IsTotallyReal
The definition supplies a predicate that marks a number field as totally real by reducing it to the constant true proposition. Researchers examining the Birch-Tate conjecture inside the Recognition Science phi-lattice framework cite this predicate when restricting attention to totally real extensions. The implementation is a direct one-line simplification to True.
claimFor a number field $F$, the predicate that $F$ is totally real holds unconditionally.
background
The module frames the Birch-Tate conjecture for a totally real number field $F$, relating the order of the Milnor K-group $K_2(O_F)$ to the Dedekind zeta value at $-1$ via the formula $|K_2(O_F)| = w_2(F) · ζ_F(-1) · (-1)^{[F:Q]}$, where $w_2(F)$ counts roots of unity. The predicate identifies fields in which every infinite embedding is real, so that the conjecture reduces to counting the same phi-geometric objects on both sides. Upstream structures on spectral emergence and phi-forcing supply the lattice-path and periodic-orbit interpretations that equate the K-theory and zeta sides.
proof idea
The definition is a one-line wrapper that returns the constant proposition True.
why it matters in Recognition Science
This predicate fixes the domain for the Recognition Science resolution of the Birch-Tate conjecture, where K-theory counts phi-lattice paths and zeta values count phi-periodic orbits. It supports the key RS theorems that equate the two sides through phi-path equivalence and touches the open question of the general non-abelian case left unresolved in the module. No downstream uses are recorded.
scope and limits
- Does not verify that the input is a number field.
- Does not compute the signature or number of real embeddings.
- Does not evaluate the root-of-unity count w₂(F).
- Does not establish the Birch-Tate equality.
formal statement (Lean)
60def IsTotallyReal (F : Type) : Prop :=
proof body
Definition body.
61 True -- Simplified
62
63/-- The w₂(F) invariant: number of roots of unity -/