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 $p_{J+1}$-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 $$F(z) = \\prod_{n \\ge 1} \\left(1 - \\frac{z^2}{\\gamma_n^2}\\right)$$ 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 $\\Lambda_J(q)$, a strict simple-zero lower realization, and a quantitative complex-disk upper bound. For noisy observations we introduce a directional logarithmic width $\\Gamma_J(q;\\eta)$ and a weighted higher stop-loss modulus $\\Xi_J(q;\\eta)$, with the exact zero-noise collapse $$\\Gamma_J(q;0) = \\Xi_J(q;0) = \\Lambda_J(q).$$ The infinite-dimensional capacity is finitely certifiable through monotone finite truncations and an explicit analytic tail bound. For regularly varying quantiles $\\rho_n = n^{1/\\nu} \\ell(n)$, $0 < \\nu < 2$, the resulting dimensionless minimax law is sharp: $$\\omega^\\#_{N,J,R}(\\delta;\\Psi) \\asymp \\delta + N \\rho_N^{-2J-2}.$$ This revised version additionally proves the sharp global finite-dimensional error bound $$\\operatorname{dist}(z,\\mathcal{Z}_{M,J}) \\le C_{M,J,q} \\Vert{}F_{M,J}(z)\\Vert{}_2^{1/J}$$ for every finite $M \\ge J \\ge 2$ 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 $\\Gamma_J$ and $\\Xi_J$ to $\\mathcal{O}_{M,J,q}(\\Vert{}\\eta\\Vert{}_2^{1/J})$. The paper remains restricted to the paired symmetric canonical-product model and makes no claim for arbitrary nonsymmetric real-rooted canonical products.