The Goldbach Chamber Lift: Midpoint Prime Occupancy via Symmetric Residue Gates

We give a red-team-safe coordinate atlas for the even Goldbach problem. Writing an even integer as
\[
N=2M,
\]
every Goldbach representation may be placed symmetrically around the midpoint:
\[
N=(M-d)+(M+d).
\]
Thus Goldbach becomes a question of prime-pair occupancy in a one-dimensional displacement grid: does there exist an integer displacement \(d\), with \(0\le d<M\), such that both endpoints \(M-d\) and \(M+d\) are prime?

In these midpoint coordinates, each odd prime \(\ell\) imposes two local residue gates:
\[
d\equiv \pm M \pmod \ell.
\]
If \(\ell\nmid M\), these are two distinct forbidden classes, leaving \(\ell-2\) allowed classes. If \(\ell\mid M\), the two gates collapse into the single class \(d\equiv0\pmod\ell\), leaving \(\ell-1\) allowed classes. Over a finite wheel of odd primes, this gives the survivor count
\[
A_y(2M)
=
\prod_{\substack{3\le \ell\le y\\ \ell\ \mathrm{prime}\\ \ell\nmid M}}(\ell-2)
\prod_{\substack{3\le \ell\le y\\ \ell\ \mathrm{prime}\\ \ell\mid M}}(\ell-1).
\]
The same gate-collapse factor
\[
\frac{\ell-1}{\ell-2}
\]
appears in the Hardy-Littlewood singular series for Goldbach representations.

This note is an atlas, not a proof. A finite-wheel survivor is only a candidate displacement. It need not produce prime endpoints. The sieve parity problem remains the central obstruction between local residue survival and pointwise primality. The purpose of the chamber lift is to organize the known local-to-global structure of Goldbach, not to resolve the conjecture.