A ridge-closed connected layer family has length at most $\binom{n}{d-1}-d$

If $F$ is a ridge-closed connected layer family of rank $d\ge1$ on $n$ symbols, then $L+d\le\binom{n}{d-1}$.

Proof. Every base of layer $t\ge1$ has a ridge that is newly active at $t$: otherwise all its ridges were present in layer $t-1$ and ridge-closure would place the base in layer $t-1$, contradicting base-disjointness. A ridge becomes newly active at most once (its active layers form an interval), so there are at least $L$ distinct newly-born ridges after layer $0$, and layer $0$ already uses at least $d$ ridges. This is the basic 'born ridge' count for ridge-closed families; it is polynomial for fixed $d$ but not uniformly, and the cone embedding shows why it cannot be sharpened by counting alone.