Conjectured algebraic laws for homotopy cardinality of types

Establish that the homotopy cardinality function |·|: U → R, where R is a non-standard model of the real numbers and |X| is defined recursively by |X| = ∑_{x∈X} 1/|x=x| for any type X in homotopy type theory, satisfies the following equalities for all types X, Y: |0| = 0, |1| = 1, |X + Y| = |X| + |Y|, |X × Y| = |X||Y|, and |X → Y| = |Y|^{|X|}; and for any dependent type P: X → U, |Σ_{x:X} P_x| = ∑_{x∈X} |P_x| / |x=x| and |Π_{x:X} P_x| = ∏_{x∈X} |P_x|^{1/|x=x|}.

Background

The paper postulates a homotopy cardinality |·|: U → R for types in homotopy type theory, defined via a recursive formula that sums over terms x of a type X with weights determined by the cardinality of the identity type (x = x). The target R is specified to be a non-standard real model to accommodate infinite or infinitesimal cardinalities.

Motivated by analogies with classical set cardinality and groupoid cardinality, the author conjectures a suite of algebraic rules describing how this cardinality interacts with standard type constructors (sum, product, function types) and with dependent sums and products. The text notes that some of these properties are easy to show in simple cases, while the dependent sum and product formulas are justified heuristically, leaving a rigorous proof of the full collection of properties as an open task.