FBT08E: Abstract Clock–Sliced Localisation and Worldline Interfaces

This paper isolates the abstract mathematical interface by which a physical configuration acquires a position on a clock slice and, under an additional transport estimate, a causal worldline. It replaces the earlier attempt to manufacture a base localisation functional from an internal thimble partition function and an unspecified mixed response. The localisation datum is now typed directly on the physical quotient: either a real functional Φτ : Cphys ×Στ → R ∪{+∞}or a probability measure µτ [C] from which the intrinsic Fréchet functional is formed. Representative independence is therefore a property of the domain, not a separate covariance hypothesis. A compact global sublevel confined to a strongly convex localisation domain gives a unique centre on the whole slice. The probability measure may have nonzero tails outside that domain: it is the full Fréchet Hessian, including the tail contribution, that must remain positive. A jointly C1 positional gradient and local uniform confinement of the minimisers give a C1 positional history by the parameterised implicit-function theorem. Causality is not assumed in the definition of that history. In clock-adapted ADM variables it follows whenever the normal velocity satisfies a strict cone-margin estimate |VX|hτ ≤v∗ + Etr <1. Thus localisation, smooth persistence, and causal upgrade are three distinct theorems with distinct minimal inputs. The current FBT13A supplies a conditional intrinsic thimble–to–base matter descent, without reinstating its retired SCIEC predecessor. FBT13E supplies a conditional dynamical realisation with exponential tails, a complete core–tail Hessian budget, and a separate causal transport margin. FBT13D supplies the principal realised matter instance: its projected Dirac current gives the slice probability measure, its Karcher centre realises the present positional interface, and its barycentre-transport estimate supplies the causal margin. FBT08F is independent and may strengthen persistence by transporting a gapped spectral branch. No twelve-slot carrier, level–24 quantisation, balanced shell, Floer class, QGT diagnostic, mass law, or effective action is required by the abstract interface.