Partially Observable Infinite-Dimensional Vectors (POIDs)

[For a technical audience. Assumes undergraduate-level knowledge of Pure Mathematics. Also available paywall-free on Substack.]

What is a POID?

By infinite-dimensional, we refer to vectors defined over an unbounded — or effectively uncountable — set of dimensions, rendering explicit enumeration of all components impractical.

By partially observable, we mean that each individual observation reveals only a subset of the vector’s dimensions.

Properties of POIDs

1. Comparison between two POIDs is undefined; mathematical notions such as equality and ordering do not apply.
2. Conventional binary logic is not applicable to set operations involving POIDs. A given POID may simultaneously belong to both a set A and its complement ¬A (“not A”), or to neither.
3. Observations are non-deterministic and non-repeatable. Subsequent observations of the same POID may reveal different subsets of dimensions, and there is no guarantee of overlap or consistency across observations.
4. Aggregation across observations is non-trivial. Constructing a fuller representation of a POID from multiple observations is inherently uncertain.
5. Metric spaces are ill-defined. Standard notions of distance or similarity cannot be meaningfully applied to POIDs, as any given comparison is based on incomplete and potentially non-overlapping…