Boundary Loss and the Born Rule: Pre-Numerical Probability-Status in Deterministic Systems

This paper is Part III of The Pre-Emergence Origin of Probability trilogy. The paper develops a boundary-loss account of pre-numerical probability-status in deterministic systems. Its central claim is that the Born rule does not need to be the first appearance of probability. Before probabilities can be assigned, a boundary-readable description may first lose guarantee. Part I, Finite Recurrent Stability Before Spacetime: Collapse, Emergence, Horizons, and Recurrence as Recoverability Boundaries, develops the structural requirements for persistent recoverable organization before spacetime-readable realization. Part II, Stability, Boundary Observability, and Emergent Probability in Deterministic Systems, asks when persistent boundary-readable structure can support conditional numerical probability-measure status. Part III examines what probability-status remains after recoverability-relevant boundary loss, before numerical weighting, and under what narrower local conditions numerical and Born-form weighting may follow. Let B:S_pre -> S_B be a boundary or readout map from a pre-boundary state space to a boundary-readable residue space. When a residue r = B(x) fails to preserve recoverability-relevant information, the compatibility class B^{-1}(r) may contain admissible alternatives that remain unresolved by r alone. The boundary-readable context then loses guarantee: it cannot determine which admissible alternative obtains, even when the pre-boundary structure is deterministic. The main result is a conditional structural theorem. Under a declared admissibility structure, if a domain D contained in B^{-1}(r) admits a scalar-neutral resolution relation ~_K with at least two admissible alternatives whose distinction is not recoverable from r alone, then the quotient context supports a representative-invariant boundary-unresolved quotient status. This status is called pre-numerical probability-status only in the restricted technical sense used in the paper: admissible alternatives remain live under the same residue, while no numerical probability rule has yet been supplied. The result does not derive numerical probability, Born probabilities, empirical frequencies, stochastic law, measurement outcomes, collapse, determinism, or physical instantiation. Numerical probability enters only through additional local scalarization. A local realization must supply nonnegative scalar residues, finite nonzero total, normalization, and a declared probability regime. The paper also explains why Born-rule numerics cannot be derived from boundary loss alone. Boundary loss is quotient-level and scalar-neutral, while Born weights require local Hilbert-space structure, including a state vector, projectors or effects, amplitudes, orthogonal additivity, and normalization. In quantum contexts, the Born rule is therefore treated as the local Hilbert-space scalarization of boundary-induced probability-status, not as the origin of probability itself. The theorem is compatible with deterministic pre-boundary structure but does not prove determinism. It is realization-neutral until a physical model supplies the relevant boundary map and admissible structures. Within the trilogy, Part III therefore begins only after the persistence, boundary-observability, scalar, measure, normalization, and realization-burden architecture developed in Parts I and II. Series order establishes bounded inheritance only. It does not supply an omitted premise, automatically establish physical realization, or convert the restricted boundary-loss result into an unrestricted probability theorem. Companion works: Finite Recurrent Stability Before Spacetime: Collapse, Emergence, Horizons, and Recurrence as Recoverability Boundaries 10.5281/zenodo.1