85. A finite uncertainty box and the role of convexity
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A robust witness should identify its uncertainty family rather than supply an unexplained tolerance. Suppose the transformed error has polynomial coefficients
All bounds are in the submitted Lorentz coordinates. For three coordinates, normalized inputs have each coordinate of magnitude at most one.
Proposition 76: Coefficient boxes and convex uncertainty
The coefficient box satisfies the output bounds of Proposition 71 with
Only coefficient arrays obeying \(\lambda\int\Delta_a(u)\,du/2=0\) are included in the normalized uncertainty family. A checked margin at least \(e_0+e_1\) certifies every such array.
Alternatively, if finitely many complete polynomial operator families satisfy a common margin \(\kappa\) on the same domain, every convex combination of those families satisfies that margin. Initial states may also range over a certified convex hull.
For a fixed geometry, the constraints \(\sigma_a(u)\geq\kappa\), exact mass, and membership in a convex data-consistency set are convex constraints on the operator coefficients and initial state. Joint optimization of geometry and operators is not asserted to be convex.
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Proof
Each output scalar is bounded by \(3r_k\) on a normalized input. Bound the spatial Euclidean norm by the sum of the absolute values of its two coordinates, obtaining \(6r_k\). Sum over coefficients using \(|u|\leq1\). Mass conservation is a separate linear constraint and follows for every family satisfying it.
For convex combinations, output mass is affine and the spatial norm is convex. Hence the margin of the combined output is at least the same combination of margins. A convex cone contains convex combinations of admissible initial states. For fixed geometry and input, the inequality affine mass minus a norm at least \(\kappa\) is convex in the coefficients. Intersect over all inputs and observations, then add the linear mass constraints and the convex data set. \(\square\)
The squared polynomial witness in Proposition 75 is a convenient verification language. Its coefficient constraints are not automatically a convex optimization formulation in unknown operators. Convexity of the underlying admissibility set must not be confused with convexity of every certificate parameterization.
Likewise, a free uncertainty box in all entries contains families that violate integrated mass. The affine restriction is essential. Removing physically valid candidates from a statistical confidence set for some other reason requires its own coverage argument.