A no-global-diagonal framework links logical diagonalization to quantum no-cloning and no-broadcasting. In causal monoidal (non-cartesian) process theories, a natural global diagonal would implement uniform broadcasting; with non-classical sectors and no-signalling, this structure is obstructed. In Hilb, the obstruction appears as the absence of nonzero natural copy maps, equivalently the no-cloning/no-broadcasting constraints.Locally, basis-dependent Frobenius copy maps and decoherence define classical slices: an idempotent comonad whose Eilenberg--Moore coalgebras capture context-dependent classical systems in a causal wedge. In AQFT, these local diagonals are causally confined and assemble in a sheaf-like local-to-global pattern that can glue on overlaps without yielding a global copier. Under explicit covariance and nonfactorization assumptions, a conditional No-Coaction Principle (no natural globally copyable label) yields a kinematic UV filter excluding preferred-frame/foliation-type backgrounds. This positions no-global-diagonal as an organizing principle across Lawvere-style self-reference, quantum no-go theorems, and relativistic locality, and suggests an operational notion of emergent classicality structure from local gluing data.
Lorand Bruhacs (Thu,) studied this question.