On the Theory of Entropicity (ToE) and the Obidi Curvature Invariant (OCI) of ln 2 and its Global Implications in Modern Theoretical Physics: Quantum Measurements, Wavefunction Collapse, Quantum Entanglement, Quantum Entanglement Formation Time, Quantum Speed, Mandelstam–Tamm (MT) and Margolus–Levitin (ML) Bounds, Szilard Engine Model, Schrödinger’s Cat, and Wigner’s Friend (Canonical Version)
Abstract
The Theory of Entropicity (ToE) proposes a radical but coherent unification of physics by positing entropy as a universal field \( S(x) \) whose curvature, gradients, and dynamics generate spacetime, matter, information, and physical law. Central to this framework is the Obidi Curvature Invariant (OCI), the constant
\[
\ln 2,
\]
which represents the minimal entropic curvature divergence required for the universe to register a physically distinguishable event, state, or bit of information. This paper develops the mathematical and conceptual foundations of the OCI and demonstrates how this single invariant resolves longstanding paradoxes and limitations in quantum mechanics, relativity, information theory, and thermodynamics. We show that quantum measurement, wavefunction collapse, entanglement, entanglement formation time, quantum speed limits (MT and ML), the Szilard engine, Schrödinger’s Cat, and Wigner’s Friend all emerge naturally from the entropic geometry defined by the ToE. The result is a unified explanatory framework in which the discrete and continuous aspects of physics arise from a single entropic substrate.
1. Introduction
Modern physics remains divided between quantum mechanics, general relativity, and information theory, each with its own mathematical structures and conceptual foundations. The Theory of Entropicity (ToE) proposes a unifying ontology: entropy is the fundamental field, and all physical phenomena arise from its geometry.
In this framework:
- spacetime is induced by the entropic metric \( g(S) \),
- matter is encoded in the entropic stress–energy tensor \( T_{\mu\nu}(S) \),
- dynamics follow the Entropic Least‑Resistance Principle, and
- distinguishability is quantized by the Obidi Curvature Invariant
\[
\text{OCI} = \ln 2.
\]
The OCI is the smallest nonzero curvature divergence the entropic field can sustain. It is the “pixel size” of physical distinguishability—not in space, but in entropic curvature. This paper shows how this single invariant explains and resolves a wide range of open problems in theoretical physics.
2. The Obidi Curvature Invariant (OCI)
2.1 Definition
The Obidi Curvature Invariant is defined as:
\[
\text{OCI} = \ln 2.
\]
It represents the minimal entropic curvature required for the universe to distinguish two states. Any entropic divergence smaller than \( \ln 2 \) is sub‑threshold, physically indistinguishable, and cannot produce a measurable event.
2.2 Derivation from Relative Entropy
Given two entropic configurations \( \rhoA(x) \) and \( \rhoB(x) \), the entropic curvature divergence is:
\[
D(\rhoA \,\|\, \rhoB)
= \int \rhoA(x)\, \ln\!\left(\frac{\rhoA(x)}{\rho_B(x)}\right)\, dV.
\]
The minimal nontrivial distinguishable case occurs when:
\[
\rhoB(x) = 2\,\rhoA(x),
\]
yielding:
\[
D(\rhoA \,\|\, \rhoB) = \ln 2.
\]
Thus, \( \ln 2 \) is the smallest curvature difference that produces a physically distinct state.
3. The Singular Axiom of ToE
The entire theory rests on one axiom:
> Entropy is a universal physical field whose curvature generates spacetime, matter, information, and dynamics.
From this axiom follow:
- the entropic metric \( g(S) \),
- the Obidi Action Principle,
- the Master Entropic Field Equation (OFE),
- the entropic stress–energy tensor,
- and the OCI.
This single axiom replaces the dualistic foundations of modern physics.
4. Quantum Measurement and Wavefunction Collapse
4.1 Collapse as an Entropic Threshold Crossing
In ToE, a quantum state collapses when the entropic curvature between alternatives exceeds the OCI:
\[
\Delta S \ge \ln 2.
\]
Before this threshold, superpositions persist because the entropic field cannot distinguish the alternatives.
4.2 Measurement as Entropic Amplification
A measurement device amplifies microscopic entropic differences until they exceed \( \ln 2 \), forcing the system into a distinct entropic extremum.
Thus, collapse is not mysterious—it is a geometric inevitability.
5. Quantum Entanglement
5.1 Entanglement as Sub‑Threshold Curvature
Two systems are entangled when their entropic separation satisfies:
\[
\Delta S < \ln 2.
\]
They share a single entropic configuration and cannot be treated as independent.
5.2 Decoherence as Threshold Crossing
Decoherence occurs when:
\[
\Delta S = \ln 2.
\]
This explains why entanglement is fragile and why classicality emerges.
6. Entanglement Formation Time and Quantum Speed
6.1 Minimal Time to Create Entanglement
To create entanglement, the entropic curvature must be reduced below \( \ln 2 \). The minimal time is therefore:
\[
\tau_{\text{ent}} \propto \frac{\ln 2}{\dot{S}},
\]
where \( \dot{S} \) is the entropic rate of change.
6.2 Quantum Speed Limits (MT and ML)
The Mandelstam–Tamm and Margolus–Levitin bounds are unified by the OCI.
MT bound:
\[
\tau \ge \frac{\hbar}{2\Delta E}.
\]
ML bound:
\[
\tau \ge \frac{\hbar}{2E}.
\]
ToE interprets both as entropic curvature constraints:
\[
\tau{\min} = \frac{\ln 2}{\dot{S}{\max}}.
\]
Quantum evolution cannot proceed faster than the entropic field can accumulate \( \ln 2 \) curvature.
7. Szilard Engine and Landauer’s Principle
The Szilard engine requires one bit of information to extract work. In ToE:
- one bit = curvature \( \ln 2 \),
- erasing one bit requires flattening curvature \( \ln 2 \),
- the energy cost is:
\[
k_B T \ln 2.
\]
Landauer’s principle becomes a geometric necessity, not a statistical rule.
8. Schrödinger’s Cat and Wigner’s Friend
8.1 Schrödinger’s Cat
The cat remains in superposition as long as the entropic curvature between “alive” and “dead” branches is:
\[
\Delta S < \ln 2.
\]
The macroscopic environment rapidly amplifies curvature beyond \( \ln 2 \), forcing collapse.
8.2 Wigner’s Friend
The friend’s observation creates an entropic curvature \( \ge \ln 2 \) in their local frame.
Wigner’s external frame does not share this curvature until interaction occurs.
Thus, the paradox dissolves:
entropic curvature is frame‑dependent until interaction synchronizes it.
9. Global Implications for Modern Physics
The OCI provides a unifying explanation for:
- discreteness in quantum mechanics,
- continuity in relativity,
- the emergence of classicality,
- the limits of computation,
- the speed of quantum evolution,
- the nature of measurement,
- the structure of information,
- and the geometry of spacetime.
Everything becomes a manifestation of entropic curvature.
10. Conclusion
The Obidi Curvature Invariant \( \ln 2 \) is not merely a constant—it is the fundamental quantum of distinguishability in the universe. By positing entropy as a universal field, the Theory of Entropicity provides a coherent, mathematically grounded explanation for quantum measurement, collapse, entanglement, quantum speed limits, thermodynamic information bounds, and foundational paradoxes such as Schrödinger’s Cat and Wigner’s Friend.
The result is a unified theoretical framework in which the discrete and continuous aspects of physics arise from a single entropic substrate, governed by a single invariant, and articulated through a single axiom.
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