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\label{def:constraint-lattice}
The \textbf{constraint lattice} is the conjunction
of the following five conditions on a quadruple
$(A, B, C, D) \in \tau\text{-Idx}^4$:
\begin{enumerate}
    \item[\textbf{(C1)}] \textbf{Prime constraint.}
          $A \in \mathbb{P}_\tau \cup \{1\}$.
          That is, $A$ is either an internal $\tau$-prime
          or the degenerate value~$1$.

    \item[\textbf{(C2)}] \textbf{Non-negativity.}
          $B \geq 0$, $C \geq 0$, $D \geq 0$.

    \item[\textbf{(C3)}] \textbf{Remainder constraint.}
          Every prime factor of $D$ is strictly less than~$A$.
          Equivalently: if $p \in \mathbb{P}_\tau$ and $p \mid D$,
          then $p < A$.

    \item[\textbf{(C4)}] \textbf{Tower constraint.}
          If $A = 1$, then $B = 0$ and $C = 0$.
          (The degenerate case collapses the tower.)

    \item[\textbf{(C5)}] \textbf{Normalization.}
          If $B \geq 1$ and $C \geq 1$,
          then the triple $(A, B, C)$ is the output
          of the greedy peel-off algorithm
          applied to the tower part.
          That is, the exponent~$B$ is maximal
          and the tetration height~$C$ is the result
          of the greedy extraction ---
          no under-extraction is permitted.
\end{enumerate}
A quadruple satisfying \textbf{(C1)--(C5)} is called
\textbf{$\tau$-admissible}.