The Natural Pre-Bounce Measure: From SU(2) Geometry to a Parameter-Free CMB Prediction (Paper 2 of the LQG–LQC Intertwiner Series)

1 Introduction Paper 1 of this series 1 established the LQG–LQC intertwiner map and derived the windowfunction W(κ) governing intertwiner-mediated corrections to the primordial power spectrum.The central result was that the Bianchi I shear scalar is uniquely identified with Σ2 = 2λ2|β|2,where |β|2 is the probability amplitude for the pre-bounce four-valent vertex to occupy theanisotropic intertwiner state |k=1⟩. Every observable prediction of Paper 1 is conditional on|β|2 > 0; the value of |β|2 was treated as an empirical parameter.The Livine–Speziale (LS) coherent intertwiner construction 2, 3 provides the missingingredient. In Paper 1 (Section 4.4–4.5), we derived the exact decomposition of the LScoherent state in the intertwiner basis and showed that classical Bianchi I geometries alwaysgive |βLS|2 = 0. The parameter |β|2 is therefore genuinely quantum—a shape excitation ofthe vertex that has no classical Bianchi I analog.The question then becomes: if |β|2 is a quantum degree of freedom, what probabilitydistribution should it carry? The answer is uniquely determined by the principle of nopreferred orientation: the pre-bounce quantum state, in the absence of additional structure,should be distributed according to the SU(2)-invariant Haar measure on the space of vertexgeometries. This paper derives the distribution that the Haar measure induces on |β|2,computes its moments exactly, and confronts the result with existing Planck 2018 data.The outcome is striking: the induced median is 1/13, agreeing with the Planck best-fitto 7.3% with no free parameters whatsoever. The framework transitions from a parametriceffective theory to a first-principles prediction.