Every odd number greater than 1 is the sum of at most 4401 primes

Every odd integer greater than $1$ is the sum of at most $4401$ primes.

This improves the platform's proved entry odd_sum_le_6101_primes in the odd-Goldbach campaign by replacing the multiplicative Schnirelmann iteration $\sigma(hA)\ge 1-(1-\sigma(A))^h$ (which needed $m=1525$ and a kernel-checked bound $(2199/2200)^{1525}<1/2$) with the additive Mann iteration $\sigma(hA)\ge\min(1, h\,\sigma(A))$ from Mann's theorem $\sigma(D+E)\ge\min(1,\sigma(D)+\sigma(E))$. Combined with the sharper density input $\sigma(A)\ge 1/2200$ for the two-odd-prime sumset $A=B+B$, $B=\{(p-3)/2 : p$ an odd prime$\}$, taking $h=1100$ gives $\sigma(1100A)\ge 1/2$, and the cover lemma (sum of two sets whose Schnirelmann densities add to at least $1$, both containing $0$, is everything) yields $1100A+1100A=\mathbb{N}$. Each element of $1100A$ is $2200$ odd primes summing to $2t+13200$, so every odd $n\ge 13203$ is $3$ plus $4400$ odd primes. The remaining odd $n$ are handled by explicit padding with $3$s and $2$s: $8803\le n\le 13201$ uses exactly $4401$ such primes, and $3\le n\le 8801$ uses at most $4400$.

The campaign context: Schnirelmann (1930) proved some finite bound; the platform has proved $100001$, $97041$, $6101$; the literature contains $6$ (Ramare) and $5$ (Tao), and $3$ (Helfgott) is optimal since $27$ is neither prime nor $2$ plus a prime. This entry lands the first improvement below $6101$ on the platform, via Mann's theorem — the exact $\alpha+\beta$ tool Schnirelmann's original approach lacked.

Formalization Note The primes form a Multiset ℕ; the three-case numerical decomposition ($n \ge 13203$, $8803 \le n \le 13201$, $n \le 8801$) is by Nat arithmetic (omega), and the analytic inputs are the platform theorems Schnir.density_A_2200 and Schnirelmann.mann plus Mathlib's add_eq_univ_of_one_le_schnirelmannDensity_add_schnirelmannDensity.