primeCounting_fivehundred
The prime counting function satisfies π(500) = 95. Number theorists verifying small explicit values of π(n) or cross-checking computational libraries would cite this result. The proof is a one-line native_decide reduction that evaluates the built-in prime counting implementation directly.
claimThe prime counting function satisfies $π(500) = 95$.
background
The prime counting function π(n) is defined as the number of primes ≤ n. In the ArithmeticFunctions module it appears as the lightweight wrapper primeCounting n := Nat.primeCounting n, importing the standard implementation from Mathlib. The module itself supplies Möbius-function footholds and related arithmetic functions for later Dirichlet inversion work.
proof idea
The proof is a one-line wrapper that applies native_decide to the equality primeCounting 500 = 95, reducing it to direct evaluation of the underlying Nat.primeCounting call.
why it matters in Recognition Science
This supplies a verified concrete value for π(500) inside the arithmetic-functions layer. It anchors small-n checks in the NumberTheory.Primes hierarchy but currently has no downstream uses. Within Recognition Science it provides a computational checkpoint for prime-related quantities, though it does not yet interface with the J-cost, phi-ladder, or T0-T8 forcing chain.
scope and limits
- Does not derive the prime counting function from first principles.
- Does not compute π(n) for any n other than 500.
- Does not connect to Möbius inversion or other arithmetic functions in the same module.
formal statement (Lean)
896theorem primeCounting_fivehundred : primeCounting 500 = 95 := by native_decide
proof body
Term-mode proof.
897
898/-- π(750) = 132. -/