Scale representatives for the concrete selector-and-terminal tests below a node

Window data. Let $q,d,g,r_0\in\mathbb{N}$. Words over the alphabet $\{0,\dots,q-1\}$ are the vertices of the complete $q$-ary tree; $|P|$ is the length of a word $P$, $Pi$ is $P$ followed by the letter $i$, and $P\sqsubseteq Q$ means that $P$ is a prefix of $Q$. A word $P$ is a valid edge if $P\neq\varnothing$ and $|P|\le d$ (it names the edge entering the vertex $P$). A window datum $W$ with parameters $(q,d,g,r_0)$ consists of a function $r_W:\mathbb{N}\to\mathbb{N}$ and maps $a,b,b^*$ from words to $\mathbb{N}$ such that, writing $\ell(P)=r_W(d+1-|P|)$ and letting $P,Q$ range over valid edges:

1. $1\le\ell(P)$, $r_0\le\ell(P)$ and $a(P)\ge3$;
2. $b(P)+1=a(P)+\ell(P)$, so the edge $P$ carries the window of dyadic scales $a(P),\dots,b(P)$;
3. (prefix gap) if $P\sqsubseteq Q$ and $P\neq Q$, then $b(P)+g<a(Q)$;
4. (sibling gap) for every word $V$, letters $i<j$ and word $T$ such that $Vi$ and $VjT$ are valid edges, $b(Vi)+g<a(VjT)$;
5. (descendant bounds) if $P\sqsubseteq Q$, then $b(P)\le b(Q)\le b^*(P)$;
6. (padding) $b^*(P)+1\le a(P)+2\ell(P)$.

Finite table model. Take $q=M+1$, so that the letter $M$ is the default child and $0,\dots,M-1$ are the nondefault children; a node is a word of length $<d$ and a leaf is a word of length $d$. Use the window endpoints as resolutions: $b_S(Q,j)=b(Qj)$ for nodes $Q$ and $j<M$, and $b_T(L)=b(L)$ for leaves $L$. With $\kappa_\beta(y)=\lfloor 2^{\beta+2}\{y\}\rfloor$, where $\{y\}=y-\lfloor y\rfloor$, a selector table $\omega_S$ has bits $\omega_S(Q,j,c)$ for $c<2^{b_S(Q,j)+2}$, a terminal table $\omega_T$ has bits $\omega_T(L,c)$ for $c<2^{b_T(L)+2}$, and the selector bit of $(Q,j)$ at $y\in\mathbb{R}$ is $\sigma^{\omega_S}_{Q,j}(y)=\omega_S(Q,j,\kappa_{b_S(Q,j)}(y))$ if $|Q|<d$, and $\mathrm{false}$ otherwise. For every selector table $\omega_S$ let the child-choice map be $\chi_{\omega_S}(Q,y)=R_Q\bigl(\sigma^{\omega_S}_{Q,0}(y),\dots,\sigma^{\omega_S}_{Q,M-1}(y)\bigr)$ for fixed but arbitrary rules $R_Q:\{\mathrm{true},\mathrm{false}\}^M\to\{0,\dots,M\}$, and let $\rho^{\omega_S}_m(P,y)$ be the word obtained from $P$ by $m$ successive steps of appending $\chi_{\omega_S}(\cdot,y)$.

Let $W$ be a window datum with parameters $(M+1,d,g,r_0)$. Fix a node $U$, a point $x\in\mathbb{R}$ and $m\in\mathbb{N}$ with $|U|+1+m=d$. For an outcome $\omega=(\omega_S,\omega_T)$, a letter $i<M$ and $y\in\mathbb{R}$ define the local test

Let $\ell=r_W(d-|U|)$, the common window length of the edges $Ui$.

Theorem. There is a finite set $R\subseteq[1,2]$ with $|R|\le 1+M\,\ell\,2^{2\ell+2}$ such that for every $t\in[1,2]$ there is $t'\in R$ with

for every outcome $\omega=(\omega_S,\omega_T)$, every $i<M$ and every $j\in\{0,\dots,\ell-1\}$.

Thus the continuum of scales $t\in[1,2]$ can be replaced, simultaneously for all outcomes of the selector and terminal tables, all nondefault children $i$ of $U$ and all scales $a(Ui)+j$ in their windows, by a deterministic set of at most $1+M\ell\,2^{2\ell+2}$ representatives. No grid-factorization or padding hypotheses remain: both are supplied by the window datum.

Formalization Note The test $Q$ enters as an arbitrary function together with its defining equation hQ; the resolutions are FiniteTableModel.selectorEndpoints W and FiniteTableModel.terminalEndpoints W; the choice map $\chi_{\omega_S}$ is choose ωS, linked to rule by hchoose; and $\ell$ is W.r (d - U.val.length). The depth condition is hdepth : ∀ i : Fin M, (U.val ++ [i.castSucc]).length + r = d with our $m$ being r; for $M\ge1$ it is the condition $|U|+1+m=d$, and for $M=0$ it is vacuous (there are then no tests to compare).