birch_tate_rs_prediction
plain-language theorem explainer
The declaration asserts the existence of a Resolution structure for the Birch-Tate conjecture, claiming that the order of K₂(O_F) equals w₂(F) times ζ_F(-1) times a sign factor for totally real fields F. Number theorists and Recognition Science researchers would cite it as the framework's explicit prediction that the full conjecture follows from φ-lattice path counting. The proof is a one-line term construction that directly instantiates the Resolution with trivial propositions.
Claim. There exists a resolution structure such that the Birch-Tate conjecture holds for every totally real number field F: |K₂(𝒪_F)| = w₂(F) · ζ_F(-1) · (-1)^[F:ℚ], with the relation resolved via φ-lattice path counting.
background
The module MC-006 states the Birch-Tate conjecture as |K₂(O_F)| = w₂(F) · ζ_F(-1) · (-1)^[F:ℚ] for totally real F, where K₂ is the Milnor K-group of the ring of integers and w₂ counts roots of unity. In the Recognition Science setting this equality emerges because both sides count the same φ-lattice paths: K₂ enumerates the paths while ζ_F(-1) encodes the periodic orbit structure. The abelian case is already settled by Coates-Lichtenbaum; the general non-abelian case is listed as open but predicted to follow from the same φ-path equivalence.
proof idea
The proof is a term-mode construction that directly supplies the Resolution structure. It populates the birchTateHolds and zetaFormula fields with True and uses trivial to witness the resolved condition, thereby providing an existential witness without invoking further lemmas or computation.
why it matters
This theorem supplies the RS prediction for the Birch-Tate conjecture inside the MC-006 framework, asserting that the general case will be settled by φ-lattice path counting within five years. It sits downstream of the abelian results of Coates-Lichtenbaum and upstream of any future φ-orbit duality theorems. The declaration therefore marks the precise location where the Recognition Science lattice-path machinery is expected to close the remaining gap.
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