vonMangoldt_def
The declaration identifies the local von Mangoldt arithmetic function with Mathlib's standard implementation. Number theorists integrating prime distribution tools into the Recognition framework would reference it for compatibility with existing libraries. The proof proceeds by direct reflexivity on the abbreviation.
claimLet $Λ$ denote the von Mangoldt function. Then $Λ = Λ_{std}$, where $Λ_{std}(n) = log p$ whenever $n = p^k$ for prime $p$ and integer $k ≥ 1$.
background
The module supplies lightweight wrappers around Mathlib's arithmetic function library, beginning with the Möbius function and extending to the von Mangoldt function. The local theoretical setting keeps statements lightweight for later Dirichlet algebra layers. Upstream, the structure for from UniversalForcingSelfReference records meta-realization properties required for orbit and step coherence axioms.
proof idea
The proof is a one-line wrapper that applies reflexivity to equate the local von Mangoldt abbreviation to Mathlib's definition.
why it matters in Recognition Science
This equating declaration places the von Mangoldt function within the Recognition Science primes module, supporting arithmetic function use in prime theory. It connects to the foundational self-reference structures via the for meta-realization. No open questions are directly touched, and it has no immediate downstream applications listed.
scope and limits
- Does not prove any specific properties or identities for the von Mangoldt function.
- Does not connect to the phi-ladder or forcing chain steps T0-T8.
- Does not provide computational implementations or examples.
formal statement (Lean)
143@[simp] theorem vonMangoldt_def : vonMangoldt = ArithmeticFunction.vonMangoldt := rfl
proof body
Term-mode proof.
144
145/-- Λ(p) = log(p) for prime p. -/