Arxe Logic

This chart illustrates the progression of exentations as the logical and temporal pillars that construct physical reality in ArXe Theory, revealing how coherence is ensured through discrete Planck Time (\( T_p \)) pulses, enabling increasing levels of complexity.

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Hierarchical Level	Fundamental Concept	Logical Definition & Character	Physical Interpretation & Function	Temporal Correlation (\( T_p \))
Level 0: Fundamental Absence	Istence (I)	Contradiction: \( (S \land \neg S) \) (Always False)	That which cannot physically exist. It represents pure logical incoherence, the “non-being” that the cosmos must negate.	Incompatible with existence: Lacks temporal correlation.
Level 1: The Unary Pulse of Being	Ex-Istence (E)	Tautology: \( (\neg S \lor S) \) (Always True) Result of the First Exentation \( \neg I \)	The elementary being. It allows for the existence of an entity in a single state (either S or ¬S), without internal contradiction. Foundation of the Principle of Non-Contradiction (PNC) at a fundamental level.	1 \( T_p \): An indivisible pulse of temporal existence. It’s the “instant” of being where no simultaneous “beginning” or “end” exists.
Level 2: Immediate Finitude & Binary Relation	Tensia (Te)	Contradiction: \( (I \land E) \) (Always False)	The impossibility of coexistence between what does not exist and what does. A second-order incoherence that must be resolved.	Incompatible with existence: Lacks temporal correlation.
	Exentia (Xt)	Tautology: \( \neg (I \land E) \) (Always True) Result of the Second Exentation \( \neg Te \)	The coherence of the relationship between contraries (e.g., “beginning” and “end”). It allows two opposing states to exist sequentially. This finitude is immediate, as the transition from one contrary to another does not include a discernible temporal “middle” at this level.	2 \( T_p \): A minimum duration that allows for the coherent transition from one state to its contrary. The two \( T_p \) are the “beginning” and “end” of a direct transition.
Level 3: Coherence of Processes & The Logic of the “Third”	Perience (Pe)	Contradiction: \( (Te \land Xt) \) (Always False)	An even deeper level of incoherence: the impossibility for the contradiction of coexistence and the coherence of succession to themselves coexist. An “excluded peripheral entity” from the domain of reality.	Incompatible with existence: Lacks temporal correlation.
	Ex-Perience (Xp)	Tautology: \( \neg (Te \land Xt) \) (Always True) Result of the Third Exentation \( \neg Pe \)	The coherence of change processes themselves. It ensures that the rules governing succession and temporality are stable and non-contradictory. It enables the validation of more complex temporal sequences.	> 2 \( T_p \): Could signify the stabilization of extended durations or processes containing multiple \( T_p \) and intermediate states. It serves as the foundation for “Extension” (a mediated Exentia).