Controlled dynamics across quaternionic sectors

Construct controlled dynamics that move coherently among distinct quaternionic subalgebras of the octonions while preserving a well-defined operational interpretation.

Background

Every associative quaternionic subalgebra of the octonions supports the quaternionic state, channel, Choi, and error-correction theory developed earlier in the paper. However, a process that moves amplitudes through several incompatible quaternionic sectors cannot be treated as a binary quaternionic channel, because associators affect sequential composition.

The open problem is therefore to construct and control dynamics that coherently connect different quaternionic sectors. Such a theory would need to specify how sector changes interact with nonassociative multiplication and how states, channels, and measurements remain operationally meaningful during the transition.

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