Difference-valuation conjecture for partial Stirling functions

Let $5\le\Delta\le8$, $0\le m<2^{e-1}$, and $e\ge4$. If

$$\nu(P_\Delta(m))<e+\varepsilon_{\Delta,m_2}-2,$$

then the difference-valuation conjecture. For every $d\ge0$,

$$\nu\bigl(P_{2^e+\Delta}(2^{e-1}(x+2^d)+m)-P_{2^e+\Delta}(2^{e-1}x+m)\bigr)=d+\nu(P_\Delta(m)).$$

This is presented as a reduction step toward the one-zero congruence-class conjecture and would control the precise 2-adic variation in each relevant residue class.