# Admissibility and Many-Term Sums Relation-aware addition must remain inside the ultrareal domain: $$ \mathbb U=[0,\infty). $$ For two terms, $$ U=u^2,\qquad V=v^2, $$ the joined value is: $$ U\oplus_d V=u^2+v^2+2duv. $$ The descriptor $d$ is admissible for this joining when: $$ u^2+v^2+2duv\ge0. $$ Equivalently: $$ U\oplus_d V\in\mathbb U. $$ This is the basic closure rule. ## Bounded Opposition In the bounded angular or field-alignment scale, $$ -1\le d\le1, $$ closure is automatic: $$ u^2+v^2+2duv\ge(u-v)^2\ge0. $$ The smallest value occurs at complete opposition: $$ d=-1. $$ Then: $$ U\oplus_{-1}V=(u-v)^2. $$ Thus opposition can cancel equal inner magnitudes to zero, but it cannot push a joined ultrareal 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: $$ A\oplus_d B=0, $$ then: $$ a^2+b^2+2dab=0. $$ Solving for $d$ gives: $$ d=-\frac{a^2+b^2}{2ab}. $$ By the arithmetic-geometric mean inequality, $$ \frac{a^2+b^2}{2ab}\ge1, $$ with equality only when $a=b$. Therefore, inside $-1\le d\le1$, exact cancellation requires: $$ a=b,\qquad d=-1. $$ 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, $$ U_i=u_i^2,\qquad i=1,\ldots,n, $$ relation-aware addition is determined by a relation table $D=(d_{ij})$. Write the joined value as: $$ \boxed{ \operatorname{Join}_D(U_1,\ldots,U_n) := \sum_i u_i^2 +2\sum_{i --- - [Preferred Frame Writing on GitHub.com](https://github.com/siran/writing) (built: 2026-04-27 18:07 EDT UTC-4)