Quasi-Local Horizon Predicates Without a Global Teleologist: Certification-Structure Classification of the Horizon Predicate, the Quasi-Local Typology, and the Teleology-Dependence Kill

A century of relativity has left a quiet asymmetry in how we speak of black-hole horizons. The event horizon — the boundary of the region from which nothing ever escapes to infinity — is a global, teleological object: to know where it is now, one must know the entire future of the spacetime, so a late infall of matter can make the horizon form and grow through a region that is, at the moment, perfectly flat. An earlier installment of this series priced that fact honestly, typing the event-horizon predicate as adopted classification data rather than a fact an embedded observer could certify from local data, and naming, but not building, a constructive successor: the quasi-local horizon notions — trapped surfaces, marginally outer trapped surfaces, trapping, isolated, and dynamical horizons — that are defined from data on a slice rather than from the whole future. This paper builds the successor at exactly one grade: it does not certify any horizon; it classifies the certification structure of each horizon predicate — whether its truth is, in principle, a function of patch-side data, or whether it irreducibly requires the global future. The authority it retires is the Global Teleologist: not the observer at infinity, but the operating assumption that horizon-hood can be certified only from the global future. The method is disciplined. A patch-side tube class is declared from local geometry alone — a foliation, a candidate closed two-surface, its null expansions, the accessible metric, a bounded search region — with future null infinity and the global-future development explicitly barred, so that horizon-hood is an output of the classification, never a smuggled input. Over a finite, declared language of horizon predicates, each read both as an evaluation on a given surface and as an existence claim over the accessible region, a two-part exterior-deformation test asks whether the predicate's truth is unmoved when the spacetime is changed only beyond the accessible boundary. The event-horizon predicate visibly moves — exhibited by a matched pair of spacetimes that agree on all accessible data and differ only in a later infalling shell — and is typed as teleological. The one-slice quasi-local predicates do not move and are classified as patch-local in structure. The tube-level predicates are classified only conditionally, under a slicing-selector that must itself be shown patch-local by a finite, decidable grammar; where the slicing can be chosen only by appeal to infinity, the predicate is local but its certification is not, and the classification records an honest kill. The verdict is deliberately modest: scoped. At least one genuinely quasi-local predicate is patch-local in its certification structure, so the Teleologist is retired on it; but the event-horizon predicate remains teleological, the physics that would actually certify a horizon in a real strong-field spacetime is left open at its own seat, and full certification is structurally out of reach at classification grade. Forgetting the machinery returns the earlier installment's classification unchanged — the event-horizon predicate is left typed as before, with the quasi-local classified rows added beside it, never overwriting it. The philosophical moral is that the teleological character of the event horizon is not a defect to be repaired but a type to be respected: one can retire the authority that treats the global future as the only certificate of horizon-hood exactly for the predicates that read local data, and no further — and saying precisely where that line falls, and why, is the work.