Bounded ridge incidence makes saturated layer families linear: $L+1\le\rho(n-d+1)$

Let $\mathcal L$ be a saturated homogeneous connected layer family of rank $d$ on $n$ symbols ($1\le d\le n$) in which every $(d-1)$-subset is contained in at most $\rho$ bases of the whole family. Then

For $\rho=2$, the incidence of a simple polytope (every ridge lies in exactly two facets, so every $(d-1)$-set of facets contains at most two vertices), this is $L\le 2(n-d)+1$: one geometric axiom collapses the quadratic bound of the saturated class to a linear one and eliminates the EHRR long-mesh reservoir. Whether the same holds without saturation is open.

Proof idea. In the charging argument of the quadratic bound every account is spent through a fixed $(d-1)$-subset, whose containing bases must be distinct across the layers it pays for; ridge incidence caps each account at $\rho$, and the initial-profile accounting leaves at most $n-d+1$ paying accounts.