What are the Key Postulates of the Theory of Entropicity (ToE)? A Beautiful and Concise Introduction to the Foundational Postulates of the Theory of Entropicity (ToE)
Here we provide a really concise, yet beautiful, list of the core postulates that define the Theory of Entropicity (ToE) as laid out in its foundational papers.[1][2][4]
## Ontological postulate: entropy as the substrate
1. Entropy is the fundamental physical field
- Entropy $$S(x)$$ is not a statistical by‑product but a continuous, dynamical field that is the **causal substrate** of reality.[1][4]
- Gradients and flows of this entropic field generate motion, gravitation, time, and information flow.[1][4]
2. Physical entities are emergent from entropy
- Mass, energy, spacetime geometry, and even consciousness arise as emergent constraints or patterns of a single entropic reality, not as independent primitives.[4]
## Dynamical postulate: Obidi Action and master equations
3. The Obidi Action governs entropic dynamics
- There exists an entropic action functional (the Obidi Action) for the entropy field; extremizing this action yields the fundamental equations of motion.[1][4]
- From this action follow:
- the Master Entropic Equation (MEE),
- Entropic Geodesics (entropy‑determined paths), and
- the Entropy Potential Equation.[1][4]
4. Information geometry underlies physical law
- The dynamical arena is an informational–entropic manifold, equipped with Fisher–Rao and Fubini–Study metrics unified via the Amari–Čencov $$\alpha$$-connection.[1][4]
- Generalized entropies (Rényi, Tsallis) correspond to deformations of this geometry, with an entropic order parameter $$\alpha$$ acting as a universal deformation index.[4]
## Relativistic postulate: speed of light as entropic rate
5. $$c$$ is the maximum rate of entropic rearrangement
- The relativistic speed of light is not a primitive postulate; it is the characteristic propagation speed of disturbances in the universal entropic field.[1][2]
- All causal processes are limited by this finite rate of entropy propagation, which yields Lorentz invariance as an entropic necessity.[2]
6. Relativistic effects are entropic resistances
- Mass increase, time dilation, and length contraction arise from how the entropic field redistributes “entropic budget” between motion and timekeeping.[1][6]
- The Entropic Resistance Principle, the Entropic Resistance Field, and the Entropic Accounting Principle encode this redistribution and produce an entropic Lorentz factor that reproduces Einstein’s transformations without assuming geometric postulates.[1][6]
## Causality and arrow‑of‑time postulate
7. No‑Rush Theorem and causal bounds
- There is a universal lower bound on causal intervals: the entropic field must first establish conditions before any interaction or information transfer can occur.[1][2]
- This forbids superluminal processes and ties causality directly to the dynamics of the entropy field.[2]
8. Fundamental irreversibility via the Vuli–Ndlela Integral
- Quantum evolution is governed by an entropy‑weighted path integral (the Vuli–Ndlela Integral), a deformation of Feynman’s path integral.[1][4]
- This weighting introduces intrinsic irreversibility and a built‑in arrow of time at the fundamental level, rather than treating irreversibility as merely statistical.[1][4]
## Unification postulate
9. Thermodynamics, relativity, and quantum theory form a single entropic continuum
- When expressed in the entropic–informational language of ToE, Einstein’s field equations appear as a limiting, geometric case of the more general entropic dynamics.[2][4]
- Other “gravity from entropy” approaches (e.g., Bianconi‑style models) are recovered as special instances within this broader entropic framework.[4]
In our next exposition, we shall undertake to wade through the sophisticated mathematical foundations of the above the above Postulates.
Mathematical Endnotes on the Postulates of the Theory of Entropicity (ToE): The Main Postulate and the Associated Auxiliary Postulates
Postulate 1 (Entropic substrate)
There exists a real scalar field $$S(x^\mu)$$, the entropic field, defined on a 4‑dimensional differentiable manifold $$\mathcal{M}$$, such that all physical phenomena are determined by the configuration and evolution of $$S(x^\mu)$$.
Postulate 2 (Emergence of physical observables)
Conventional physical observables—mass $$m$$, energy $$E$$, momentum $$p^\mu$$, and spacetime metric $$g_{\mu\nu}$$—are emergent functionals of the entropic field and its derivatives,
$$
m = m[S],\quad E = E[S],\quad p^\mu = p^\mu[S],\quad g_{\mu\nu} = g_{\mu\nu}[S].
$$
Postulate 3 (Entropic action principle)
The dynamics of $$S(x^\mu)$$ are obtained from a variational principle on an entropic action [this is the potent and elegant Obidi Action of the Theory of Entropicity (ToE), from which the Local Obidi Action (LOA) and the Spectral Obidi Action (SOA) both arise]:
$$
\mathcal{A}_E[S] = \int_{\mathcal{M}} \mathcal{L}_E\big(S,\partial_\mu S,g_{\mu\nu}\big)\,\sqrt{-g}\,d^4x,
$$
where the physical evolution satisfies
$$
\delta \mathcal{A}_E = 0,
$$
yielding a Master Entropic Equation— MEE of the form
$$
\mathcal{E}[S,g_{\mu\nu}] = 0.
$$
This set of equations constitutes the beautiful Obidi Field Equations (OFE) of the Theory of Entropicity (ToE).
Postulate 4 (Entropic geodesics)
The physical trajectory $$\gamma$$ of any system between two macrostates corresponds to an extremal of an entropic length functional
$$
\mathcal{L}[\gamma] = \int_{\gamma} \sqrt{G_{ij}(\theta)\,d\theta^i d\theta^j},
$$
where $$\theta^i$$ are informational/entropic coordinates and $$G_{ij}$$ is an information‑geometric metric derived from $$S$$. Entropic geodesics satisfy
$$
\frac{d^2\theta^k}{d\lambda^2} + \Gamma^{k}{}_{ij}\frac{d\theta^i}{d\lambda}\frac{d\theta^j}{d\lambda} = 0,
$$
with $$\Gamma^{k}{}_{ij}$$ an $$\alpha$$-connection on the underlying information manifold.
Postulate 5 (Information geometry and generalized entropy)
The state space $$\mathcal{P}$$ of probability distributions $$p(x)$$ is an information manifold endowed with a metric $$g_{ij}$$ and an $$\alpha$$-connection $$\nabla^{(\alpha)}$$ such that
$$
g_{ij} = \frac{\partial^2}{\partial\theta^i\partial\theta^j} \mathcal{S}_\alpha(p),
$$
where $$\mathcal{S}_\alpha$$ is a generalized entropy (e.g., Rényi/Tsallis) and $$\alpha$$ is an entropic order parameter that deforms the geometry.
Postulate 6 (Maximum entropic rearrangement rate)
There exists a universal constant $$c$$ such that for any physical process the norm of the 4‑current of entropic change $$J_S^\mu$$ satisfies
$$
J_S^\mu J_{S\mu} \leq c^2 J_S^0 J_S^0,
$$
i.e., $$c$$ is the maximum propagation speed of disturbances in the entropic field, and thus the maximal speed for any causal influence.
Postulate 7 (Entropic origin of relativistic effects)
For a system of rest entropic capacity $$S_0$$ moving with velocity $$v$$, the effective temporal and spatial entropic budgets satisfy
$$
S_0 = S_t(v) + S_x(v),
$$
with
$$
S_t(v) = \gamma(v)\,S_0,\quad S_x(v) = \big(\gamma(v)-1\big)S_0,
$$
where the entropic Lorentz factor $$\gamma(v)$$ is therefore inevitably derived to be:
$$
\gamma(v) = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}},
$$
which is exactly in alignment with Einstein's Relativistic Kinematics, but arrived at here without any recourse or resort to the geometric postulates or axiom of the Theory of Relativity (ToR).
Time dilation, length contraction, and mass increase are thus manifestations of the entropic redistribution encoded in $$\gamma(v)$$.
Postulate 8 (Causality and No‑Rush bound)
Between any two causally related events $$A,B \in \mathcal{M}$$, the entropic field must establish the necessary configuration before interaction, imposing a minimal causal interval $$\Delta \tau_{\min} > 0$$ such that
$$
\Delta \tau(A,B) \geq \Delta \tau_{\min},
$$
and no process can transmit entropic, energetic, or informational influence with effective speed exceeding $$c$$. This is the ToE formulation of the Entropic Cone (EC) of causality, analogous to Einstein's Light Cone of his beautiful and elegant Theory of Relativity (ToR).
Postulate 9 (Entropic path integral and irreversibility)
Quantum evolution is governed by an entropy‑weighted path integral [this is the potent Vuli-Ndlela Integral of the Theory of Entropicity (ToE)]: the amplitude for a transition between configurations $$S_i$$ and $$S_f$$ is
$$
\mathcal{Z}(S_f,S_i) = \int \mathcal{D}S\,\exp\!\left[\frac{i}{\hbar}\mathcal{A}_E[S] - \Lambda\,\Delta \mathcal{S}[S]\right],
$$
where $$\Delta \mathcal{S}[S]$$ is the net entropy production along a path and $$\Lambda > 0$$ is an entropic deformation parameter. The nonzero $$\Lambda$$ term breaks microscopic time‑reversal symmetry and implements a fundamental arrow of time.
Postulate 10 (Entropic unification of gravitation)
There exists a functional $$\mathcal{G}[S,g_{\mu\nu}]$$ such that the effective spacetime dynamics satisfy
$$
\mathcal{G}[S,g_{\mu\nu}] = 8\pi G\,T_{\mu\nu}[S],
$$
where $$T_{\mu\nu}[S]$$ is the emergent stress–energy tensor derived from the entropic field. In an appropriate limit of weak entropic deformation and near‑equilibrium conditions, $$\mathcal{G}[S,g_{\mu\nu}] \to G_{\mu\nu}$$, recovering Einstein’s field equations
$$
G_{\mu\nu} = 8\pi G\,T_{\mu\nu}.
$$
Citations:
[1] The Theory of Entropicity (ToE) Derives and Explains Mass Increase ... https://client.prod.orp.cambridge.org/engage/coe/article-details/6900d89c113cc7cfff94ef3a
[2] The Theory of Entropicity (ToE) Derives Einstein's ... https://www.cambridge.org/engage/coe/article-details/6908aca0113cc7cfffd949e3
[3] Entropy - Wikipedia https://en.wikipedia.org/wiki/Entropy
[4] On the Conceptual and Mathematical Foundations of ... https://client.prod.orp.cambridge.org/engage/coe/article-details/68ea8b61bc2ac3a0e07a6f2c
[5] Entropy (information theory) - Wikipedia https://en.wikipedia.org/wiki/Entropy_(information_theory)
[6] The Theory of Entropicity (ToE) Derives and Explains Mass ... https://www.authorea.com/users/896400/articles/1351230-the-theory-of-entropicity-toe-derives-and-explains-mass-increase-time-dilation-and-length-contraction-in-einstein-s-theory-of-relativity-tor-toe-applies-logical-entropic-concepts-and-principles-to-verify-einstein-s-relativity
[7] The Theory of Entropicity (ToE) Derives and Explains Mass ... https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5673430
[8] Three Postulates to Understand Entropic Gravity https://www.scienceforums.com/topic/38749-three-postulates-to-understand-entropic-gravity/
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