-\begin{theorem} \label{thm:named-form}
-TODO: Without defining a named form, prove that it is unique.
-\end{theorem}
-
-\begin{proof}
-TODO
-\end{proof}
-
-\begin{defi} \label{defi:named-form}
-TODO: Define a named form as the oldest reduced form of an equivalence class determined by similarity.
-\end{defi}
-
-% ==================================================================================================
-% ==================================================================================================
-% ==================================================================================================
-
-Between \autoref{thm:strong-self-relation} and Definition
-\autoref{defi:named-form} which follows from \autoref{thm:named-form}, we
-finally possesss the tools necessary to finish our Go program and use it to
-generate and examine a surreal number line.
+Note that reduced forms are not unique since $\surreal{}{} \similar
+\surreal{-1}{1}$ but $\surreal{}{} \neq \surreal{-1}{1}$.
+
+Note also that uniqueness is not trivially achieved by applying the generation
+function to an equivalence class. Although all elements are comparable, the set
+is finite, and the natural relation inherits the reflexive and transitive
+properties from $\mathbb{N}$, the relation is not antisymmetric since
+$\surreal{-1}{0} \neq \surreal{-1}{}$ and yet both are from generation-2. Thus,
+stealing well ordering from $\mathbb{N}$ and using it to define a unique
+element to represent the equivalence class turns out to be harder than I had
+hoped.
+
+I suspect the first time a reduced form number from a given equivalence class
+is generated, it will be alone. I base that on viewing the growing set as being
+comprised of two types of numbers, first a primary set forming a sparse line
+bookended by the empty set, making the set longer with every generation.
+
+$$\set{}$$
+$$\set{} < 0 < \set{}$$
+$$\set{} < -1 < 0 < 1 < \set{}$$
+
+Within this set a secondary set is forming, sitting directly in the middle
+between every existing number, making the set denser with every generation.
+Although harder to whip up in \LaTeX, I visualize this like a house of cards
+viewed end-on, a honeycomb lattice linking two parents to each child, adding a
+full spatial dimension to the honeycomb with each generation.
+
+If this visualization is true, then uniqueness of the reduced form representing
+an equivalence class in the first generation said class is encountered, follows
+naturally, even though this result doesn't hold for any following generations.
+
+Regardless, for the purpose of writing a program that generates numbers and
+orders them, the truth of this conjecture is irrelevant. We can simply identify
+the first number encountered in an equivalence class as the canonical
+representation of that class. The naturally serial nature of the program will
+serialize creation of numbers, adding a finer grained age relation and making
+it trivial to choose a `first' element to represent each class.
+
+All this started because I wanted a program to generate some numbers and
+investigate the relation. We finally have the requisite tools to return to that
+task, namely the strong self relation in \autoref{thm:strong-self-relation}
+which solves our comparison problem and the uniqueness of reduced forms in the
+subset of numbers(/equivalence classes) our program will actually generate.