Ordered-Box Moment Geometry and Sharp Finite-Information Stability for Symmetric Real-Rooted Canonical Products

We study two linked finite-information problems. The first is an obstacle-constrained equal-weight moment problem with coordinate-dependent bounds and fixed first J power sums. We prove that every nonempty finite ordered-box fiber has unique upper and lower pJ+1-principal profiles, and that the same two profiles simultaneously extremize every separable objective with nonnegative $(J+1)$-st derivative. The theorem extends to countably infinite nonnegative ordered boxes dominated by a summable upper sequence. The second problem concerns normalized symmetric real-rooted canonical products F(z) = ∏n ≥ 1 (1 - z²/γₙ²) with simple positive zeros satisfying a counting envelope. Observing the first N positive zeros and the first J reciprocal even moments reduces the hidden tail to an ordered-capacity moment class. A general-J strictification theorem realizes this geometry inside the original simple-zero class. This yields an exact arbitrary-envelope minimax capacity ΛJ(q), a strict simple-zero lower realization, and a quantitative complex-disk upper bound. For noisy observations we introduce a directional logarithmic width ΓJ(q;η) and a weighted higher stop-loss modulus ΞJ(q;η), with the exact zero-noise collapse ΓJ(q;0) = ΞJ(q;0) = ΛJ(q). The infinite-dimensional capacity is finitely certifiable through monotone finite truncations and an explicit analytic tail bound. For regularly varying quantiles ρₙ = n1/ν (n), 0 < ν < 2, the resulting dimensionless minimax law is sharp: ω^#N,J,R(δ;Ψ) δ + N ρN-2J-2. This revised version additionally proves the sharp global finite-dimensional error bound dist(z,ZM,J) ≤ CM,J,q FM,J(z)₂1/J for every finite M ≥ J ≥ 2 in the rank-dependent ordered-capacity class. The exponent $1/J$ is optimal for every fixed positive finite ordered box. This closes the previously open regime $M > J > 2$ and improves the finite positive-noise excess of ΓJ and ΞJ to OM,J,q(η₂1/J). The paper remains restricted to the paired symmetric canonical-product model and makes no claim for arbitrary nonsymmetric real-rooted canonical products.