A Geometric Route to the Born Rule in the Stochastic Rupture Framework: Reduction to a Single Tangent-Orientation Hypothesis

We present an exploratory geometric route, within the Stochastic Rupture (SR/ICR) framework, that reduces the Born rule P=|a|² to a single central hypothesis: that the physical quantum phase is the holographic tangent orientation of the fundamental 2D screens. We show that conditional on this hypothesis, the remaining structure follows: the complex conjugate a ∗ arises from the Z2 front/back orientation inversion of a screen (a geometric fact), the continuous U(1) phase arises from the tangent orientation of a holographic surface, the temporal evolution e−iEt/ℏ follows from the wave dispersion already present in the SR master equation, and the modulussquared weight |a|² = a · a∗follows from counting both faces of a screen. We make explicit what is derived, what is hypothesised, and what remains open. The central hypothesis is supported by a uniqueness argument (of the local geometric variables of a 2D surface, only the tangent orientation lives in S1 ≃ U(1)) and by a consistency check against the four defining properties of the quantum phase, but it is not proven: we do not establish that the phase must be the tangent orientation. We further identify the connection to the Wallstrom problem (the quantisationof-circulation condition that inequivalence-proofs show Madelung-type reconstructions require) and argue that the topological defects of the SR network (π1 = Z) are a natural candidate to supply it. This is a priority-timestamp record of an open research programme, not a closed derivation.