Far‑Reaching Implications of the Entropic Cone (EC) in the Theory of Entropicity (ToE)

The concept of the Entropic Cone (EC) is a central structural element in the Theory of Entropicity (ToE), formulated by John Onimisi Obidi. It generalizes and subsumes the traditional light cone (LC) of relativity by embedding causal structure within a deeper, entropy‑based framework. In this formulation, the fundamental causal boundaries of the universe are determined not merely by geometric constraints on spacetime, but by the propagation and reconfiguration properties of a fundamental entropy field \( S(x) \). The implications of this construction extend across relativity, quantum mechanics, causality, and the very status of spacetime as an emergent rather than primitive entity.

1. Reformulation of causality as entropic causality

In General Relativity (GR), causal structure is encoded by the light cone (LC), which restricts physical influence to propagate at or below the invariant speed of light. The Theory of Entropicity (ToE) refines this picture by introducing the notion of entropic causality, in which physical influence is mediated by changes in the entropy field \( S(x) \). Within this framework, the Entropic Cone (EC) defines the domain in which entropic reconfigurations can propagate and become physically effective.

An event is regarded as physically realized only when its entropic signature has propagated through the entropy field to the relevant region or observer, subject to the constraints imposed by the Entropic Cone (EC). No influence, signal, or measurable effect can arise outside this cone. In this sense, the familiar relativistic prohibition of superluminal signaling is reinterpreted as a deeper statement: causality is fundamentally thermodynamic rather than purely geometric. The causal order of events is determined by the allowed patterns of entropic propagation and reconfiguration in \( S(x) \).

2. The light cone (LC) as a special case of the Entropic Cone (EC)

The light cone (LC) of relativity appears in the Theory of Entropicity (ToE) as a special, limiting case of the more general Entropic Cone (EC). This occurs when the entropic metric \( G_{\alpha \mu\nu}(S) \), which encodes the information‑geometric and entropic structure of the field, becomes proportional to the usual spacetime metric \( g_{\mu\nu} \). A representative condition is

\[ G_{\alpha \mu\nu}(S) \propto g_{\mu\nu}, \]

which typically arises in regimes where generalized entropies (such as Tsallis or Rényi entropies) reduce to the standard Shannon entropy, for example in the limit \( \alpha \to 1 \). In this situation, the usual relativistic causal condition

\[ g_{\mu\nu} v^\mu v^\nu \leq 0 \]

is recovered as a special case of the more general entropic condition

\[ G_{\alpha \mu\nu}(S) v^\mu v^\nu \leq 0. \]

This demonstrates that the familiar spacetime geometry and its associated light cone (LC) structure are not fundamental in the ToE framework, but rather emergent from the underlying entropic geometry. The Entropic Cone (EC) is primary; the light cone (LC) is a derived, equilibrium‑like limit of this more general entropic causal structure.

3. Origin of the speed of light limit in the Theory of Entropicity (ToE)

Within the Theory of Entropicity (ToE), the invariance of the speed of light \( c \) is not postulated as an axiom but is derived as a consequence of the dynamics of the entropy field. The Entropic Cone (EC) encodes a maximum rate at which entropic reconfigurations can propagate through the field. This maximal propagation speed is identified with the observed constant \( c \).

In this view, electromagnetic waves (light) propagate at speed \( c \) because they are constrained by the same entropic propagation limit that governs all physical processes. The speed of light is therefore interpreted as the maximum speed of entropic propagation, and the relativistic invariance of \( c \) acquires a thermodynamic justification: it reflects the fundamental bound on how rapidly the entropy field \( S(x) \) can reorganize and transmit information.

4. Non‑coincidence of Light Cone (LC) and Entropic Cone (EC) away from equilibrium

In non‑equilibrium regimes, or in domains where non‑extensive entropies dominate, the Entropic Cone (EC) generally does not coincide with the standard relativistic light cone (LC). The entropic metric \( G_{\alpha \mu\nu}(S) \) can deviate significantly from any simple proportionality to \( g_{\mu\nu} \), leading to modified causal and kinematic structures.

Such deviations can manifest as modified dispersion relations, direction‑dependent causal boundaries, and potential breakdown or deformation of Lorentz symmetry in regions of strong entropic gradients. These effects are, in principle, empirically testable and may become relevant in high‑energy astrophysical environments, early‑universe cosmology, or quantum gravity regimes, where the entropic structure of the field is far from equilibrium.

5. Quantum measurement and entanglement within the Entropic Cone (EC)

The Entropic Cone (EC) also plays a central role in the quantum domain within the Theory of Entropicity (ToE). Quantum processes are reinterpreted as constrained entropic reconfigurations rather than instantaneous, nonlocal events. In particular, wavefunction collapse is modeled as a finite‑time process governed by an Entropic Time Limit (ETL), reflecting the finite speed at which entropic information can propagate.

Quantum entanglement is likewise constrained by the Entropic Cone (EC). Correlations between entangled systems are understood as arising from a shared entropic configuration whose reorganization must respect the entropic causal structure. This allows the theory to account for strong quantum correlations without violating causality: no usable signal or influence propagates outside the Entropic Cone (EC), even though entangled correlations may appear nonlocal in conventional spacetime terms.

Measurement outcomes are therefore determined not by instantaneous collapse across spacelike separations, but by the propagation of entropic information within the allowed cone. Within this framework, the realization of any quantum event is governed by the finite-speed reconfiguration of the entropy field, rather than by abrupt, non-dynamical state-vector updates. This perspective provides a unified route toward resolving long‑standing quantum paradoxes, including Schrödinger’s Cat and Wigner’s Friend. In the Theory of Entropicity (ToE), these two thought experiments are not distinct puzzles but manifestations of the same entropic mechanism: the “cat” and the “friend” occupy the same entropic role because the observer–system boundary is defined entirely by whether the entropic signature of the event has entered the observer’s Entropic Cone. Thus, ToE concludes that Schrödinger’s Cat is, in fact, Wigner’s Friend—both are simply entropic observers awaiting the arrival of the same entropic information. The apparent paradox dissolves once the propagation of entropic information, rather than instantaneous collapse, is recognized as the fundamental determinant of when and how quantum outcomes become physically realized.

6. Spacetime as an emergent structure in the Theory of Entropicity (ToE)

A key ontological claim of the Theory of Entropicity (ToE) is that spacetime is not fundamental. Instead, spacetime geometry is treated as an emergent construct arising from the curvature, flow, and information‑geometric structure of the entropy field \( S(x) \). The Entropic Cone (EC) is then the primary causal object, defining the admissible patterns of influence and correlation prior to any specific metric representation.

Observers and physical systems are said to “live inside” the Entropic Cone (EC), in the sense that their possible histories and futures are constrained by the entropic causal structure. The familiar spacetime manifold, with its metric \( g_{\mu\nu} \), arises as a coarse‑grained or effective description of this deeper entropic substrate. This viewpoint aligns with, but also extends, various holographic and thermodynamic gravity approaches by making entropy itself the primary ontological field rather than a derived quantity.

7. Operational and philosophical implications of the Entropic Cone (EC)

The Entropic Cone (EC) carries significant operational and philosophical implications. From an operational standpoint, what is regarded as “real” for a given observer depends on whether the entropic signature of an event has entered that observer’s Entropic Cone (EC). Reality is thus constrained by entropic accessibility rather than by abstract simultaneity surfaces.

The arrow of time is naturally encoded in this framework through the unidirectional flow and increase of entropy. Temporal asymmetry arises from the irreversible evolution of the entropy field and the associated expansion of accessible entropic configurations, rather than from ad hoc boundary conditions. At the same time, quantum mechanics, relativity, and thermodynamics are unified as different manifestations of a single entropic dynamics.

Conceptually, the Theory of Entropicity (ToE) thereby reframes physics as an entropic accounting mechanism (EAM), in which every interaction, process, or transformation incurs an Entropic Cost determined by the structure of the Entropic Cone (EC) and the finite speed of entropic propagation. The Entropic Cone (EC) thus functions not only as a generalized causal boundary but also as the organizing principle for understanding efficiency limits, irreversibility, and the fundamental constraints on information flow in the universe.

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