Saturated homogeneous connected layer families have length at most $d(n-d)$

Let $\mathcal L_0,\dots,\mathcal L_L$ be a connected layer family of rank $d$ on $n$ symbols, $1\le d\le n$, which is saturated homogeneous with respect to a fixed partition of the symbols into blocks: every layer has a single profile $p$ (number of symbols in each block) and contains a base above every subset of fewer than $d$ symbols whose profile is pointwise $\le p$. Then

This subclass contains the mesh constructions of Eisenbrand--Hähnle--Razborov--Rothvoß, whose length is $\Omega(n^2/\log n)$; so on this subclass the exponent two is optimal up to a logarithm, and the almost-quadratic mechanism of the known lower bounds cannot be pushed further by more blocks or by incomplete saturated phases.

Proof idea. For each block $g$ and threshold $1\le r<d$, saturation makes the set of layers whose profile has $g$-coordinate $\ge r$ equal to the interval of layers containing a fixed $r$-subset of the block, hence an interval. A maximal run of layers with a fixed mixed profile is paid for by a newly crossed threshold $(g,r)$ and has at most $m_g-r+1$ layers, because all of them must contain a fixed $(d-1)$-subset and differ in the remaining symbol of block $g$. Pure layers of a block are paid from one reservoir of size $m_g-d+1$ by the same completion count. Summing the disjoint budgets and subtracting the initial profile's credit gives $L\le d(n-d)$.

Formalization Note IsSaturatedHomogeneous blk F is the definition in Hirsch_clf; the block set G is an arbitrary finite type.