Dimension prediction for the modification process

Let $X$ be a $2$-dimensional simplicial complex, let ${\mathcal{F}}$ be a sheaf on $X$, and use the notation of the modification process, including the subspaces $E_r$, $E'_r$, $V_r$, and $U_r$. Assume $E_r$ has just been constructed, so that $\dim E'_r=\dim\ker\omega_r$ is determined. If

$$\dim E_r\leq \frac{(|X(2)|-|X(1)|+|X(0)|-1)\dim{\mathcal{F}}-(h^1({\mathcal{F}})-h^0({\mathcal{F}})+1)}{2|X(2)|-|X(1)|},$$

then dimension prediction conjecture.

$$\dim E'_r=\dim V_r-\dim U_r$$

with probability $1-o(1)$ as a function of $|\mathbb{F}|$. If $M$ is the difference between the right- and left-hand sides of the displayed inequality, the same equality holds with probability $1-o(1)$ as a function of $M$. In particular, if $V_r=U_r$, the iterative process stops at the $r$-th step with probability $1-o(1)$ in either sense. The conjecture is supported by all of the authors' simulations and is used to motivate the affine-building quick-termination conjecture.