Decidability of the disruption frontier

Determine whether, for a given total computable transformation \(\Phi\) and a specified non-trivial behavioural property \(\widetilde P\), it is decidable that \(\Phi\) is uniformly disruptive with respect to \(\widetilde P\), and, if not, prove that the frontier between disruptive and non-disruptive transformations is undecidable.

Background

The paper defines uniform disruption as a syntactic property of a total computable transformation: a computable family of wrappers must preserve a fixed semantic witness while the transformed wrappers encode the halting status of an arbitrary computation. It proves that every transformation in the syntactic-instrumentation class $R(P)$ is uniformly disruptive.

The converse characterization is unresolved. Specifically, the paper asks whether one can decide from an index for $\Phi$ whether it is uniformly disruptive relative to a given $\widetilde P$, and conjectures that this recognition problem is itself undecidable, plausibly by reduction from the halting set.