Theorem: Exact phase-integrity residual

For the arrays in Record rec:phase,

Thus the v4 regression assertion $a^{(0)}=-f^{\mathrm{reg}}$ is false at precisely Python index 5. There is no single scalar $\lambda$ with $a^{(0)}=\lambda f^{\mathrm{reg}}$.

If these arrays are encoded as the corresponding complex reports, the cross-artifact coefficient discrepancy is $4/3$. At common bound $B=32$, its half-diamond distance is $1/24$.

Proof. All entries cancel except the sixth, where $4/3+4/3=8/3$. The first entry forces $\lambda =-1$, which fails at the sixth. Under the convention $v=u/\mathrm{i}$, compare $\mathrm{i}{f}^{\mathrm{reg}}$ with $-\mathrm{i}{a}^{(0)}$; their difference is $(8\mathrm{i}/3)e_6$. Apply Theorem thm:error.