Entropic rigidity is not a universal geometry theorem

In finite dimension, the exact normal form gives

The additive syndrome entropy is independent of the logical input. This follows directly from tensor-product entropy and isometric invariance.

Conversely, a selected entropy does not locate the code. For

both radiation entropies vanish on every pure code input. Their radiation recovery errors are $0$ and $a_d$. An entropy identity alone is not the complete complementary-channel criterion.

Cao's area-operator result concerns stabilizer codes with the specified complementary operator-algebra structure, not arbitrary Clifford channels Cao2024. The skewed-code analysis of Cao and collaborators relates leading proto-area entropy corrections to a specially defined perturbative tripartite Choi magic under its subsystem and perturbation assumptions CaoEtAl2026. Neither statement equates generic complete residual non-constancy with geometric backreaction.

The syndrome counterexample in Theorem thm:NLcounterexample has bipartite magic but no logical response. It therefore does not contradict a theorem whose resource is a more restricted matter–geometry tripartite coupling. It demonstrates why those restrictions matter.