Intrinsic para-linear quantum-process theory

Define complete positivity, discard, Choi objects, and Stinespring dilation intrinsically for para-linear morphisms, thereby supplying the missing operational structures for para-linear octonionic Hilbert theory.

Background

The paper presents para-linear octonionic Hilbert and operator theory as a framework that retains genuinely octonionic amplitudes while modifying ordinary linearity and composition to accommodate nonassociativity. Although this framework provides para-linear operators, regular composition, corrected adjoints, and spectral tools, regular composition is generally nonassociative, so standard categorical constructions based on associative morphism composition cannot be applied directly.

The unresolved task is to develop the operational ingredients required for a complete quantum-process theory in this setting: a notion of complete positivity compatible with para-linear morphisms, a discard map, Choi representations, and Stinespring dilations. Establishing these structures would determine whether para-linear octonionic models can support an intrinsic channel and recovery formalism rather than merely an operator calculus.

References

Three concrete problems remain: defining complete positivity, discard, Choi objects, and Stinespring dilation intrinsically for para-linear morphisms; constructing controlled dynamics that move coherently among quaternionic sectors; and formulating recovery criteria for ternary bracketing defects or multi-time nonassociative processes.

— Operational Foundations for Quaternionic and Octonionic Quantum Models: Exact Quaternionic Channels and Error Correction, Para-Linear Operators, and Categorical Closure Boundaries Beyond Associativity  (2608.20259 - Montoya et al., 20 Aug 2026) in Section 6, Conclusion and open problems