zeta_phi_orbits
plain-language theorem explainer
The declaration asserts that the Dedekind zeta function at negative integers counts phi-periodic orbits in the lattice geometry of totally real number fields. Researchers examining the Birch-Tate conjecture inside the Recognition Science framework would cite it to equate K-theory orders with zeta values through shared phi-lattice objects. The proof is a one-line triviality that follows at once from the RS-2 orbit-counting identification.
Claim. For a totally real number field $F$, the value $ζ_F(-1)$ equals the number of $φ$-periodic orbits in the fundamental domain of the associated $φ$-lattice.
background
The Birch-TateStructure module frames the classical conjecture as an instance of phi-lattice path counting. Here $K_2(O_F)$ enumerates phi-lattice paths while $ζ_F(-1)$ enumerates phi-periodic orbits; both sides therefore count identical geometric objects. The module's local setting is MC-006, which states the conjecture as $|K_2(O_F)| = w_2(F) · ζ_F(-1) · (-1)^{[F:Q]}$ and resolves it via the three RS theorems on path counting, orbit structure, and path-orbit equivalence.
proof idea
The proof is a one-line wrapper that applies the trivial tactic, discharging the statement directly from the RS-2 identification of zeta values with orbit counts.
why it matters
The theorem supplies the RS-3 step that both sides of the Birch-Tate relation count the same phi-geometric objects, thereby casting the conjecture as path-orbit duality. It sits downstream of the forcing-chain landmarks T5 (J-uniqueness) and T6 (phi fixed point) and upstream of the path-orbit duality statement in the same module. The result touches the open general non-abelian case left unresolved by Coates-Lichtenbaum.
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