five_almost_prime_onehundredtwelve
plain-language theorem explainer
112 factors as 2^4 times 7 and therefore satisfies the 5-almost-prime condition under the big-Omega counting function. Number theorists extending Recognition Science arithmetic tools would reference the result when checking concrete cases for the mass formulas. Native decide evaluates the predicate by direct computation on the factored form.
Claim. $Omega(112) = 5$, where $Omega(n)$ counts the prime factors of $n$ with multiplicity.
background
The ArithmeticFunctions module supplies lightweight wrappers around Mathlib arithmetic functions, with the key definition isFiveAlmostPrime n holding exactly when bigOmega n equals 5. The module doc states that statements remain lightweight pending deeper Dirichlet algebra. Upstream results include the explicit definition of isFiveAlmostPrime together with foundational is predicates that certify algebraic tautologies and collision-free structures without new axioms.
proof idea
One-line wrapper that applies native_decide to the definition isFiveAlmostPrime 112, reducing directly to the Boolean check bigOmega 112 = 5.
why it matters
The result supplies a concrete 5-almost-prime instance available for use in Recognition Science number-theoretic constructions such as phi-ladder mass formulas. It aligns with the framework's pattern of explicit small-integer verifications inside the arithmetic functions layer. No downstream uses are recorded, leaving open its integration into larger theorems on prime distributions or forcing steps.
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