Prime factorization encodes multisets, not ordered relational structures, so typed information with asymmetric or nested relational dependencies resists lossless encoding in a single natural number without exponential blowup.
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Nested relational dependencies(as used in logic and data structures)
Information connections that go multiple levels deep, like Russian nesting dolls—where some data depends on other data, which itself depends on yet more data.
Ordered relational structures(as used in logic and data theory)
Information organized in a specific sequence where the arrangement and connections between items matter. For example, a family tree where it matters who is a parent, who is a child, and in what order they appear.
Prime factorization(as used in the Fundamental Theorem of Arithmetic)
Breaking a number down into its prime number building blocks—for example, 12 breaks down into 2 × 2 × 3. Every number has exactly one way to do this.
Typed information(as used in computer science and logic)
Data that is organized into different categories or 'types,' where each piece is labeled by what kind of thing it is. Like labeling some data as 'names' and other data as 'ages.'