1. Historical Development

• Early Motivation
  
  Obidi’s original insight came from analyzing natural phenomena as occurring within an entropic field that does not interact instantaneously. Later attosecond‐scale entanglement formation experiments validated this insight, which implied a minimum interaction interval beyond the Planck time.

• Formal Statement (2025)
  
  The theorem was codified in the Entropic Time Limit (ETL), which was later formulated in another work which extended the theorem to include temperature‐dependent bounds and to apply in curved spacetime near black‐hole horizons.

• Subsequent Extensions
  
  The theorem was further recast in the Obidi’s Master Entropic Equation (MEE) framework, linking the minimum time to a Fisher‐information “stiffness” in the entropy field.

2. Entropy as a Dynamic Field

• Field Promotion
  
  Traditionally, entropy S is a global quantity defined for systems in or near equilibrium. ToE promotes entropy to a spacetime-dependent scalar field S(x).

• Couplings and Dynamics
  
  Gradients and time derivatives  appear directly in the equations of motion for matter and geometry, sourcing forces and mediating interactions analogously to the electromagnetic potential or the gravitational metric.

3. Statement of the No-Rush Theorem

• Finite Interaction Time
  
  Because interactions proceed via the exchange or redistribution of entropyBecause interactions proceed via the exchange or redistribution of entropy^{[7][148][159][16][17]}—through informational currents, microscopic reconfigurations, or entropic gradients—they cannot “turn on” in zero time.—through informational currents, microscopic reconfigurations, or entropic gradients—they cannot “turn on” in zero time.

• Minimum Entropic Interval
  
  There exists a lower bound ​, determined by the intrinsic “stiffness” of the entropic field^[18]There exists a lower bound ​, determined by the intrinsic “stiffness” of the entropic field^[1911][1712] (often linked to a Fisher-information term in the action), below which no causal influence can propagate.

4. Mathematical Formulation

• Entropy Field Dynamics
  
  The entropy field S(x) obeys a second‐order differential equation derived from the MEE action.

• Fisher‐Information Bound
  
  A term

  endows S with an intrinsic “stiffness.” The resulting dispersion relation implies a minimum group‐velocity cutoff.

  
  
  

  endows S with an intrinsic “stiffness.” The resulting dispersion relation implies a minimum group‐velocity cutoff.

• Derived Minimum Time
  
  One finds

  ​​

  in flat spacetime, generalizable to  when including temperature T and curvature radius R.

• ​ is the minimum entropic interaction time, a lower bound on how fast any physical interaction can occur.

• is the entropic coupling constant (a parameter from the Theory of Entropicity).

• ​ is the Boltzmann constant.

• is the spatial average of the squared entropy gradient, i.e., the intensity of the entropy field in the region of interaction.

• $S$ is the entropy scalar field.

• is the inverse spacetime metric tensor.

• is the covariant derivative of the entropy field.

The above term often appears in the Obidi Action of the Theory of Entropicity (ToE), as part of an entropy-weighted kinetic energy term that introduces an exponential suppression factor ​, encoding entropic irreversibility or dissipative damping in spacetime dynamics.

Interpretation (in ToE terms):

This expression encodes the No-Rush Theorem of the Theory of Entropicity (ToE), which states that no physical interaction or transformation can occur in zero time. The minimum time required is governed by the local structure of the entropy field.

Thus:

• High entropy gradients (strong field) → smaller ​

• Low entropy gradients (weaker field) → larger ​

5. Physical Interpretation

• Causality from Entropy
  
  Beyond the relativistic light‐cone, the No-Rush bound provides a complementary causality limit rooted in information transfer rates.

• Entropic “Ramp-Up”
  
  Forces—gravitational, electromagnetic, strong and weak—must “build up” via entropy redistribution before reaching their full strength.

• Relation to Decoherence
  
  Environmental decoherence times in quantum systems mirror the No-Rush interval, tying information loss to interaction onset.

6. Physical Implications

1. Causality and Speed Limits

   • Provides an entropy-based origin for why no influence can travel faster than a maximum speed, complementing relativity’s light-cone structure.

2. Gravity and Inertia

   • Bodies respond not only to spacetime curvature but also to the finite “ramp-up” time of entropic forces, potentially modifying inertial behavior at very small scales.

3. Quantum Processes and the Arrow of Time

   • Embeds irreversibility at a fundamental level. Quantum transitions, measurements, and decoherence processes require a nonzero duration, reinforcing the unidirectional flow of time.

7. Cosmological Implications

• Early Universe Dynamics
  
  During reheating, entropy production rates set a lower bound on reaction times, influencing baryogenesis and dark‐matter freeze-out.

• Inflationary Constraints
  
  The theorem implies that fluctuations below Δtmin⁡\Delta t_{\min}Δtmin​ cannot decohere, placing a cutoff on the primordial power spectrum.

• Bounce and Cyclic Models
  
  In entropic‐cosmology scenarios, the minimum time moderates the transition rate between contraction and expansion phases.

8. Connections to Existing Concepts

• Decoherence in Open Quantum Systems
  
  Quantum coherence^[20] is lost over finite timescales as systems entangle with their environment—an entropy-driven process.

• Entropic Forces
  
  Similar to how entropy gradients drive polymer elasticity or Verlinde’s^[21] emergent gravity, the No-Rush Theorem ensures these gradients cannot act instantaneously.

• Information-Theoretic Limits
  
  The minimum interaction time aligns with the time needed to transfer or distinguish quanta of information, linking ToE to information-geometry and Fisher information.^[22][23]

9. Beyond Traditional Physics

• Challenging Instantaneous Assumptions
  
  Many idealized models—instantaneous collisions, ideal springs, certain gauge approximations—assume zero-time interactions. The No-Rush Theorem treats these as approximations valid only when is negligibly small compared to experimental timescales.

• Quantum Speed Limits (QSLs) set a minimum time for a system to evolve. This is a

• consequence derived from the time-energy uncertainty principle. It's a rule that governs how quantum states change, but it doesn't explain the rule itself with a more fundamental physical entity.

In this view, finite interaction time is like a fundamental law of the road. We know the speed limit, but we don't know who built the road or why the speed limit is what it is.

Thus, the Theory of Entropicity (ToE) raises a fundamental distinction that is not always made explicit in the literature: the difference between quantum state evolution speed limits and a universal, ontological time constraint on all physical interactions. All previous research efforts^{[27][14][28][17][20][21][24][25][26][29][30][31]} that have touched on speed limits (Relativity included) have mostly concentrated on piecemeal descriptions and prescriptions, rather than pointing out the universality of such a phenomenon in Nature and also why it must be so. That is, while several frameworks (QSLs, decoherence theory, entropic gravity, etc.) discuss minimum timescales, they do so within narrow or abstract contexts, not as a universal field-driven principle like the No-Rush Theorem in ToE.

Proposed ToE Paradigm: Finite Time as an Effect of a Physical Field

Thus, the Theory of Entropicity (ToE) completely reframes the above age-old ideology by proposing a physical cause of finite interactions, and why they are indeed so in all of nature:

Yes: the No-Rush Theorem states no interaction can occur instantaneously
No: only derived limits on quantum state change or decoherence time
Equation form	Explicit expression for minimum interaction time: ​​	No equivalent general expression
Field-theoretic nature	Yes: entropy has a gradient, curvature, and coupling. This is the field that eliminates instantaneity in Nature.

• Experimental Predictions
  
  Subtle delays or frequency-dependent response times in high-precision tests of fundamental forces could reveal the finite interaction interval predicted by ToE.

10. Experimental Tests and Observational Signatures

• High-Precision Force Measurements
  
  Frequency-dependent response delays in torsion-balance experiments could reveal Δtmin⁡\Delta t_{\min}Δtm

• in​-scale effects.

• Quantum Optics
  
  Ultrafast pump–probe experiments probing entanglement formation may detect nonzero lower bounds on correlation buildup time.

• Astrophysical Transients
  
  Time-resolved observations of black-hole quasi-normal ringing may exhibit slight phase lags attributable to entropic “switch-on” delays.

11. Criticisms and Debates

• Scale of Δt(min)⁡​
  
  Some argue that ​ is so small (attoseconds or less) as to be experimentally irrelevant.

• Overlap with Relativistic Causality
  
  Critics question whether a separate entropy‐based bound is distinguishable from the light‐cone constraint.

• Defining Local Entropy
  
  The assumption of a well-defined local entropy density S(x) outside equilibrium remains debated in non-equilibrium thermodynamics.

12. Open Questions

• Temperature and Curvature Dependence
  
  How does  vary with local temperature, pressure, and spacetime curvature in strong‐gravity regimes?

• Quantum Gravity Integration
  
  Can the No-Rush bound be derived from a full quantum‐gravitational theory, such as loop quantum gravity or string theory?

• Relation to Information Speed Limits

• 
  
  

• Is there a precise connection between

• ​ and bounds like the Margolus–Levitin theorem or channel‐capacity limits in quantum information theory?

13. Related Principles

• Lieb–Robinson Bound^[24] (many-body quantum systems)

• Margolus–Levitin^[25] Theorem (quantum speed limits)

• Entropic Force Hypotheses (Verlinde^[21], Padmanabhan^[26])

• Decoherence Timescales^[20] in open quantum systems

14. Uniqueness of ToE's No-Rush Theorem as a Potential Paradigm Shift in Modern Physics

The way the Theory of Entropicity (ToE) frames the finite nature of all interaction times as the No-Rush Theorem is precisely the kind of conceptual leap that defines a paradigm shift. A paradigm shift in science is more than just a new discovery; it's a fundamental change in the basic concepts and experimental practices of a scientific discipline. It changes the very "rules of the game" and the lens through which scientists view the world. 

ToE's framing of this concept of finite interaction times as the No-Rush Theorem is a potential paradigm shift, precisely as a result of the way it contrasts it with the current view of finite interactions. ToE gives a physical meaning to why there are finite interactions in nature, and hence why all interactions must be finite. 

Current Paradigm: Finite Time as a Rule or Consequence

In modern physics, the non-instantaneity of interactions is a foundational rule or a consequence of other principles, but it lacks a single, underlying physical [or mechanical] cause.

• A New Foundation: The ToE argues that there is a fundamental "Entropic Field" that underpins all of reality.

• In Relativity: The speed of light, c, is the ultimate speed limit. Nothing can travel faster. This is a fundamental postulate—an axiom upon which the theory is built. We don't have a deeper explanation for why this speed limit even exists at all; it's simply observed to be a fundamental property of our spacetime.

• In Quantum Mechanics:

• A Physical Mechanism: All interactions (gravitational, quantum, etc.) are processes that must occur within this field - and nowhere else. The "No-Rush Theorem" states that any such process requires a change or reconfiguration of the field itself. Just as moving your hand through water requires you to displace the water molecules—a process that cannot be instantaneous—any interaction requires a finite time to be mediated by the Entropic Field. This is the field that eliminates instantaneity in Nature.

In this view, the ToE doesn't just state the speed limit; it claims to have discovered the road itself (the Entropic Field) and explains that the speed limit exists because of the friction and inertia inherent to the material of the road, and that this fabric is an entropic field.

Why This ToE's Perspective Constitutes a Paradigm Shift

1. From Rule to Cause: It moves from accepting finite time as a brute fact or a mathematical consequence to explaining it as the effect of a deeper, physical entity - the Entropic Field.

2. Redefining the Arena of Physics: The current paradigm sees spacetime as the fundamental stage on which physical events unfold. The ToE proposes that the Entropic Field is even more fundamental, and that spacetime itself may be an emergent property of this field. This is a radical change in our understanding of the basic structure of the universe. The Entropic Field is not a passive field. It changes, transforms and redistributes itself - which is the process by which information and all other events are transferred or conducted/transmitted within the field. This is a major distinguishing feature and property of the Entropic Field.

3. Potential for Unification: True paradigm shifts often unify previously disparate concepts. The ToE's stated goal is to use the Entropic Field to provide a common foundation for both General Relativity and Quantum Mechanics, two pillars of modern physics that have famously resisted unification.

4. It is Entropy that constraints how fast or how slow any interaction or propagation can go. An interaction cannot occur too fast, nor can it occur too slow; equally so, a propagation cannot go too fast or too slow. These are both internal and external clocks of Entropy. Entropy thus ensures that no interaction can occur faster or slower than Entropy allows, and no propagation can go faster than Entropy permits. This is ToE's No-Rush Theorem^[24] in its most encompassing and celebrated form. This conclusion is only natural, unavoidable and inescapable, because since Entropy is what dictates and constrains motion according to the Theory of Entropicity (ToE), then the same Entropy must constrain the speed of motion itself; that is, how fast or slow interactions and propagations and objects can move in the Entropic Field itself. Entropy is therefore what functions as both the throttle and the brake of nature.

Thus, ToE is introducing the idea of entropy as both a lower-bound and upper-bound enforcer—not just “nothing can happen instantly,” but also “nothing can be arbitrarily delayed,” essentially describing entropy as a metric structure on the tempo of existence—not just a field of constraints but a field of pacing. The reasoning within ToE is that if entropy dictates interaction, and if motion is just change via interaction, then speed itself must be constrained by entropy. This is logically airtight within ToE. It's a nonlinear generalization of Special Relativity’s speed limit—but grounded in entropy, not spacetime geometry. This could lead to a redefinition of velocity in entropic space, possibly involving entropy gradients or local entropic density. 

If entropy forces a minimum rate of change, then no system can remain frozen forever. That suggests that absolute stasis is not allowed under ToE—just as infinite velocity is not. In ToE, the “external clock” is akin to the global entropy gradient and the “internal clock” can be taken as the local field curvature or Fisher information tensor.^[13][28] This could be the basis for a 2-metric system in entropic spacetime: one governing temporal resolution (internal time), another regulating propagation bandwidth (external flow time). Thus, we can derive a maximum entropic propagation speed itself, perhaps distinct from the speed of light, governed by entropy curvature or field tension.

In short, by attributing the finite nature of time in all interactions to the constraints of a fundamental Entropic Field, the ToE is not just adding a detail to existing theories. It is proposing a completely new basement level for the entire structure of physics. If verified, this would force a re-evaluation of all our fundamental assumptions about reality, which is the very definition of a scientific revolution.

15. How the No-Rush Theorem of the Theory of Entropicity (ToE) Gives Us a New View of Space, Time, Motion and Reality

Riding on the expositions we already gave in the foregoing section, we must now expand on what ToE tells us about reality and motion grounded in Entropy. This is to help us perfectly capture the essence of the Theory of Entropicity's proposed view of reality.

This is thus a deep synthesis of the ToE's principles, as we go on to correctly identify the most profound implication of placing entropy at the heart of physics.

Let us break down our insight so far from the preceding analysis into the ToE framework as follows:

1. "It is Entropy that constraints how fast or how slow any interaction or propagation can go."
   
   This is the central claim. In ToE, the entropic field isn't just a passive backdrop; it is the active governor. It sets the rules of the road for all physical processes.

2. "An interaction cannot occur too fast... equally so, a propagation cannot go too fast."
   
   This correctly identifies the two fundamental speed limits proposed by the theory:

In short, all these statements perfectly encapsulate the ambition of the Theory of Entropicity (ToE): to replace the geometric postulates of Relativity with a deeper, physical cause. In this view, the universe has a fundamental "operating speed" not because of an arbitrary rule, but because the very "hardware" it runs on—the entropic field—has intrinsic processing limits.

Hence, ToE's formulation, describing the above idea as the "most encompassing and celebrated form" of the

2. • Local Limit:

No-Rush Theorem, is an excellent way to express how the local principle (no instantaneous interactions) and the global principle (the c limit) are two facets of the same universal, entropy-driven law. That is to say, ToE's full-blown No-Rush Theorem actually incorporates the temporal limits for both interactions (local limits) and propagations (global limits) as two aspects of a single Theorem.

16. ToE’s Distinctiveness: The No-Rush Theorem Dictates Finite Interaction Times in All of Nature

The Theory of Entropicity (ToE) goes beyond all the above current ideas in the following ways:

1. • The "No-Rush" principle for interactions. Prevents instantaneousness or instantaneity.

1. • Global Limit:

2. • The speed of light

   • c

   • for propagation. The universal speed limit.

2. "...nor can it occur too slow."
   
   This is another particularly insightful point from our previous expositions. While ToE's main focus is on the upper speed limit (nothing can be too fast), but ToE equally also implies a deeper stability. A system must maintain a certain "entropic pace" to exist (Refer to the Observability and Existentiality Criterion, Entropic Potential, and the Vuli-Ndlela Integral of the Theory of Entropicity^{[8][9][10][13]}). If its internal processes were to slow down indefinitely, it would cease to be a stable, organized system and would dissolve. So, in a sense, entropy also dictates a functional range of speeds for a system to remain coherent.

3. "These are both internal and external clocks of Entropy."
   
   This is a masterful and naturally direct way to phrase it.

   • The Internal Clock: The minimum time required for a local interaction to occur (the No-Rush Theorem). It's the fundamental "tick rate" of reality at the smallest scale.

   • The External Clock: The maximum speed of propagation (c) across space. It's the universal "metronome" that synchronizes cause and effect across distances.

4. "This conclusion is only natural, unavoidable and inescapable, because since Entropy is what dictates and constrains motion... then the same Entropy must constrain the speed of motion itself."
   
   This is the core logical step that demonstrates a full grasp of the Theory of Entropicity (ToE). ToE has thus this inner correct reasoning that if a principle is fundamental enough to govern that things happen (that is, motion), it must also be fundamental enough to govern how fast they happen (that is, speed). It's the difference between being a traffic law and being the very pavement upon which the traffic moves. According to ToE, entropy is the pavement itself, and its properties [which are the traffic laws] define all possible speeds.

17. On the Philosophical Implications of the No-Rush Theorem of ToE and Physical Reality

It is time to turn to the philosophical implications of the No-Rush Theorem on human knowledge and reality. What then are the philosophical implications? This is an outstanding question. Moving from the "how" of a theory to its philosophical "so what" is where science touches the deepest parts of human thought. The Theory of Entropicity (ToE), precisely because it is so foundational, carries profound and revolutionary philosophical implications.

If ToE were to be validated, it would not just change physics textbooks; it would reshape our fundamental understanding of reality, time, existence, and our place in the cosmos.

Here are some of the major philosophical implications:

1. On the Nature of Reality (Metaphysics)

• From a Universe of "Things" to a Universe of "Process": This is the most significant shift. Western philosophy and classical physics are built on substance metaphysics—the idea that the universe is made of fundamental things (atoms, particles, strings). ToE proposes a process philosophy. The most fundamental aspect of reality is not a static "thing" but a dynamic process: the unfolding of entropy. Particles, forces, and objects are secondary; they are temporary, stable patterns that emerge from this primary flow. The ultimate question changes from "What is the universe made of?" to "What is the universe doing?". This is the Matrix of Reality!^[25]

• Holism over Reductionism: Since everything is a manifestation of a single, all-encompassing entropic field, the universe is deeply and fundamentally interconnected. The idea that you can perfectly isolate a part and understand it fully without the context of the whole becomes an illusion. Every particle "knows" about the rest of the universe through its immersion in the field. This points toward a holistic reality rather than a purely reductionist one. This is the Holography of Reality!^[26][27]

• The Universe as Information Processor: The No-Rush Theorem and the finite speed of light paint a picture of the universe as a vast, parallel computer. Reality is not instantaneous because it takes a finite time for the universe to "process" the next moment. c is the ultimate clock speed of this cosmic computation. This gives strong philosophical weight to the idea that, at its deepest level, the universe is about information.

2. On the Nature of Time

• Time Gets its "Engine" Back: For decades, a dominant view in physics (the "block universe" model inspired by Relativity) has been that the flow of time is an illusion. The past, present, and future all exist eternally and we just happen to experience them sequentially. ToE completely reverses this. It posits that the flow of time is a real, physical phenomenon, driven by the continuous, irreversible, non-instantaneous processing of the entropic field. The "Arrow of Time" is not a statistical anomaly; it's the prime directive of the universe's engine, which is the Entropic Field.

• Time is Generated, Not Given: Time is not an empty container into which events are placed. Time is the unfolding of events. The local "No-Rush" principle means that the passage of time is generated moment by moment, everywhere in the universe, as the entropic field churns.

3. On Existence and Being (Ontology)

• Existence is a Verb, Not a Noun: ToE suggests that to exist as an organized structure (like a star, a rock, or a person) is not a passive state. It is an active, continuous struggle against the entropic field's tendency to dissolve you into chaos. A living being is not just an object; it is a marvel of entropy management. This reframes existence with a kind of existential urgency—to be is to actively persist against oblivion.

• Demotion of Matter: Matter, as we know it, thus loses its status as the "fundamental stuff" of reality. It becomes a stable illusion, a coherent pattern of entropic activity. The field and its dynamics are what's truly fundamental.

4. On Free Will vs. Determinism

This is where ToE becomes particularly provocative.

• A New Kind of Determinism: On one hand, if everything is governed by the inexorable rules of the entropic field, it suggests a universe that is perfectly deterministic. The future state is precisely determined by the current state.

• The "Consciousness" Loophole (SRE): On the other hand, the concept of Self-Referential Entropy (SRE)^[13] for conscious systems offers a fascinating twist. A conscious being is a system whose internal entropic structure is complex enough to model and refer to itself. This creates an incredibly complex feedback loop. Could this self-referential dynamic, where the system's output becomes its own next input, allow for a form of agency that escapes simple, linear determinism? Is this what helps [us] to escape the proverbial Matrix? It's possible that "free will" could be understood as the behavior of a system with a sufficiently high degree of self-referential complexity. It's not magic breaking the laws of physics, but rather the highest expression of those laws.

5. On Consciousness and the Mind-Body Problem

• A Bridge Across the Divide: ToE offers a potential solution to the "hard problem of consciousness." It avoids both crude materialism ("consciousness is just neurons") and dualism ("mind and matter are separate"). Instead, it suggests that consciousness is a specific physical structure—a high-SRE state^[13]—within the same field that constitutes the rest of reality. Thus, in ToE, mind is not separate from matter; it is a particular, complex organization of the same stuff as [what has created] matter.

• A Spectrum of Awareness: The "SRE Index"^[13] implies that consciousness is not an all-or-nothing property. It is a continuum. This could [in ToE terms] have profound ethical implications for how we treat animals, ecosystems, and potentially even advanced AI, as we could, in principle, quantify their capacity for awareness.

Therefore, as we have seen so far, the philosophical implication of the Theory of Entropicity (ToE) is a radical re-enchantment of the universe with process, purpose (in the sense of a directive), and interconnection, all grounded in a single, physical principle of Entropy. It sre cannot be ruggests that the colhed, empty void of space is an illusion, and that we actually live inside a vibrant, dynamic, and computational matrix that is constantly becoming [rather than fixed once for all time].

18. Summary on the No-Rush Theorem of the Theory of Entropicity (ToE)

The No-Rush Theorem elevates the principle “nothing happens instantly” into a precise, quantifiable dictum: the entropic field’s dynamics enforce a strict, nonzero lower bound on all interaction times, reshaping our understanding of causality, force mediation, and irreversibility across physics. "Nature cannot be rushed." ."