Schanuel Integration and Euler Characteristic of Semi-algebraic Sets

We extend the Schanuel integration framework, originally introduced for finite unions of convex sets, to arbitrary semi-algebraic sets. We prove that the resulting Schanuel integral of indicator functions is independent of the choice of ordered linear bases and therefore defines a well-defined Euler characteristic in the semi-algebraic category. We further show that this Schanuel--Euler characteristic coincides with the classical Euler characteristic defined via Borel--Moore homology and cylindrical algebraic decomposition. The recursive fiberwise structure of Schanuel integration provides an elementary and geometric interpretation of Euler characteristic and yields simplified proofs of several classical properties, including invariance under semi-algebraic isomorphisms.