The two $k=5$ cyclotomic blocks are coprime when $p \equiv 1 \pmod 3$

The two non-linear cyclotomic blocks that appear in the $k=5$ Dris equation are

together with the linear factor $(p+1)/2$. In the two-prime square-free-index case one has

so the odd-multiplicity prime support of every block must be contained in $\{q, r\}$. The counting theorem used to make that inference, two_prime_block_is_prime_mul_sq, requires the two blocks to be coprime, and the first-block instance of that coprimality has already been proved: $\gcd\bigl(C, \tfrac{p+1}{2} D\bigr) = 1$ for every odd prime $p$.

The second-block instance is subtler. Writing $p = 3k+2$ gives

and the bracket is $1 \bmod 3$, so $v_3(D) = 1$; simultaneously $3 \mid p+1$, so $3 \mid \tfrac{p+1}{2}$. Hence the factor $3$ is common to $D$ and to $\tfrac{p+1}{2} C$, and indeed one has $3 \mid \gcd\bigl(\tfrac{p+1}{2} C,\, D\bigr)$ precisely when $p \equiv 2 \pmod 3$; numerically over all primes below $4000$ the gcd is $3$ for every $p \equiv 2 \pmod 3$ and $1$ otherwise. So the unconditional second-block coprimality is false, and the hypothesis $p \equiv 1 \pmod 3$ is exactly what removes the obstruction.

Under that hypothesis $3 \nmid p+1$, hence $3 \nmid \tfrac{p+1}{2}$ and $3 \nmid D$; and every other prime dividing both blocks is ruled out by the same short Euclidean argument that proves the first-block case.

Consequence. This supplies the coprimality hypothesis that the second-block split five_two_prime_cyclotomic_split_second (target 793eb5c3-69c0-425d-b3b1-7f4719cfa0f1) needs in the $p \equiv 1 \pmod 3$ case. In the $p \equiv 2 \pmod 3$ case the factor $3$ must instead be handled by valuation, using the children three_val_second_cyclotomic and three_kernel_prime_when_p_two_mod_three.