46. A norm that measures probability mass before conditioning
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Assume \(C\) is a closed, pointed, generating cone and \(\lambda\) is strictly positive on its nonzero elements. Its base \(K\) is compact. Define the base norm
The infimum is finite because the cone is generating. Compactness of \(K\) makes this a genuine norm. Its unit ball is \(\operatorname{conv}(K\cup(-K))\). In particular \(\|u\|_C=\lambda u\) on \(C\), and \(|\lambda x|\leq\|x\|_C\). On the ordinary probability orthant this is the \(\ell^1\) norm. These are standard ordered-space constructions [R15], used here to avoid mixing Euclidean coefficient magnitude with probability mass.
Proposition 42: Instrument contraction before normalization
If \(\{B_{a,o}\}_o\) is cone preserving and mass conserving for a fixed action, then
For a polyhedral base \(K=\operatorname{conv}\{v_j\}\), define the discrepancy between two instruments by
The supremum equals the maximum of the same expression at the finitely many generators \(v_j\). When both instruments are valid on \(C\), \(\epsilon_a\leq2\).
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Proof
For any \(x=u-v\) with \(u,v\in C\), cone preservation gives
Take the infimum. For the generator statement, the discrepancy is convex in \(z\). At a convex combination its value is at most the same combination of generator values, hence at most their maximum. The generators themselves belong to \(K\), proving equality. Finally, for \(z\in K\), the triangle inequality bounds the sum by the total masses of both instruments, which are one each. \(\square\)
For a polyhedral certificate, the norm itself has a finite linear-program formulation:
If \(V\) is square and invertible, this reduces to \(\|V^{-1}x\|_1\). The exact worked interval certificate is of this type. With dependent generators, choosing one pseudoinverse does not in general compute the base norm.