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We lay the groundwork in this first installment of a series of papers aimed at developing a theory of Hrushovski–Kazhdan style motivic integration for certain types of nonarchimedean 
$o$-minimal fields, namely power-bounded 
$T$-convex valued fields, and closely related structures. The main result of the present paper is a canonical homomorphism between the Grothendieck semirings of certain categories of definable sets that are associated with the 
$\text{VF}$-sort and the 
$\text{RV}$-sort of the language 
${\mathcal{L}}_{T\text{RV}}$. Many aspects of this homomorphism can be described explicitly. Since these categories do not carry volume forms, the formal groupification of the said homomorphism is understood as a universal additive invariant or a generalized Euler characteristic. It admits not just one, but two specializations to 
$\unicode[STIX]{x2124}$. The overall structure of the construction is modeled on that of the original Hrushovski–Kazhdan construction.
