Born's Rule as Simplex-Martingale Absorption: A Conditional Derivation from Reduced Substrate Dynamics

We reduce the Born-rule problem $P(i \mid |\psi\rangle) = |a_i|^2$ to a classical absorption theorem on the probability simplex $\Delta^{n-1}$, under a small set of explicit structural assumptions on a reduced effective dynamics motivated by an $n$-mode discrete self-trapping (DST) measurement network. If the reduced basin-mass process $M(t) = (M_1(t), \ldots, M_n(t))$ is a bounded martingale and is almost surely absorbed at a simplex vertex, then the optional stopping theorem yields $P(M(\infty) = \hat e_i) = M_i(0) = |a_i|^2$ by a short argument. Within this reduction we partially justify the martingale property: the Hamiltonian inter-basin flux is shown to vanish in conditional expectation from $U(1)^n$-equivariance of the DST flow; the dissipative basin-mass drift is shown to vanish from amplitude-local dissipation plus norm conservation. Two remaining ingredients are stated as explicit assumptions rather than derived from substrate physics: a martingale-preserving structure of the substrate noise beyond pure $O_h$-isotropy, and almost-sure absorption at simplex vertices (supported for $n = 2$ by a small-noise stochastic-stability argument). The microscopic derivation of these two ingredients, together with the quasi-stationary fluctuation-dissipation relation underlying noise isotropy, is left for future work. The argument does not invoke non-contextuality or any probabilistic axiom on Hilbert space; an alternative functional-equation route is also formulated as an independent algebraic cross-check. This manuscript was drafted in May 2026 and is uploaded here in May 2026 as Version 1.0 of the preprint.