45. Fitting inside a certified domain
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Once \(V\) is fixed, certificate-constrained fitting is a convex problem. Unknowns include the entries of \(\widetilde B_s\), \(A_s\), \(\widetilde z_0\) and \(\alpha_0\). The constraints
are linear. Entrywise confidence intervals and fixed-input regression tolerances add linear inequalities. Norm balls add convex constraints. Jointly optimizing \(V\) and \(A_s\) is different: the product \(VA_s\) is then bilinear. No general convexity is claimed for learning the domain and dynamics together.
Proposition 41: Certified correction and its error budget
For fixed \(V\), every feasible solution of the displayed constraints is an internally valid model. Suppose a center estimate \(\widehat B_s\) and a feasible corrected estimate \(\widetilde B_s\) satisfy
Then \(\|\widetilde B_s-B_s^*\|\leq2\epsilon_s\). The same conclusion holds for initial states in their specified norm. If the true model lies in these confidence sets and itself admits the fixed-domain certificate, the convex constraints are feasible. Feasibility alone does not establish that the true model preserves this domain.
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Proof
Validity is Proposition 39. The error bound is the triangle inequality through the center estimate. Under the final assumptions, insert the true operators, true initial state and their nonnegative witnesses into the constraints. They provide a feasible point. The converse fails because existence of one valid point does not say that every point in a confidence set, or its true member, has the same invariant domain. \(\square\)
This is why existential correction and robust certification must be reported separately. A corrected model can be a well-defined approximation even when no common true-model cone has been proved. The trajectory bounds below require both models to be valid on a common specified cone.
45.1 An explicit positivity repair in a simplicial cone
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When \(V\) is square and invertible, put \(A_s=V^{-1}\widehat B_sV\). Assume mass conservation has already been imposed. Let \(N_s\) be an entrywise positive reference instrument with the same per-action mass identities. For \(0\leq\eta\leq1\), the mixture
is nonnegative provided
An empty maximum is zero. If every negative entry has magnitude at most \(d\) and every reference entry is at least \(m_0>0\), the simpler sufficient choice is \(\eta=d/(m_0+d)\). These statements follow by solving each scalar inequality. Mass conservation is unchanged by convex mixing.
This is an explicit model modification, not coefficient clipping or an exact coordinate change. Relative to a true instrument on the same cone, the one-step error is at most \((1-\eta)\epsilon+2\eta\) in the summed base-norm metric defined next, when the original error was at most \(\epsilon\). The factor two follows because two valid instruments each distribute unit total mass. The reference example retains the repair amount as part of its approximation record.