Endpoint-support peeling: $h(n,d)\le h(n-d,d)+h(n-1,d-1)$ for connected layer families

Let $h(n,d)$ be any monotone (in $n$) upper bound for the height $L+1$ of connected layer families of rank $d$ on at most $n$ symbols. Then for every family of rank $d\ge1$ on $n$ symbols,

Proof. Fix a base $T$ of the last layer and let $t$ be the first layer containing any symbol of $T$. The $t$ earlier layers avoid all $d$ symbols of $T$, hence form a family on at most $n-d$ symbols: $t\le h(n-d,d)$. A symbol $x\in T$ present in layer $t$ is present in the last layer, so by the interval axiom its star meets every layer from $t$ onward; restricting to the star and deleting $x$ yields a family of rank $d-1$ on at most $n-1$ symbols of height $L-t+1\le h(n-1,d-1)$.

This is the justified one-sided analogue of the Kalai--Kleitman recursion. Its exact majorant is a cumulative restricted-partition function, which is superpolynomial, so the recurrence alone does not give a polynomial bound.

Formalization Note The bound h is quantified over all families via hmax; V is restricted to Type so that families on subtypes of V can be formed.