The Entropic Invariant (EI) and the Structure of the Entropic Cone (EC) of the Theory of Entropicity (ToE)

At the heart of the Theory of Entropicity (ToE) lies a remarkably powerful and conceptually elegant construct: the Entropic Cone. This cone is defined not by geometric postulates or assumptions about spacetime, but by the intrinsic dynamics of the entropy field \( S(x) \). The key insight is that the entropy field cannot reorganize itself arbitrarily fast; its propagation is constrained by a finite upper bound. This single constraint—the entropic propagation limit—contains within it the seeds of the entire relativistic structure. Once this limit is imposed, the mathematical and physical architecture of Einsteinian kinematics follows with remarkable inevitability.

The fundamental mathematical expression that encodes this causal and dynamical structure is the entropic invariant:

\[ (c_{e} s_{0})^{2} - (v s)^{2} = \text{constant}, \]

where \( c_{e} \) denotes the maximum entropic propagation speed, \( s_{0} \) is the rest‑entropy density, \( v \) is the velocity of the system, and \( s \) is the entropy density in motion. This invariant plays the same structural role in ToE that the Minkowski invariant plays in special relativity. It defines the boundary of the Entropic Cone, within which entropic influence can propagate and outside of which no entropic update is physically realizable.

Once one imposes that the entropy field \( S(x) \) cannot update outside its cone, the rest of the relativistic structure follows naturally. The Lorentz factor emerges directly from the entropic invariant:

\[ \gamma_{e} = \left(1 - \frac{v^{2}}{c_{e}^{2}}\right)^{-1/2}, \]

demonstrating that relativistic time dilation and length contraction are not geometric axioms but consequences of the finite rate at which the entropy field can reorganize itself. The velocity‑addition law likewise falls out of the entropic metric \( G_{\alpha\mu\nu}(S) \), which governs how entropic configurations transform under boosts. The familiar relativistic composition rule,

\[ \beta_{\text{tot}} = \frac{\beta_{1} + \beta_{2}}{1 + \beta_{1}\beta_{2}}, \]

is recovered as a direct consequence of the entropic transformation laws. In this way, the causal structure of relativity becomes a special case of entropic causality, obtained when the entropic metric reduces to the spacetime metric:

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

In this limit, the standard relativistic causal condition,

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

is recovered from the more general entropic condition,

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

The elegance of this derivation lies in its conceptual economy. Instead of assuming the geometry of spacetime, the invariance of the speed of light, or Lorentz symmetry, the Entropic Cone (EC) of the Theory of Entropicity (ToE) derives all of these from a single physical principle: the finite rate at which the entropy field can reorganize itself. The geometry of spacetime appears not as a primitive structure but as the physical consequence of the entropic dynamics. The causal boundary, the Lorentz factor, the velocity‑addition law (VAL), and the emergence of spacetime geometry all follow from the entropic invariant and the propagation limit encoded in the cone.

In this sense, the Entropic Cone (EC) provides a unified, physically grounded, and technically straightforward route to Einsteinian kinematics. It reveals that the relativistic structure of the universe is not an arbitrary geometric imposition but a manifestation of the deeper entropic substrate that governs all physical processes. The entropic invariant (EI) therefore stands as one of the most profound and structurally powerful components of the Theory of Entropicity (ToE), demonstrating how a single entropic constraint can generate the full machinery of relativistic physics.

Distinction Between the Information‑Theoretic Entropic Cone and the Entropic Cone of ToE

The term entropic cone appears in multiple scientific contexts, but it refers to mathematically and conceptually distinct objects in information theory and in the Theory of Entropicity (ToE). It is therefore essential to clearly distinguish between the information‑theoretic entropic cone studied in the literature on entropy vectors and information inequalities, and the Entropic Cone introduced in ToE as a causal structure generated by the dynamics of a fundamental entropy field. Although both constructions are rooted in entropy, they operate in different domains, serve different purposes, and possess fundamentally different mathematical structures.

1. The information‑theoretic entropic cone

In information theory, the entropic cone is defined as the closure of the set of all entropy vectors associated with collections of random variables. Given random variables \( X_{1}, \dots, X_{k} \), one considers the joint entropies of all nonempty subsets and arranges them into an entropy vector in a finite‑dimensional real vector space. The set of all such vectors, together with its closure, forms a closed convex cone known as the entropic cone. This object is constrained by Shannon inequalities and, for \( k \geq 4 \), by additional non‑Shannon inequalities.

The information‑theoretic entropic cone is therefore a static geometric object in a finite‑dimensional vector space. It encodes which combinations of entropies are realizable by random variables and which are forbidden. Its study involves convex geometry, linear inequalities, tropical probability theory, and asymptotic equivalence classes of diagrams of probability spaces. Crucially, this cone does not encode causal propagation, finite speeds, field dynamics, or spacetime structure. It is a tool for characterizing the space of entropy profiles, not a structure governing physical evolution.

2. The Entropic Cone in the Theory of Entropicity (ToE)

In the Theory of Entropicity (ToE), the Entropic Cone is a fundamentally different construct. It is a causal cone defined by the dynamics of a fundamental entropy field \( S(x) \) and by a finite entropic propagation limit \( c_{e} \). The central object is an entropic invariant of the form

\[ (c_{e} s_{0})^{2} - (v s)^{2} = \text{constant}, \]

where \( c_{e} \) is the maximum entropic propagation speed, \( s_{0} \) is a rest‑entropy density, \( v \) is a velocity, and \( s \) is an entropy density in motion. This invariant defines the boundary of the Entropic Cone, inside which entropic influence can propagate and outside of which no entropic update is physically realizable. The cone thus encodes a causal structure analogous to, but more fundamental than, the light cone of relativity.

From this entropic invariant, ToE derives an entropic Lorentz factor,

\[ \gamma_{e} = \left(1 - \frac{v^{2}}{c_{e}^{2}}\right)^{-1/2}, \]

and recovers Einsteinian kinematics—including the velocity‑addition law and the relativistic causal structure—as consequences of the finite rate at which the entropy field can reorganize itself. In this framework, spacetime geometry emerges as the shadow of entropic dynamics, and the usual relativistic light cone appears as a special case of the more general Entropic Cone when the entropic metric \( G_{\alpha\mu\nu}(S) \) reduces to the spacetime metric \( g_{\mu\nu} \).

3. Fundamental differences in structure and purpose

The information‑theoretic entropic cone and the Entropic Cone of ToE differ in both mathematical structure and physical interpretation. The former is a convex cone of entropy vectors in a finite‑dimensional vector space, used to study information inequalities and entropy optimization problems. It is static, non‑causal, and non‑dynamical. The latter is a causal cone defined by a finite entropic propagation speed and an entropic invariant, used to derive relativistic kinematics and to encode the causal structure of an underlying entropy field.

While both constructions involve entropy and both can be associated with geometric structures, they operate at different levels of description. The information‑theoretic entropic cone constrains which entropy profiles are possible for random variables; the ToE Entropic Cone constrains which entropic influences and physical events can be causally connected. The former is a tool of information geometry in a combinatorial setting; the latter is a tool of entropic field theory and causal dynamics.

4. Conceptual relation and independence

Conceptually, it is noteworthy that both information theory and ToE naturally give rise to cones built from entropy. This convergence suggests that entropy has an intrinsic tendency to induce geometric and convex structures. However, the Entropic Cone of ToE does not depend on, nor is it derived from, the information‑theoretic entropic cone. It is an independent construct, rooted in the postulate that entropy is a fundamental dynamical field with a finite propagation speed, and that causality and kinematics emerge from its dynamics.

The information‑theoretic entropic cone may provide conceptual inspiration and mathematical analogies, particularly in the use of convexity and information‑geometric methods, but it cannot serve as the causal or dynamical structure of ToE. The Entropic Cone in ToE remains a genuinely new and distinct entity: a causal entropic structure from which Einsteinian kinematics and spacetime geometry are derived as emergent phenomena.

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