2.
Shannon Entropy as an Ontological Field
2.1
Continuous Shannon entropy
Shannon’s
entropy for a discrete probability distribution 
is 
. Its natural continuum analog on a Riemannian or
Lorentzian manifold 
with volume element 
is
where is a probability density normalized as 
.
In
standard information theory, represents an epistemic weight. ToE makes the
ontological leap: is identified with the entropic field , a real scalar field that encodes the local intensity of
Being [1]. The simplest invertible identification that respects 
and maps the functional form of Shannon entropy to a
local density is
The
normalization condition becomes 
, which we impose as a global constraint; for local
dynamics we may work with the unconstrained field and later restore
normalization through a Lagrange multiplier when necessary.
Inserting
(2) into (1) yields
2.2
Entropy as a potential term
The
integrand 
is a scalar function of the field and the metric. The
functional
with
a positive constant 
of dimension 
to give the action its correct units, already
exhibits the essential features of a field‑theoretic potential term:
- It is a
coordinate scalar, transforming as a density of weight 1.
- It
depends locally on and the background metric.
- It can
be minimized, giving an Euler–Lagrange equation that determines the
entropic ground state.
Moreover,
the form has a unique minimum at 
(i.e., 
), corresponding to the configuration of maximal Shannon
entropy under normalization—the entropic vacuum.
Thus,
Shannon entropy itself supplies the potential term of the universal action. The
remaining task is to furnish a kinetic term that encodes the dynamical
evolution of .
3.
Information Geometry and the Kinetic Term
3.1
Fisher information metric
Information
geometry endows the space of probability distributions with a Riemannian
structure—the Fisher–Rao metric [4]. For a parametric family 
, the metric is
In
the field‑theoretic setting, the “parameters” are the values of the field at each point, and the metric on the infinite‑dimensional
space of distributions reduces to a local kinetic term. With 
, we have 
, and the gradient of the log‑probability becomes the
gradient of the entropic field. The Fisher information density therefore gives
rise to the simplest scalar built from derivatives of :
where 
is another positive constant with dimensions 
. This is precisely the standard kinetic term for a scalar
field, but its origin is entirely information‑geometric.
3.2
Interpretation
In
conventional field theory, a kinetic term 
is posited to give the field inertia. Here, it arises
from the Fisher information associated with changes in the probability density.
Distinguishability of nearby field configurations is measured by the Fisher–Rao
metric; the physical action is simply the “energy” of that distinguishability.
Thus, the kinetic term is not added by hand but is forced by the
geometry of information once entropy is promoted to a field.
4.
The Proto‑Obidi Action
Assembling
the potential term (4) and the kinetic term (5), we obtain the minimal action
functional that encodes both the static and dynamic aspects of the entropic
field:
We
call this the proto‑Obidi Action. It is the simplest entropic
action capable of producing nontrivial dynamics, and it already contains the
entire structural skeleton of the full Obidi Action.
Action
(6) is invariant under general coordinate transformations and, when the metric
is held fixed, under global shifts 
provided a boundary term is adjusted—a symmetry
reflecting the underlying scale invariance of Shannon entropy up to
normalization.
At
this stage the metric is a fixed background. The proto‑action governs how
the entropic field evolves on that background. The full theory will
eventually make itself a dynamical consequence of the entropic field,
but for the derivation of the field equation we treat as external.
5.
Derivation of the Master Entropic Equation
5.1
Variational principle
The
dynamics of follow from the Euler–Lagrange equations for the
Lagrangian density
For
a scalar field on a curved background, the Euler–Lagrange equation reads
5.2
Variation
Compute
the necessary derivatives:
Inserting
into (8):
which
simplifies to
Rearranging
gives the Master Entropic Equation in its canonical form:
5.3
Properties of the Master Entropic Equation
Equation
(12) is a nonlinear, covariant, second‑order partial differential equation for
the entropic field. The right‑hand side is an entropic self‑coupling derived
entirely from the Shannon potential. The fixed point (the maximum Shannon entropy state) is a trivial
solution, 
, corresponding to a massless free field at the entropic
vacuum. Small fluctuations 
around this vacuum obey, to linear order,
with 
, indicating that the entropic vacuum is a stable, massive
phase—an entropic “mass gap” generated without any symmetry breaking.
The
nonlinearity of (12) encodes the self‑gravitation of the entropic field: the
more the field deviates from its vacuum, the stronger the local entropic
curvature, leading to the clustering that eventually appears as matter and
spacetime curvature in the emergent description.
6.
From Proto‑Obidi to the Full Obidi Action: Curvature Transfer
6.1
The need for curvature terms
The
proto‑action (6) treats the metric as a fixed background. However, in ToE, the metric of
physical spacetime is not primitive but emergent from the
information geometry of the entropic field. The Fisher–Rao metric on the space
of distributions induces a Riemannian metric on the underlying manifold through
the field gradients. The intrinsic curvature of this induced metric must become
part of the action to close the system dynamically.
6.2
Curvature transfer
The
process of curvature transfer [2, 5] is the core of the full
Obidi Action. The Fisher–Rao metric for the entropic field is
where
the angle brackets denote the local expectation with respect to the
distribution (i.e., a functional of ). The physical metric is identified, up to a conformal factor, with the
coarse‑grained Fisher–Rao metric:
Once
this identification is made, the curvature invariants of are no longer background structures but functionals
of . The full action must therefore include the
Einstein–Hilbert term (and possibly higher‑order curvature terms) to account
for the energy of this induced geometry.
The full
Obidi Action is then
where 
is the emergent gravitational coupling, 
is the Ricci scalar of , and 
contains terms like 
, entropic‑curvature couplings, and the Vuli‑Ndlela
entropic geodesic integral that enforces the No‑Rush Theorem [3]. The
constants 
are interrelated through the curvature‑transfer
mechanism, fixing the scales of mass, length, and gravitational interaction in
terms of a single entropic scale.
6.3
Proto‑action as the primitive core
The
proto‑Obidi Action (6) is the minimal, irreducible core of the full Obidi
Action (15). Every term in the full action is either directly derived from
Shannon entropy (the potential and kinetic terms) or from the geometric
implications of treating entropy as a field (the curvature terms). In the limit
where curvature transfer is weak and spacetime is nearly flat, the full Obidi
Action reduces to the proto‑action, confirming that the Shannon‑based action is
the foundational seed.
7.
Emergence of Spacetime and Einstein’s Equations
Varying
the full Obidi Action (15) with respect to the metric yields the emergent Einstein equations:
where
the stress‑energy tensor of the entropic field is
with
the ellipsis denoting contributions from higher‑order terms. These equations
are not postulated; they are derived from the entropic action.
General relativity thus emerges as a thermodynamic limit of the information‑geometric
dynamics of the entropic field.
The
Master Entropic Equation (12), when generalized to include the back‑reaction of
the emergent metric, becomes a coupled system:
This
system unifies the pre‑geometric dynamics of Being () with the geometric dynamics of Becoming (spacetime
curvature) under a single entropic principle.
8.
Discussion
The
derivation presented here establishes that the action of the universe
can be built from nothing more than Shannon entropy and the geometry it induces.
No additional physical constants, symmetries, or metaphysical principles are
required; the potential term is Shannon’s formula, the kinetic term is the
Fisher information metric, and the gravitational sector emerges from the
curvature of the Fisher–Rao manifold. This constitutes the most parsimonious
foundation for physics currently known.
The
proto‑Obidi Action is the seed. Its simplicity is its power: it generates a
nonlinear wave equation with a mass gap, stable vacuum, and clustering
behavior. When the identification is lifted from a mathematical convenience to an
ontological statement, the distinction between probability and physical field
evaporates. Information geometry ceases to be a mere branch of statistics and
becomes the geometry of spacetime itself.
Philosophically,
this derivation answers the “why” questions that standard physics leaves open:
Why is there a universe? Because entropy is real. Why does it change? Because
the entropic field seeks its maximal entropy configuration, but the No‑Rush
Theorem [3] prevents instantaneous relaxation, forcing graded Becoming. Why is
spacetime curved? Because the information geometry of the entropic field has
intrinsic curvature, which in the thermodynamic limit manifests as the
spacetime metric. The Obidi Action is the Lagrangian of Ontodynamics.
9.
Conclusion
We
have shown, in a rigorous and self‑contained manner, how the Obidi Action—the
universal dynamical principle of the Theory of Entropicity—emerges directly
from Shannon entropy via information geometry. The chain of reasoning is:
- Shannon
entropy for a continuous distribution is a scalar functional of .
- Promoting to an ontological field converts Shannon entropy
into a potential term .
- The
Fisher information metric associated with supplies the kinetic term
.
- The
resulting proto‑Obidi Action
yields, upon variation, the Master Entropic
Equation 
.
- Including
the curvature invariants of the induced Fisher–Rao metric completes the
action to the full Obidi Action, from which Einstein’s equations emerge
through curvature transfer.
This
derivation unifies information, geometry, and dynamics, and places Shannon’s
1948 formula at the very core of physical law. The Obidi Action is not merely
analogous to an information‑theoretic functional—it is the
action of the universe, written in the language of entropy. Future Letters will
extend this framework to include gauge interactions, quantum statistics, and
cosmological solutions, all flowing from the same entropic foundation.
Acknowledgments
The
author acknowledges the foundational insights of C. E. Shannon and R. A.
Fisher, whose mathematical discoveries lie at the heart of physical law. This
work is part of the ToE Living Review Letters Series (ToE LRLS), Letters I–IIF.
References
[1]
J. O. Obidi, Entropicity: The Ontology of Entropy, ToE Living
Review Letters I (2024).
[2] J. O. Obidi, Curvature Transfer and the Emergence of Spacetime,
ToE Living Review Letters IIB (2025).
[3] J. O. Obidi, The No‑Rush Theorem: Why the Universe Cannot Change
Instantaneously, ToE Living Review Letters IIC (2025).
[4] S. Amari, Information Geometry and Its Applications, Springer
(2016).
[5] J. O. Obidi, The Obidi Action and the Master Entropic Equation,
ToE Living Review Letters IIA (2025).
[6] C. E. Shannon, A Mathematical Theory of Communication, Bell
Syst. Tech. J. 27, 379 (1948).
[7] R. A. Fisher, Theory of Statistical Estimation, Proc. Camb.
Phil. Soc. 22, 700 (1925).
[8] A. Einstein, Die Grundlage der allgemeinen Relativitätstheorie,
Ann. Phys. 49, 769 (1916).
