Theorem: IX-E-3-ST stage release

Part IV proves exact support of $W_g$, surface non-cancellation, the null-sphere lower-support theorem and two-way bridge under (TC), and verification of (TC) for the explicitly specified flat realization. It also disproves the unnecessary solid-ball lower-support assertion in that realization.

For the weak inherited operator contract, verification of (TC) remains Unresolved. If the same-adjoint identification is supplied, the commutator bridge is proved; the overall causal-product release still remains Unresolved.

Proof. The positive claims are proved in Sections sec:surface–sec:flat. Proposition prop:sameadjoint gives the two version-consistent bridge verdicts. Neither branch supplies the analytical construction required in Requirement req:analytic.