The Probability Sector of PT-Symmetric Quaternionic Spacetime

We investigate the probability sector of PT-symmetric quaternionic spacetime (PTQ) from a structural and geometric perspective. The aim of this work is not to claim a universal derivation of the Born rule or a solution of the measurement problem, but to identify a constrained and internally coherent route by which the Born-rule form becomes admissible within the projected pseudo-Hermitian sector of PTQ.The analysis proceeds in three steps. First, we show that the probability sector must be defined only after restriction to a real, PT-even, projectively invariant observable scalar sector. Second, we demonstrate that consistency of the projected dynamics selects a pseudo-Hermitian metric structure, leading to a physically distinguished G-inner product rather than a naive kinematic norm. Third, we show that the same G-metric induces a conserved probability current, whose density is j⁰ = ψ†Gψ and whose spatial integral is preserved under the projected evolution.These ingredients jointly imply a constrained route: projected observability → physical G-inner product → conserved probability current → Pᵢ = |⟨ψ | φᵢ⟩ᴳ|². Within this route, the Born-rule form is not introduced as an independent postulate, but appears as the probability assignment compatible with observability, metric compatibility, and norm-preserving dynamics in the PT-even sector. We clarify the scope of the result: the construction is restricted to the unbroken PT-even projected sector and does not address collapse, branch selection, or fully general measurement theory. Within these boundaries, however, PTQ provides a disciplined and auditable framework in which the probability sector is structurally constrained.