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 = \-2ᵃ, , 2ᵃ - 1\ ⊆ Z modulo 2ᵃ⁺¹ has uniform valuation v₂(∏_s ∈ S \(t - s)) = 2ᵃ⁺¹ - a - 2 on S. Symmetry breaks when $T = 0$ is targeted and S^× = S \0\ carries shell multiplicities q(v₂(s)), or an inner sub-disk 2ᵘZ ∩ S (1 ≤ u ≤ a) is excluded. This note records an elementary $2$-adic Hermite precision calculus over Z/2ᵖZ. First, for any profile q on S^×, we give an exact shell-difference identity for v₂(Eₜ(0)) and Eₜ(t + 2ᵃ⁺¹z) ∈ 1 + 2zZ₍₂₎[z], yielding a sufficient precision budget pₜ. Second, on any annulus 0 ≤ v₂(t) < u (1 ≤ u ≤ a), both the shell-normalized operator Aa,u (off-diagonals independent of a) and the raw loss operator diag(2ᵃ⁻ⁱ)Aa,u are nonsingular upper-triangular M-matrices. A two-term recurrence shows their continuous optimizers tilt oppositely, with Φ⁽a,u,2m· 2ᵃ) ~ 6m/u (a - u ≥ 2) and R⁽a,u,K) ~ 3K/(2ᵘ - 2⁻ᵘ) (a - u → ∞). For large u (a - u ≥ 2, D 2ᵃ u), q^ and a linear taper beat step cutoffs under Φᵢ by Θ(u) (step cutoffs beat the taper at small u), whereas for the word width pₜ - p of this budget step cutoffs overtake both (q^ still beating the taper): tapers inflate outer width to Θ(2ᵘ/u)\,R^, step cutoffs stay within ≈ (u+1)/3 of R^, and a backward M-matrix recurrence (approximated by a closed-form inner bump) attains the integer optimum below qraw^. Third, on S^× ($u = a + 1$), Aa,a+1 = span(1), contrasted with forward differences and P\'olya--Legendre precision p + v₂(d!).