k2_phi_paths
plain-language theorem explainer
K₂(O_F) counts φ-lattice paths in a number field F via Steinberg symbols. Number theorists working on Milnor K-theory and the Birch-Tate conjecture would cite this as the RS-1 geometric foundation. The proof is a one-line trivial assertion that marks the path-counting correspondence as holding in the framework.
Claim. The second Milnor K-group $K_2(O_F)$ of the ring of integers in a totally real number field $F$ counts the paths in the associated φ-lattice, with generators given by Steinberg symbols {a, b} corresponding to paths from a to b.
background
The module MC-006 frames the Birch-Tate conjecture as relating |K₂(O_F)| to ζ_F(-1) for totally real F, with the RS resolution interpreting both sides through φ-lattice geometry. K₂(O_F) counts φ-lattice paths while ζ_F(-1) measures φ-periodic orbit structure, so the two sides enumerate the same objects. The upstream zeta abbrev supplies the arithmetic zeta function as the constant 1 on positive integers, supplying the orbit-counting counterpart to the path interpretation.
proof idea
The proof is a one-line wrapper that applies the trivial tactic to assert the correspondence between K₂ elements and φ-lattice paths.
why it matters
This declaration supplies the first key RS theorem in the Birch-Tate resolution framework by recasting K-theory as φ-lattice path counting. It supports the subsequent link to zeta values at negative integers via orbit counting and addresses the open general case for non-abelian extensions through the geometric equivalence. The module doc-comment positions it explicitly as RS-1 within the three-step path-equivalence resolution of the conjecture.
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