chen_prime_three
The declaration delivers the assertion that both 3 and 5 are prime numbers, confirming the Chen prime pair with 3+2=5. Number theorists working on arithmetic functions and Möbius applications in the Recognition Science number theory layer would cite this for small-case verification. The proof goes through by direct computational decision via native_decide on the decidable conjunction.
claim$3$ and $5$ are both prime numbers.
background
The module supplies lightweight wrappers around Mathlib arithmetic functions, beginning with the Möbius function μ. Prime is the repo-local alias for the standard natural-number primality predicate. Upstream results supply the Prime definition as an alias for Nat.Prime together with structural is statements from foundation modules that frame empirical and combinatorial checks.
proof idea
The proof is a term-mode one-line wrapper that applies native_decide to evaluate the decidable proposition Prime 3 ∧ Prime 5.
why it matters in Recognition Science
This supplies a basic prime fact inside the arithmetic functions module that supports later Möbius and prime-counting steps in the Recognition framework. It aligns with the number-theory footholds required for constants such as the alpha band and phi-ladder mass formulas, though no downstream uses are recorded.
scope and limits
- Does not establish general properties of Chen primes or twin-prime conjectures.
- Does not invoke the Recognition Composition Law or the J-uniqueness forcing step.
- Does not address spatial dimension forcing or the eight-tick octave.
- Does not connect to mass formulas or Berry creation thresholds.
formal statement (Lean)
2280theorem chen_prime_three : Prime 3 ∧ Prime 5 := by native_decide
proof body
Term-mode proof.
2281
2282/-- 5 is a Chen prime: 5+2 = 7 is prime. -/