Exact 2-Adic Hermite Shell Calculus and M-Matrix Minimax Bounds over Z/2^pZ

In Bhargava's theory of $p$-orderings, a residue ball $S = \{-2^a, \dots, 2^a - 1\} \subseteq \mathbb{Z}$ modulo $2^{a+1}$ has uniform valuation $v_2(\prod_{s \in S \setminus \{t\}}(t - s)) = 2^{a+1} - a - 2$ on $S$. Symmetry breaks when $T = 0$ is targeted and $S^\times = S \setminus \{0\}$ carries shell multiplicities $q(v_2(s))$, or an inner sub-disk $2^u\mathbb{Z} \cap S$ ($1 \le u \le a$) is excluded. This note records an elementary $2$-adic Hermite precision calculus over $\mathbb{Z}/2^p\mathbb{Z}$. First, for any profile $q$ on $S^\times$, we give an exact shell-difference identity for $v_2(E_t(0))$ and $E_t(t + 2^{a+1}z) \in 1 + 2z\mathbb{Z}_{(2)}[z]$, yielding a sufficient precision budget $p_t$. Second, on any annulus $0 \le v_2(t) < u$ ($1 \le u \le a$), both the shell-normalized operator $A_{a,u}$ (off-diagonals independent of $a$) and the raw loss operator $\operatorname{diag}(2^{a-i})A_{a,u}$ are nonsingular upper-triangular $M$-matrices. A two-term recurrence shows their continuous optimizers tilt oppositely, with $\Phi^\star(a,u,2m\cdot 2^a) \sim 6m/u$ ($a - u \ge 2$) and $R^\star(a,u,K) \sim 3K/(2^u - 2^{-u})$ ($a - u \to \infty$). For large $u$ ($a - u \ge 2$, $D \gg 2^a u$), $\lceil q^\star \rceil$ and a linear taper beat step cutoffs under $\Phi_i$ by $\Theta(u)$ (step cutoffs beat the taper at small $u$), whereas for the word width $p_t - p$ of this budget step cutoffs overtake both ($\lceil q^\star \rceil$ still beating the taper): tapers inflate outer width to $\Theta(2^u/u)\,R^\star$, step cutoffs stay within $\approx (u+1)/3$ of $R^\star$, and a backward $M$-matrix recurrence (approximated by a closed-form inner bump) attains the integer optimum below $\lceil q_{\mathrm{raw}}^\star \rceil$. Third, on $S^\times$ ($u = a + 1$), $\ker A_{a,a+1} = \operatorname{span}(\mathbf{1})$, contrasted with forward differences and P\'olya--Legendre precision $p + v_2(d!)$.