Probability Functor Models

This work introduces the probability functor model, a unified framework for combining deterministic and probabilistic computation within a single structure. The model represents outputs as a pair consisting of a probability distribution and an associated uncertainty measure.

Deterministic computation is included as a special case with zero uncertainty, while probabilistic behavior is explicitly modeled in regions where uncertainty is unavoidable. A routing mechanism determines whether evaluation proceeds through deterministic or probabilistic components.

Within this framework, error is precisely characterized: it is zero in regions where exact computation is available and corresponds directly to the intrinsic uncertainty of the probabilistic model elsewhere. This leads to an uncertainty localization property, ensuring that uncertainty is confined to explicitly defined parts of the system and does not propagate implicitly. 

The formulation provides a consistent mathematical foundation for hybrid systems that integrate exact computation and probabilistic reasoning, with clear control over uncertainty, error attribution, and model behavior. It is particularly relevant for applications requiring transparency, auditability, and precise handling of uncertainty. Patent pending.