# Rotation, Opposition, and Signs

The natural inner state of an ultrareal is $u$:

$$
U=u^2,\qquad u\ge0.
$$

Rotation is not part of that basic definition. It is an optional presentation of
the inner state when orientation matters:

$$
z=ue^{i\alpha}.
$$

The ultrareal value recovered from this presentation is:

$$
z\bar z=u^2.
$$

A single oriented presentation therefore returns to the same positive
ultrareal. Orientation becomes operational when two or more presentations are
joined.

## Opposition

Two oriented inner states may differ by a half-turn:

$$
\Delta=\pi.
$$

Then:

$$
\cos\Delta=-1.
$$

Their joined value is:

$$
|u+ve^{i\pi}|^2
=u^2+v^2-2uv
=(u-v)^2.
$$

Both incoming ultrareals are positive. The opposition is the relative turn
between their inner presentations.

If $u=v$, the value cancels to zero:

$$
|u+ue^{i\pi}|^2=0.
$$

The result is absence at the zero boundary, not a negative ultrareal.

## Minus-Signed Presentations

A minus sign can describe a rotated presentation of a square-value. It does not
name a negative member of $\mathbb U$.

Euler notation gives:

$$
i=e^{i\pi/2}.
$$

The symbol $i$ marks a quarter-turn. Squaring doubles that turn:

$$
(iu)^2=(e^{i\pi/2}u)^2=e^{i\pi}u^2.
$$

Ordinary notation writes the half-turn $e^{i\pi}$ as $-1$:

$$
(iu)^2=-u^2.
$$

This expression is useful ordinary notation, but $-u^2$ is not an ultrareal
value. It is a half-turned square-presentation written with the conventional
minus sign.

The ultrareal statement remains:

$$
u^2\in\mathbb U,\qquad -u^2\notin\mathbb U\quad(u>0).
$$

## Presentation Versus Relation

A minus-signed presentation and an opposed joining are different.

The expression:

$$
-u^2
$$

is a signed presentation of a single square-value outside $\mathbb U$.

The expression:

$$
u^2\oplus_{-1}v^2=(u-v)^2
$$

is an admissible joining of two positive ultrareals through opposition.

In the first case, the sign belongs to presentation. In the second case, the
negative coefficient belongs to the relation descriptor.

Neither case creates a negative ultrareal.

## General Rotation

For an oriented presentation:

$$
z=ue^{i\alpha},
$$

the raw square is:

$$
z^2=u^2e^{i2\alpha}.
$$

This raw square is generally an oriented square-presentation, not the
ultrareal value itself. The positive ultrareal value is recovered by return:

$$
z\bar z=u^2.
$$

For two oriented presentations,

$$
a=ue^{i\alpha},\qquad b=ve^{i\beta},
$$

their joined positive value is:

$$
|a+b|^2
=u^2+v^2+2uv\cos(\alpha-\beta).
$$

The relative difference $\Delta=\alpha-\beta$ determines the angular
descriptor:

$$
d=\cos\Delta.
$$

This is why $\Delta$ is the right symbol for opposition and difference. It names
the relation between presentations, not a hidden phase attached to every
ultrareal.