Epistemic note
This project constitutes a deductive structure based on constraints and invariants, not a philosophical hypothesis nor a physical theory. Its contents are not evaluable in isolation: each individual page presupposes the full conceptual structure of the site. Any judgment formulated without examination of the complete context is necessarily partial and non-conclusive. The distinction between structural deduction and interpretation is an integral part of the model.

Minimal Mathematics of Horizons

This appendix provides minimal mathematical tools to make the idea of an “horizon” readable as a regime limit: the equator (compression) and the poles (expansion) both tend toward the same operational infinity.

1) Zero and infinity as dual limits

Zero and infinity are not reachable states, but limits. A fundamental example:

$$x \to 0^+ \Rightarrow \frac{1}{x} \to \infty$$

In the model: at the equator time tends toward zero; at the poles order tends toward infinity. In both cases, the system approaches the limit without ever reaching it.

2) One single infinity, two directions

Let us consider the function:

$$f(x)=\frac{1}{|x|}$$

Whether \(x \to 0^+\) or \(x \to 0^-\), the result is always:

$$f(x)\to \infty$$

The infinity is the same: what changes is the direction, not the horizon.

3) The horizon as an asymptote (not as a point)

A horizon is an asymptote: the limit is approached without collapsing into it.

$$g(n)=\frac{1}{n}\quad \text{with}\quad n\to\infty \Rightarrow g(n)\to 0$$

For this reason: the equator does not annihilate existence (it makes it indescribable), and the poles do not annihilate operational conflict (they asymptotically reduce it).

4) Symmetry: compression and expansion

  • Equator → infinity by compression (maximum operational conflict, \(t \to 0\))
  • Poles → infinity by expansion (minimum operational conflict, increasing order)

It follows that the residual rule at the limit is the same.

5) Residual rule

When a system tends toward infinity, objects, states, and forms do not survive: only invariant rules remain. This conclusion is consistent with the mathematics of limits.