Constrained Polynomial Probabilistic Zonotopes: When Can One Compressed Model Serve Many Tasks?

When can one compressed probability model meet several prediction requirements simultaneously? We introduce Constrained Polynomial Probabilistic Zonotopes (CPPZs), which combine polynomial constraint geometry, prescribed intrinsic probability laws, and explicit observation semantics. We study shared finite-coordinate continuous representations for declared tasks under worst-case squared prediction risk. For a class of elliptic task families of common order with scalar principal symbols, task-mixture constraints exactly characterize the jointly attainable leading-order risk region, without a common eigenbasis. For the stated nonlinear CPPZ family with smooth, strictly convex geometry, requiring one shared reconstruction with its prescribed intrinsic law preserves the optimal leading-order risk region attainable by unrestricted task-specific decoders of shared codes. This holds for each fixed finite task family differing only in context distribution and every fixed source radius strictly inside the geometric validity domain; each target admits a shared code. The construction separates coarse geometry from residual information and compiles recovery into a valid polynomial probability object. Within the same family, one geometry-dependent measurement update preserves the leading performance of deterministic adaptive linear acquisition, with sublinear additional queries. For two compact linear tasks on a real Hilbert source ball, continuous, linear, and deterministic adaptive-linear encoding have identical attainable risk regions at every coordinate budget, without ellipticity assumptions. Yet complete pairwise information does not determine collective feasibility: two CPPZ task triples with a common finite-context detector can have identical pairwise risk regions at every budget while differing on whether one code meets all three tasks' requirements at the same budget.