Two improved lower bounds for the 2-adic valuation of the constant of the Prouhet-Tarry-Escott problem

For an ideal solution of the Prouhet–Tarry–Escott problem of size $n$ the associated constant $C$ is divisible by a large power of $2$. Restricting to symmetric solutions, we prove two new lower bounds for $v_2(C)$: $v_2(C)\ge14$ at $n=15$, improving $12$, and $v_2(C)\ge19$ at $n=20$, improving $18$; both earlier bounds are those of the companion article hexagon:2611.00057. The first rests on two identities satisfied by every symmetric solution of odd size, one relating $v_2(C)$ to a difference of two products and one turning the index relation into an equality, together with an arithmetic criterion for a block of entries of equal valuation. The second rests on an identity internal to each of the two sides, from which we deduce that the vanishing of the cross invariant $R$ forces $k_0\equiv0\pmod4$ when $k_0\ge4$. Two self-contained verifiers accompany the note.