uence ${\mathscr{I}} \to {\mathbb{S}}$ of steps (usually ${\mathscr{I}}$ will be the natural numbers ${\mathbb{N}}$).

We revisit the three fundamental P’s (path, procedure, and process – §\[S:THREE\_P\]). A walk in step space decomposes into these three sequences: $\textrm{walk}_{\,i} = (\textrm{path}_{\,i}, \textrm{process}_{\,i}, \textrm{procedure}_{\,i})$. This triple is not logically independent; being so, shorter characterizations of step space exist. However we retain the present representation, favoring its three-element formulation, which covers all possibilities in simple fashion.

[\[Extended projection\]]{} \[D:EXTENDED\_PROJECTION\] Let $\lbrace {\mathit{s}}_n \rbrace$ be a walk in step space ${\mathbb{S}} = \Lambda \times {\mathscr{F}} \times {\mathbf{F}}$ and let ${\mathscr{I}}$ be a denumerable index set (usually ${\mathbb{N}}$). Use the locus projection $\mho_\Lambda \colon {\mathbb{S}} \to \Lambda$ of definition \[D:STEP\_SPACE\_PROJECTION\] to construct the sequential *path* projection $\overline{\mho}_\Lambda \colon {\mathbb{S}}^{\mathscr{I}} \to \Lambda^{\mathscr{I}}$ via setting $\overline{\mho}_\Lambda(\lbrace {\mathit{s}}_n \rbrace) = \lbrace (i,\mho_\Lambda({\mathit{s}})) \colon
    (i,{\mathit{s}}) \in \lbrace {\mathit{s}}_n \rbrace \rbrace$. Similarly define the sequential *process* and *procedure* projections $\overline{\mho}_{\mathbf{F}} \colon {\mathbb{S}}^{\mathscr{I}} \to {\mathbf{F}}^{\mathscr{I}}$ and $\overline{\mho}_{\mathscr{F}} \colon {\mathbb{S}}^{\mathscr{I}} \to {\mathscr{F}}^{\mathscr{I}}$.

With $\lbrace x_n \rbrace \colon {\mathscr{I}} \to X$ a sequence in some set $X$, we alternatively denote the sequence’s ${{i}^{\text{th}}}$ term by $x_i = \lbrace x_n \rbrace(i)$.

\[L:RELATED\_PROJECTION\] Let $\langle \Psi, \Phi \rangle$ and locus set $\Lambda$ be the bases for step space ${\mathbb{S}} = \Lambda \times {\mathscr{F}} \times {\mathbf{F}}$. Let $\lbrace {\mathit{s}}_n \rbrace$ be a walk ${\mathscr{I}} \to {\mathbb{S}}$. For each $i \in {\mathscr{I}}$, $(\overli