The Second Exentation: The Logical Pulse of Reality

Within the ambitious framework of ArXe Theory, which interweaves logic and physics at the universe’s most fundamental scale, we’ve established that exentation (\( \neg C \)), defined as the negation of a contradiction, is the origin of a tautology (\( E \equiv T \)). This process, in turn, is directly equated with Planck Time (\( T_p \)), the most basic pulse of physical existence. This foundational principle not only defines what can exist but also establishes an essential framework of logical coherence for reality.

Now, we delve deeper into an implication of the theory that reinforces its internal consistency: the exentation of the coexistence between what does not exist and what does.

Recapping Fundamental Concepts

To understand this second exentation, let’s recall our key definitions in ArXe:

Istence (I): Represents the logical contradiction (\( I \equiv (S \land \neg S) \)). It is what is logically impossible and, therefore, does not physically exist. Istence is always false (F).
Ex-Istence or Existence (E): Represents the logical tautology (\( E \equiv (\neg S \lor S) \)). It is the result of the negation of a contradiction and, therefore, what does physically exist. Ex-Istence is always true (V).

Our central axiom posits that the process of exentation is fundamentally equivalent to one Planck Time:

\( \neg C \equiv T_p \)

The Impossibility of Coexistence: Tensia

Let’s now consider the conjunction of Istence and Ex-Istence: \( (I \land E) \). This expression hypothesizes that what is contradictory (and non-existent) could simultaneously coexist with what is tautological (and existent).

Logically, if we substitute the truth values:

\( (I \land E) \equiv (F \land V) \)

The result of a conjunction between a false value and a true value is always false:

\( (F \land V) \equiv F \)

This reveals that the conjunction \( (I \land E) \) is itself a logical contradiction. We call this new inherent contradiction, which represents impossible coexistence, Tensia.

\( Tensia \equiv (I \land E) \)

The implication is profound: the coexistence of non-existence with existence is, paradoxically, a logical impossibility.

The Second Exentation: Exentia, the Affirmation of Coherence

Given that Tensia (\( (I \land E) \)) is a contradiction, ArXe Theory postulates that it must also be “exentated” for the universe to maintain its fundamental coherence. Applying the concept of exentation to Tensia:

\( \neg Tensia \equiv \neg (I \land E) \)

Since Tensia is always false, its negation \( \neg (I \land E) \) will always be true:

\( \neg F \equiv V \)

Therefore, \( \neg (I \land E) \) is a tautology. We call this tautology, resulting from the negation of Tensia, Exentia.

\( Exentia \equiv \neg Tensia \equiv \neg (I \land E) \)

Exentia represents the affirmation of coherence, the inescapable truth that non-existence and existence cannot coexist.

Physical Implication: The Pulse that Sustains Ontological Distinction

According to the central axiom of ArXe Theory (\( \neg C \equiv T_p \)), this “second exentation”—Exentia—must also be correlated with a Planck Time:

\( Exentia \equiv T_p \)

This suggests that, in addition to the exentation of any primary contradiction that gives rise to Existence, there is a constant pulse of \( T_p \) dedicated to the validation and maintenance of the universe’s fundamental coherence. This \( T_p \) ensures that Istence (what cannot exist) never overlaps or coexists with Ex-Istence (what does exist).

In other words, the ontological distinction between what is and what is not in the ArXe universe is not static; it is actively sustained by a fundamental logico-physical operation that consumes one Planck Time. This pulse guarantees that the universe is logically consistent at its innermost level, resolving any potential collision between “being” and “non-being” in a discrete temporal flow.

Conclusion

The Second Exentation (Exentia), by demonstrating that the coexistence of Istence and Ex-Istence (Tensia) is a contradiction that, when negated, produces a tautology (and thus a \( T_p \)), is a powerful reaffirmation of the consistency of ArXe Theory. The ArXe universe is not only defined by what exists but is also a universe that actively avoids incoherence through discrete temporal pulses defined by Planck Time. This adds a layer of logical dynamism to the very fabric of spacetime, revealing that coherence is an active principle, not merely a descriptive one.

Arxe Logic