Triangular threshold conjecture for simplex layers in the partition graph

Proof of all triangular thresholds and the maximal-dimension formula

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

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

Put $a_i=m_i-1$, $b_i=g_i-1$, and $T_k=k(k+1)/2$. Direct summation gives the exact weight identity

$$|\lambda| = T_k+ \sum_{i=1}^k(k-i+1)a_i + \sum_{i=1}^k i b_i + \sum_{1\le i\le j\le k}a_i b_j. \tag{2}$$

All terms after $T_k$ are nonnegative.

Fix $r\ge2$. If $k\le r-2$, (1) rules out local dimension at least $r$. If $k\ge r+1$, (2) gives

$$|\lambda|\ge T_{r+1}>T_r+1.$$

If $k=r$, the staircase itself has dimension $r-1$, so dimension at least $r$ requires

$$|\lambda|\ge T_r+1.$$

Equality occurs only for $b_1=1$ or $a_r=1$, with all other excess parameters zero.

If $k=r-1$, dimension at least $r$ requires $a_i,b_i\ge1$ for a common $i$. Formula (2) then gives

$$|\lambda|-T_k \ge(k-i+1)+i+1=k+2,$$

hence again

$$|\lambda|\ge T_k+k+2=T_r+1.$$

Equality occurs exactly when $a_i=b_i=1$ for one $1\le i\le k$, with all other excess parameters zero.

Both equality constructions exist. Therefore

$$\boxed{\tau(r)=\frac{r(r+1)}2+1\qquad(r\ge2).}$$

The exceptional initial layers satisfy $\tau(0)=1$ and $\tau(1)=2$, as required.

Moreover, for every $n\ge T_r+1$, the partition

$$(n-T_r+r,r-1,r-2,\ldots,1)$$

has exactly $r$ distinct part sizes and local dimension $r$. Conversely, the argument above rules out dimension $r$ at every smaller size. Hence

$$\boxed{ \Delta(n)=\max\left\{ r\ge0:\frac{r(r+1)}2+1\le n \right\}. }$$

This proves every assertion of the triangular-threshold conjecture.