Reciprocity laws and K-theory

We associate to a full flag F in an n-dimensional variety X over a field k, a "symbol map" μ_F : K(F_X)→Σ^n K(k). Here, F_X is the field of rational functions on X, and K(⋅) is the K-theory spectrum. We prove a "reciprocity law" for these symbols: Given a partial flag, the sum of all symbols of full flags refining it is 0. Examining this result on the level of K-groups, we re-obtain various "reciprocity laws". Namely, when X is a smooth complete curve, we obtain degree of a principal divisor is zero, Weil reciprocity, Residue theorem, Contou-Carrère reciprocity. When X is higher-dimensional, we obtain Parshin reciprocity.