Triangular n Postmaster: From the Discrete Primitive to the Completed Source Frontier

CerebralGraphix.com

Folding the two axes of $s=\sigma+it$:

Define
$$ q=q(s)=s(1-s) =\sigma(1-\sigma)+t^2+it(1-2\sigma)$$
The centered identity is
$$
\boxed{q=\frac14-\left(s-\frac12\right)^2}
$$
This is the analytic continuation of the centered split in (TDP.5): the signed coordinate $2s-1$ is squared and the two sheets $s$ and $1-s$ fold to the same $q$. Three specializations should be kept side by side.

There is no omitted real-zero branch. For $0<\sigma<1$, the alternating Dirichlet eta function satisfies
$$
\eta(\sigma)=(1-2^{1-\sigma})\zeta(\sigma)>0
$$
Since $1-2^{1-\sigma}<0$, one has $\zeta(\sigma)<0$. Thus every nontrivial zero $\rho=\beta+i\gamma$ has $\gamma\ne0$, and
$$
q_\rho\in\mathbb R
\iff \gamma(1-2\beta)=0
\iff \beta=\frac12
$$
Therefore RH is exactly the assertion that all folded zero orbits land on the positive real $q$-axis.