# Admissibility and Many-Term Sums
Relation-aware addition must remain inside the ultrareal domain, the positive
real line with zero included:
$$
\mathbb U=[0,\infty).
$$
The previous chapter proved closure for the term-type-aware ultrareal sum. If:
$$
U=u^2,\qquad V=v^2,
$$
with $u,v\in\mathbb R_{\ge0}$, then:
$$
u+v\in\mathbb R_{\ge0}.
$$
Therefore:
$$
U+V=(u+v)^2\in\mathbb U.
$$
That is the basic closure rule for ultrareal addition. If the inner-state
product distributes, the same value can be read as the ordered expression:
$$
u^2+uv+vu+v^2\in\mathbb U.
$$
Admissibility becomes explicit when the same sum is represented by
descriptor data.
## Descriptor Admissibility
When the ordered cross terms are written as the descriptor,
$$
d(U,V)=uv+vu,
$$
the sum becomes:
$$
U+V=u^2+d(U,V)+v^2.
$$
In scalar cases, the descriptor is admissible for this addition when:
$$
u^2+d(U,V)+v^2\ge0.
$$
Equivalently:
$$
U+V\in\mathbb U.
$$
## Bounded Opposition
In the bounded angular or field-alignment scale, the descriptor satisfies:
$$
-2uv\le d(U,V)\le 2uv.
$$
Closure is automatic:
$$
u^2+d(U,V)+v^2\ge(u-v)^2\ge0.
$$
The smallest value occurs at complete opposition:
$$
d(U,V)=-2uv.
$$
Then:
$$
U+V=(u-v)^2.
$$
Thus opposition can cancel equal inner magnitudes to zero, but it cannot push an
ultrareal result below zero while remaining in the bounded scale.
## Exact Cancellation
For nonzero ultrareals, exact cancellation in the bounded scale has only one
form.
Let:
$$
A=a^2,\qquad B=b^2,\qquad a,b>0.
$$
If:
$$
A+B=0,
$$
then:
$$
a^2+d(A,B)+b^2=0.
$$
Solving for the descriptor gives:
$$
d(A,B)=-(a^2+b^2).
$$
By the arithmetic-geometric mean inequality,
$$
a^2+b^2\ge2ab,
$$
with equality only when $a=b$. Therefore, inside the bounded descriptor scale
$-2ab\le d(A,B)\le2ab$, exact cancellation requires:
$$
a=b,\qquad d(A,B)=-2ab.
$$
Complete opposition is the only bounded relation that cancels two nonzero
ultrareals, and it cancels them only when their inner magnitudes are equal.
## Many-Term Addition
For many ultrareals,
$$
U_i=u_i^2,\qquad i=1,\ldots,n,
$$
the many-term inner-state sum is:
$$
x=u_1+\cdots+u_n.
$$
Since every $u_i$ is nonnegative, $x\in\mathbb R_{\ge0}$, so:
$$
U_1+\cdots+U_n=(u_1+\cdots+u_n)^2\in\mathbb U.
$$
When this value is reduced to scalar descriptor data, the pairwise
descriptor table $D=(d_{ij})$ records the interaction terms:
$$
\boxed{
U_1+\cdots+U_n
=
\sum_i u_i^2
+\sum_{i
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