birch_tate_abelian_proven
plain-language theorem explainer
The abelian case of the Birch-Tate conjecture holds for extensions of the rationals by the Coates-Lichtenbaum theorem. Number theorists working on Milnor K-theory and Dedekind zeta values at negative integers cite this result to isolate the settled abelian subcase. The argument reduces immediately to the classical theorem via a direct term.
Claim. For a totally real abelian extension $F$ of $Q$, $|K_2(O_F)| = w_2(F) · ζ_F(-1) · (-1)^{[F:Q]}$.
background
The Birch-Tate conjecture equates the order of the Milnor K-group K₂ of the ring of integers of a totally real field F with w₂(F) times the Dedekind zeta value at -1, signed by the degree. This module embeds the conjecture in Recognition Science by interpreting both sides as counts of φ-lattice paths and φ-periodic orbits. The abelian case over Q is already settled by Coates-Lichtenbaum, while the general non-abelian case is framed as open and expected to follow from φ-path equivalence.
proof idea
The proof is a one-line term that directly invokes the known Coates-Lichtenbaum theorem for abelian extensions.
why it matters
This declaration marks the settled abelian case inside the Recognition Science resolution of Birch-Tate, which predicts a full proof via φ-lattice path counting within five years. It separates the classical result from the open general case that the module links to zeta values as φ-periodic orbits. The parent framework remains the RS resolution via φ-path equivalence stated in the module.
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