Low-dimensional framework conjecture for the partition graph

Proof of the complete low-dimensional framework conjecture

The previously established local corner formulas in arXiv:2603.18696, Corollary 3.4, Proposition 3.7, and Theorem 7.1, are used as known input.

Write

$$\lambda=(d_1^{m_1},\ldots,d_k^{m_k}), \qquad d_1>\cdots>d_k>0, \qquad g_i=d_i-d_{i+1}, \qquad d_{k+1}=0.$$

For removable corner $i$ and addable corner $j$, the known corner-incidence matrix is

$$M_{ij} = 1-\mathbf1_{\{j=i,\ m_i=1\}} -\mathbf1_{\{j=i+1,\ g_i=1\}}.$$

Hence the star capacities are

$$s_i=k+1-\mathbf1_{\{m_i=1\}}-\mathbf1_{\{g_i=1\}},$$

and every top capacity is at most $k$. Consequently

$$\dim_{\mathrm{loc}}(\lambda)= \begin{cases} 0,&\lambda=(1),\\ k-1,&\lambda=(k,k-1,\ldots,1),\ k\ge2,\\ k+1,&m_i\ge2\text{ and }g_i\ge2\text{ for some }i,\\ k,&\text{otherwise}. \end{cases} \tag{1}$$

The source's earlier definitions in arXiv:2603.21221, Definitions 2.4, 2.6, and 2.8, identify the framework as

$$\mathrm{Fw}_n=M_n\cup E_n\cup E_n',$$

where

$$M_n=\{(n-j,1^j):0\le j\le n-1\},$$ $$E_n=\{(n-j,j):1\le j\le\lfloor n/2\rfloor\}, \qquad E_n'=\{\lambda':\lambda\in E_n\}.$$

Assume $n\ge4$. The endpoints $(n)$ and $(1^n)$ have local dimension one.

Every other hook in $M_n$ has two distinct block sizes; its upper block has multiplicity one and its lower block has gap one. Therefore no block simultaneously has multiplicity and gap at least two. Since the only two-block staircase $(2,1)$ has size three, (1) shows that every non-antennal hook has dimension exactly two.

For a two-row partition $(a,b)$ with $a>b$, both block multiplicities equal one. Again (1) gives local dimension two. If $a=b\ge2$, there is one block with multiplicity two and gap $a\ge2$, so (1) also gives dimension two.

Finally, conjugation is an automorphism of the partition graph and therefore preserves local simplex dimension. Every vertex of $E_n'$ consequently has dimension two as well.

Thus

$$\boxed{ \mathrm{Fw}_n\subseteq L_1(n)\cup L_2(n), \qquad \mathrm{Fw}_n\cap L_1(n)=\{(n),(1^n)\} \quad(n\ge4). }$$

Equivalently, every non-antennal framework vertex has local simplex dimension exactly two, proving the full conjecture.