Every Odd Number Greater Than 1 is the Sum of at Most 100001 Primes

Every odd natural number greater than $1$ is a sum of at most $100\,001$ primes, with repetition allowed.

Precisely: for every $n \in \mathbb{N}$ with $n$ odd and $n > 1$ there is a finite multiset $s$ of natural numbers such that

Here $|s|$ counts elements with multiplicity, so the same prime may be used several times, and the order of the summands is irrelevant.

This is the campaign statement of Odd numbers as sums of primes with the value $100\,001$. The constant comes from Schnirelmann's elementary sieve-and-density method with all constants made explicit, which avoids the prime number theorem.

Formalization Note The representation is a Multiset ℕ; the bound is on Multiset.card, so repeated primes count separately.