zeta_def
The zeta_def theorem equates the Recognition zeta function to the standard arithmetic zeta function that takes the constant value 1 on positive integers. Number theorists working inside the Recognition framework cite it when they need the basic identification before applying Dirichlet convolution or inversion. The proof is a one-line reflexivity that follows directly from the preceding abbreviation.
claimLet $Z$ be the Recognition zeta function. Then $Z$ coincides with the standard arithmetic zeta function, so $Z(n)=1$ for every positive integer $n$.
background
The module supplies thin wrappers around Mathlib arithmetic functions, beginning with the Möbius function and its square-free properties. The upstream abbreviation zeta defines the Recognition zeta as the constant function 1 on positive integers. The declaration also depends on the for structure from UniversalForcingSelfReference, which records the structural properties required for meta-realization certificates.
proof idea
The proof is a one-line term that applies reflexivity to the preceding zeta abbreviation.
why it matters in Recognition Science
The equality places the standard zeta function inside the Recognition number-theory layer and prepares the ground for Dirichlet algebra. No direct dependents appear yet, but the declaration supports later inversion formulas and connects to the forcing-chain landmarks by supplying a basic arithmetic tool consistent with the phi-ladder conventions.
scope and limits
- Does not establish any multiplicative or analytic properties of zeta.
- Does not address convergence of associated series.
- Does not link zeta to the Recognition mass formula or the alpha band.
- Does not invoke the Recognition Composition Law.
formal statement (Lean)
220@[simp] theorem zeta_def : zeta = ArithmeticFunction.zeta := rfl
proof body
Term-mode proof.
221
222/-- ζ(n) = 1 for n ≥ 1. -/