Born Rule: The Born Rule for a Structural Class of Projective Measurements: Instrument-Effect Characterization, First Records, and Persistent Readout

We prove a Born-rule characterization for a natural family of finite-dimensional projective measurement data. The preceding reconstruction of quadratic channel mass, finite contextual closure, and complex scalar type yields a Hilbert-space PVM together with a complex structure unique up to conjugation. For a time-labelled quantum instrument, the Born law for every state is equivalent to the effect factorization By(Δ) = ν(Δ)Py, with a common time measure ν; the terminal law is equivalently By = aPy. Pure states already suffice for both effect classifications. The recovered complex structure also makes the context-generating flows unitary. We establish quantitative rigidity of the Born class and, for normalized terminal effects, characterize the Lüders update by repeatability and preservation of compatible observables. We then construct the corresponding completely positive first-record instrument. A scalar-fibre Hamiltonian model reproduces its source effects and synchronizes the classical outcome with a spatial pointer and a persistent material record. At finite parameters the sector-time readout has the exact conditional Born law on bright records, while the material-label error is quantitatively controlled. A no-go theorem shows why an exactly persistent record cannot be produced by a single closed finite-dimensional apparatus.