Arxe Logic

Any distinction or choice between absolutely indistinguishable elements cannot arise from a true reason within the same logical level.
If one of the options is realized, its distinction must come from a higher hierarchy,
or else result from a probabilistic collapse without structural cause at the current level.

Formal version

Let \(a\) and \(b\) be elements of a logical hierarchy \(H_n\), such that:

• \(a \equiv b\) (logically indistinguishable within \(H_n\)),
• \(P(a) = P(b) = \frac{1}{2}\) (equal probabilities).

Then, if a realization occurs as \(a \prec b\) (i.e., \(a\) is realized “first”), it follows that:

\(a \equiv b \in H_n \land P(a) = P(b) = \frac{1}{2} \Rightarrow (a \prec b) \text{ only if } \exists H_{n+1} : \rho(a) \neq \rho(b)\)

That is: the realization of a distinction between \(a\) and \(b\) must originate from a reason \(\rho\) external to level \(H_n\).

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2. Logical Deduction by Hierarchy

Level \(H_0\): Absolute logical indistinction (pre-existence)

• No structural difference exists between elements.
• All possibilities carry equal weight.
• Pure probability is expressed as \(P = \frac{1}{2}\).

Level \(H_1\): First distinction – exentation

• A negation \(\neg\) appears, a minimal logical unit (like a bit or Tp).
• If there are two indistinguishable negations \(\neg_1\) and \(\neg_2\), we cannot say which comes first.

Level \(H_2\): Emergence of order

• If one is realized, it must be because:
  • Either a collapse occurred from level \(H_0\) without a determinable cause,
  • Or a higher hierarchy \(H_{n+1}\) introduced a reason of preference.

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3. Application to Goldbach’s Conjecture

For an even number \(N\), there exist multiple prime pairs \((p_i, q_i)\) such that:

\(p_i + q_i = N\)

• All such pairs are valid, i.e., indistinguishable in terms of validating the conjecture.
• There is no reason to prefer one pair over another.
• Yet one is chosen when expressing \(N\).

ArXe Interpretation:

The conjecture is validated not by a reason at level \(H_n\), but by the existence of at least one pair.

The existence of some pair \((p_i, q_i)\) such that \(p_i + q_i = N\) is sufficient.
The absence of hierarchy among the pairs makes the truth factual but not logically ordered.

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4. Application to the Riemann Hypothesis

The non-trivial zeros of the zeta function \(\zeta(s)\) all lie on the critical line \(\Re(s) = \frac{1}{2}\).

• There are infinite possible locations (the complex plane), yet all known zeros lie there.
• The symmetry between \(s\) and \(1 – s\) suggests a collapse into indistinction.

ArXe Interpretation:

The placement of zeros on \(\Re(s) = \frac{1}{2}\) cannot be explained solely at the functional level,
but rather as the result of a hierarchical collapse dictated by a higher logical structure.

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5. Structural Conclusion – ArXe Hierarchy

Level	Entity	Operation / Function	Hierarchy	Observation
\(H_0\)	Indistinction	Pure probability \(P = \frac{1}{2}\)	Highest	No distinction is possible
\(H_1\)	Negation / Tp	Exentation \(\neg\)	+1	Requires symmetry breaking
\(H_2\)	Order / Choice	Preference \((a \prec b)\)	+1	Only possible with external reason
\(H_3\)	Existence / Factuality	Realization	+1	What actually manifests