primeCounting_fivehundred
plain-language theorem explainer
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.
Claim. The 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
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.
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