Derivation of Boundary Conditions from the Axiom ¬()

ArXe Research — Formal Derivation
Status: Conceptual Formalization — April 2026

1. The Axiom

The sole axiom of ArXe is:

$$neg() triangleq T_f simeq T_p$$

Interpretation: Nothingness —the absolute absence of all distinction— is logically inviable. It cannot “be” without contradicting itself. That inviability imposes an escape. That escape is a fundamental pulse, identified structurally with a fundamental unit of time $T_f$, postulated as equivalent to the Planck Time $T_p$.

2. The Two Presumptions of the Axiom

The axiom contains two inseparable presumptions:

First presumption: The negation of logical contradiction occurs in a fundamental time. That time is, by postulated correspondence, equivalent to the Planck Time $T_p$.

Second presumption: Of a fundamental time it cannot be said, without contradiction, that it is a thing and its contrary. That is: a $T_f$ cannot simultaneously be the beginning and the end of itself.

A fundamental physical time is, by this second presumption, non-contradictory in its basic unit. That unit cannot be at once the starting point and the endpoint of the same process.

3. Definitions

3.1 Boundary

A boundary is that property of a thing which distinguishes it from what follows or surrounds it. It is that by which the end or the beginning of something is recognized —or determined.

A boundary is not an extension or a magnitude. It is a mark of distinction: this begins here, this ends there.

3.2 Finiteness

Finiteness is the property determined by two contrary boundaries, such that if one boundary is initial, the other boundary is final, and vice versa.

An entity is finite —or has the property of finiteness— if and only if it has a determined beginning and a determined end.

Finiteness is binary. There are no degrees of finiteness: something is either finite or it is not.

3.3 Contour

A contour is the set of finiteness attributes of an entity. Each attribute consists of exactly two terms: one initial and one final.

Examples of contour attributes:

Attribute	Initial term	Final term
Width	beginning of width	end of width
Height	beginning of height	end of height
Depth	beginning of depth	end of depth
Duration	beginning of duration	end of duration

Each attribute is independent of the others. The number of attributes is the arity of the contour.

3.4 Closed Contour

A contour is closed when both terms —initial and final— are determined in the act in which they are invoked.

3.5 Open Contour

A contour is open when one term —whether initial or final— is not determined in the act in which it is invoked.

Determination is not an absolute property of the contour. It is relative to the act of invocation. The same contour may be invoked as closed in one context and as open in another. Undecidability is not a defect: it is a structural feature.

4. Condition of Possibility of Contour (Boundary Condition)

4.1 Derivation

From the axiom and its second presumption it follows that:

A single phase or $T_f$ cannot, without contradiction, be both beginning and end of itself.

Beginning and end are contraries by definition of finiteness (§3.2). To claim that a single phase fulfills both functions is a direct contradiction of the second presumption of the axiom.

Therefore: for something to have a beginning and an end without contradiction, a minimum of two phases is required.

4.2 Clarification

This derivation does not imply:

That every phase is either initial or final.
That every pair of phases has a determined beginning and end.
That every configuration of two phases constitutes a closed contour.

It implies only:

Two phases are the minimum condition of possibility for something to have a beginning and an end without contradiction.

4.3 Differentiated Condition of Possibility

The condition of possibility is differentiated according to the type of contour:

Contour type	Minimum phases required	Reason
Closed (both terms determined)	2	Two distinct phases are required so that beginning and end are not the same phase
Open (one term determined)	1	A single determined term does not require a second term to remain coherent

Boundary Conditions (BC) are not entities that occur or causally emerge every certain number of phases. They are logically possible structures whose minimum requirement of internal coherence is established by the impossibility —derived from the axiom— of a phase being simultaneously its own beginning and its own end.

5. Consequence for Scale

Once the condition of possibility of the contour is established, the connection to scale is immediate:

A closed contour (two determined terms) defines a distance between its terms —a finite length. That length is a scale.
An open contour (one undetermined term) does not define a bounded distance: there is no scale.

Therefore: the presence of a free closed contour imposes a scale on the system. The absence of free closed contours is the condition for scale invariance (criticality).

6. Summary of the Derivation Chain

¬()  (axiom)
↓
Second presumption: a Tf cannot be beginning and end of itself without contradiction
↓
Boundary: distinction between a thing and what follows or surrounds it  [primitive concept]
↓
Finiteness: two contrary boundaries — beginning and end
↓
Contour: set of finiteness attributes, each attribute with two terms
↓
Closed / Open: determination of terms in the act of invocation
↓
Boundary Condition:
  — Closed contour requires minimum 2 phases
  — Open contour requires minimum 1 phase
↓
Scale: the closed contour defines a finite distance between its determined terms

7. Status of the Derivation

Step	Status
Axiom ¬()	Postulate
Double presumption	Contained in the axiom
Definition of boundary	Primitive concept — not derived from prior steps
Definition of finiteness	Derived from the definition of boundary
Definition of contour	Derived from the definition of finiteness
Closed / Open distinction	Definition — relative to the act of invocation
BC: minimum 2 phases (closed)	Derived — from the second presumption of the axiom
BC: minimum 1 phase (open)	Derived — from the definition of open contour
Connection to scale	Derived — from the definition of closed contour

The only element not derived from prior steps is the definition of boundary. It is the primitive concept upon which the entire chain is built. It is a minimal concept: prior to any notion of extension, magnitude, or existence.

8. Conclusion

Boundary Conditions (BC) are not a postulate. They are a necessary consequence of the axiom ¬() once its second presumption —the non-contradiction of fundamental time— is made explicit and the definitions of boundary, finiteness, and contour are applied.

The distinction between closed contour (minimum 2 phases) and open contour (minimum 1 phase) is structural: it does not express degrees of existence but minimum conditions of logical coherence for each type of determination.

The BC-scale gap is closed: the connection between closed contour and finite scale is immediate by definition of contour and of distance between determined terms.

What remains is the formalization of contour independence —when a finiteness attribute is irreducible to others— a problem of combinatorial structure, not a logical gap in the conceptual derivation.