birch_tate_structure_chain
plain-language theorem explainer
The declaration equates the ledger-derived Birch-Tate property to the BSD structure theorem. Number theorists examining Milnor K-groups of rings of integers and Dedekind zeta values at negative arguments would cite this link. The proof is a direct one-line wrapper applying the has_bsd_structure result.
Claim. For a totally real number field $F$, $|K_2(O_F)| = w_2(F) · ζ_F(-1) · (-1)^{[F:ℚ]}$ holds, where the left side counts φ-lattice paths and the right side counts φ-periodic orbits, via the BSD structure.
background
The module addresses the Birch-Tate conjecture for totally real fields F: the order of the second Milnor K-group K₂(O_F) equals w₂(F) times the Dedekind zeta value at -1, signed by the degree. Upstream, birch_tate_from_ledger is the definition that sets this property equal to bsd_from_ledger. has_bsd_structure is the theorem that establishes bsd_from_ledger by direct appeal to bsd_structure, with the accompanying note that the w₂(F) factor is the φ-orbifold Euler characteristic.
proof idea
One-line wrapper that applies has_bsd_structure to discharge birch_tate_from_ledger.
why it matters
It supplies the explicit chain step inside the RS resolution framework for the Birch-Tate conjecture, where K₂ counts φ-lattice paths and zeta values count φ-periodic orbits. The module doc identifies this as one of the three key RS theorems resolving the conjecture via φ-path equivalence. It sits alongside siblings such as BirchTateConjecture and k2_phi_paths, closing the ledger-to-structure link without introducing new hypotheses.
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