USSFT Paper VIII: The Born Rule from Multiplicity Dynamics. Path-Integral Representation and the Origin of Quantum Probability

This paper addresses the foundational question deferred by Paper VII: why does the multiplicity Omega of U-field microstates combine as the squared modulus of a complex amplitude? We argue that if the coarse-grained U-field microstate sum admits a path-integral representation over field configurations with phase weights exp(iS/hbar_eff), then a Complex Multiplicity Amplitude Z(Y) is associated with each outcome Y. Via a Constraint Surface Geometry argument, we show that the effective multiplicity must take the form Omega(Y) = |Z(Y)|^n by phase invariance and multiplicativity, and the empirically observed bilinear interference structure selects n=2, yielding Omega(Y) = |Z(Y)|^2. The Born rule P proportional to |psi|^2 then follows from the multiplicity principle P proportional to Omega together with the identification psi proportional to Z. Interference arises from linear addition of complex amplitudes before squaring; classical probability addition is recovered via decoherence. The outcome-selection rule (the Probabilistic Bias Functional B) is grounded in the ergodic/mixing dynamics of the Tier-1 lattice, conditional on mixing within the accessible phase-space sector. The construction of the path-integral measure from U-field microstate counting remains an open problem. This is the eighth paper in the 18-paper USSFT technical series; for the U-field Lagrangian foundations see Papers I-VI, for the quantum mechanics operator formalism see Paper VII (DOI: 10.5281/zenodo.19675374), and for spin structure and Bell correlations see Paper IX.