Proposition: Local isometry is insufficient

A local developing isometry, by itself, supplies neither:

1. a global Minkowski plane-wave or TT representation on $M$;

2. an intrinsic Newton inverse on a conical spatial slice;

3. a local implementation of a globally specified controlled unitary.

Proof. On the angular cover, changing $\varphi$ by $2\pi$ rotates the developed spatial point by $2\pi \alpha$. A generic function $e^{ik\cdot y}$ is not invariant under that rotation. Its pullback therefore does not descend to a single-valued function on the cone. The same issue applies to tensor modes, with the additional tensor rotation. A global mode representation needs compatible gluing data.

For the inverse statement, two smooth flat sources may agree on a developing neighborhood but differ outside it. Their Newton potentials on the neighborhood can differ by a nonconstant harmonic function. For example, a positive source bump outside the neighborhood produces a nonzero potential there. Consequently, restriction to the neighborhood does not determine the Newton response.

For the implementation statement, a unitary on an abstract global Hilbert space has no specified spatial coupling, intermediate dynamics, or accessible local algebra. Adding a time label supplies none of these missing data.