Stage D: the conditional relational alternative

This is a first-order coefficient statement, not an all-orders nonlinear gauge theorem. The chosen baseline values $a_0,r_0$ do not separately prove physical gauge invariance.

Freeze $\sigma >0$ and ${\sigma }^{\prime }=(r_{\epsilon }/c_{\epsilon })\sigma \ne 0$. Consider the abstract preparation

Apply CNOT, first qubit controlling the second, then Hadamard on the first, then destructively read both. Report odd-parity outcomes as $+$ and $-$, and merge both even-parity outcomes into a retained failure $F$.

Let