Purification and the Structure of Operational Theories: Unrestricted Effects, Quantum Reconstruction, and Complete Composition Classification

Abstract

In a finite-dimensional causal convex operational theory with closed physical effect sets and essentially unique purification, we derive pure products, general purity preservation and effect non-restriction. Exact binary steering follows from an informative noisy pointer; support order intervals, pure dilations and probabilistic returns then force unrestricted effects without local tomography, ideal compression, product domination or a bound on purifying partners. Finite conditioned control is also derived. Spectrality, pure support projections and symmetric transitions yield a Jordan representation. For an essentially small catalogue, we then classify compositions without local tomography. Exact pure-ancilla faces restrict simple factors to a common real or complex matrix field. Sector translations embed in a locally finite abelian group. Complete event representations and simultaneous phase normalizations identify the full sequential and symmetric monoidal structure. At fixed field, equivalence classes are determined by the effective charge group, the actual product-closed object catalogue and a parity character. The proof retains all rectangular event spaces and does not add charge objects or compressions to an incomplete catalogue. Local tomography selects the ordinary complex quantum case on the available multiplicative catalogue. Exact one-way teleportation supplies an operational comparison. In the specified protocol class, one state is shared on two copies of the input system, without an additional correlated record. Exact preservation of all reference correlations forces a pure balanced resource and reversible corrections on nonzero outcomes. A system with three rank-one sectors and Z3 fusion has exact one-copy outcome minima twelve over the reals and nine over the complexes, with corresponding message entropies, although both alphabets fit in four binary digits. Joint protocols have the same regularized rate. Alternative geometric proofs, restricted-model obstructions, optimal physical returns, a quantum-pointer application and an open-effect counter-theory delimit the assumptions and conclusions. Preparation note: This preprint builds on Yoann Dumont’s research programme, earlier manuscript, and research notes. ChatGPT assisted with proof development and checking, comparison with these prior materials, bibliographic searches, English drafting, and typesetting. AI-assisted checks do not constitute independent peer review. As author, Yoann Dumont takes responsibility for the claims, references, and any necessary corrections to the published text.