Boundary Loss and the Born Rule: The Origin of Probability

Boundary Loss and the Born Rule: The Origin of Probability This paper is Part III of the three-paper series The Origin of Probability: A Complete Derivation in Three Parts. Part I develops the structural requirements for persistent recoverable organization and finite recurrent stability before dimensional realization is assumed. Part II asks when persistent boundary-readable structure can support conditional numerical probability. Part III asks what lawful structure remains when recoverability-relevant Boundary Loss prevents a later readable description from recovering a determining distinction, and how that remaining structure can support local predictive probability and the exact local source-linked Born form. The central claim is not that determinism fails when probability appears. The paper instead separates deterministic evolution from recoverability. Distinct admissible pre-boundary states may produce the same boundary-readable residue while differing with respect to a physically relevant distinction. The later description may therefore remain lawful while no longer preserving enough information to recover which admissible precursor distinction obtained. Boundary Loss is consequently defined as a loss of recoverable guarantee, not as randomness, stochastic dynamics, ignorance, or probability itself. For a pre-boundary carrier and boundary/readout map, a common residue defines a compatibility class of states sharing that same readable result. When a declared recoverability-relevant proposition varies across that common-residue class, the residue no longer guarantees which admissible distinction obtains. Generic noninjectivity is not enough. The lost distinction must be relevant to the declared physical resolution context. A separately supplied scalar-neutral resolution relation is then required before the unresolved distinctions form a lawful alternative structure. Under the complete quotient-stage conditions, the result is a finite, nontrivial, representative-invariant, scalar-neutral unresolved context, PreProbContext_B, whose operative class distinction is not recoverable from the same boundary-readable residue. This is pre-numerical probability-status only in the restricted technical sense used in the paper. It supplies lawful unresolved alternatives before scalar weighting. It is not a probability distribution, not an incomplete measure, and not an unnormalized probability assignment. Finite unresolved alternatives do not imply equal probabilities, stochasticity, physical occupancy, or physical quantization. The paper then develops a separately staged physical realization architecture. Adopted-scope physical recurrence is established without identifying recurrence with physical quantization. A physical-state classifier is separately descended, followed by physical co-typing, a pre-boundary carrier, a physical boundary/readout map, recoverability-relevant proposition variation, a scalar-neutral physical resolution relation, independently established quotient nontriviality and finiteness, and physical presentation invariance. At that scope, the paper obtains a finite, nontrivial, representative-invariant, scalar-neutral physical unresolved quotient. The result still contains no numerical probability. Numerical structure enters only afterward. Within the admitted source regime, the nonnegative scalar density ρ = |Ψ|² is connected to the unresolved alternatives through a separately supplied carrier-grounding relation. This yields source-grounded residues sᵢ = ∫Dᵢ |Ψ|² dx ≥ 0. Finiteness alone initially establishes only 0 ≤ S < ∞. A separate positive-witness condition is required before the stronger result 0 < S < ∞ is available. Only then is normalization lawful. The resulting finite additive scalar assignment is converted into a normalized finite measure, wᵢ = sᵢ / S, but normalization alone is still not predictive probability. The normalized measure receives local predictive-probability status only after a separately declared finite predictive-admission criterion is satisfied. This ordering is essential: Boundary Loss↓lawful unresolved alternatives↓source-grounded scalar residues↓strictly positive finite total↓finite measure and normalization↓local predictive probability. Probability is therefore present at the declared local predictive scope before the final Hilbert-space compatibility step. The paper next constructs two independent source-Hilbert bridges. A source-state bridge supplies the linked Hilbert state J(Ψ), while a separate source-event bridge maps each relevant source event to a closed Hilbert channel, Aᵢ → Hᵢ, from which the corresponding orthogonal projector Pᵢ follows by ordinary Hilbert-space projection theory. These two bridges are independent. Their coexistence does not imply that the source-grounded predictive probability pᵢ equals the Hilbert projector quantity ⟨J(Ψ), PᵢJ(Ψ)⟩. Common indexing, nonnegativity, normalization, finite additivity, coarse-graining, presentation invariance, and projector structure do not manufacture that equality. The final substantive compatibility burden is therefore isolated explicitly. At the exact finite linked source/Hilbert scope, the theory adopts the minimum substantive compatibility commitment required to connect the two independently constructed sides. Adoption is distinguished from application, and the compatibility law is not presented as a source-neutral consequence of the preceding Boundary-Loss, scalar, probability, state, event, or projector architecture. When the explicitly adopted compatibility law is lawfully applied to the already established linked package, the paper obtains, for every relevant event in the current finite family, pᵢ = ⟨J(Ψ), PᵢJ(Ψ)⟩ = ‖PᵢJ(Ψ)‖². This is the paper's terminal positive result: the exact local source-linked Born form at the declared adopted-theory application scope. The compatibility equality is a lawful consequence of the explicitly adopted minimum compatibility law at that scope. It is not claimed to have been forced source-neutrally by the antecedent structures. The paper therefore develops the complete bounded route recoverability-relevant Boundary Loss↓lawful unresolved alternatives↓source-grounded finite scalar measure↓locally admitted predictive probability↓independent source-state and source-event Hilbert bridges↓explicit minimum compatibility commitment↓lawful application↓exact local source-linked Born form. The result does not establish an unrestricted global Born rule, measurement theory, collapse dynamics, detector dynamics, event occupancy, empirical confirmation, physical quantization, necessity, fundamentality, source-native predictive semantics, source-native status for the compatibility law, or final-theory status. The paper's narrower result is that deterministic recurrent organization can coexist with a boundary at which recoverability of a determining distinction fails, while enough lawful structure remains to support unresolved alternatives, source-grounded scalar structure, a finite normalized measure, local predictive probability, and, after an explicit additional compatibility commitment, the exact local source-linked Born form. Version 2 revision note Version 2 is a substantial reconstruction and completion of Part III within the final trilogy The Origin of Probability: A Complete Derivation in Three Parts. The revision extends the earlier Boundary-Loss and pre-numerical probability-status architecture into a complete bounded probability-origin chain. The realization-neutral Boundary-Loss theorem and scalar-neutral unresolved quotient remain foundational, but the manuscript now separately develops their physical realization, source-grounded scalar architecture, finite measure, normalization, local predictive-probability admission, source-Hilbert state and event bridges, and the final local Born-form compatibility result. The revised physical Boundary-Loss architecture explicitly separates physical recurrence, classifier descent, physical co-typing, boundary/readout realization, proposition variation, scalar-neutral relation formation, quotient construction, nontriviality, finiteness, presentation invariance, and physical occupancy. A discrete classifier image is not promoted into physical quantization, and a finite unresolved quotient is not promoted into numerical probability. The scalar stage now makes explicit that source-native ρ = |Ψ|² does not automatically ground the unresolved alternatives. Alternative-to-carrier grounding, nonnegative residues, strict positivity of the total, scalar-preserving presentation transport, finite-measure recognition, normalization, and predictive interpretation are treated as separate burdens in dependency order. The revision further distinguishes an abstract Hilbert representation of an already known probability vector from a source-linked Hilbert state. Independent source-state and source-event bridges are therefore constructed before the compatibility question is posed. Their coexistence does not establish the Born equality. The final compatibility burden is isolated as an explicit minimum theory commitment. The compatibility law is adopted at the exact linked source/Hilbert scope and then separately applied to the locally admitted predictive probabilities and independently established Hilbert structures. That application yields the exact local source-linked Born form. Version 2 therefore closes the declared probability-origin compatibility objective at its exact bounded scope while retaining strict nonpromotion boundaries. The manuscript does not claim a source-neutral derivation of the compatibility law, an unrestricted global Born rule, measurement or collapse dynamics, empirical realization, physical quantization, necessity, fundamentality, or final-theory status. Companion works Part I: Finite Recurrent Stability Before Spacetime: Collapse, Emergence, Horizons, and Recurrence as Rec