CMB Constraints on the Pre-Bounce Quantum State: Planck 2018 Data Analysis and an Exact Haar-Percentile Framework for Loop Quantum Cosmology (Paper 20 in the Intertwiner Series)

We test predictions from loop quantum cosmology (Papers 1–4) against Planck 2018 Cosmic Microwave Background temperature data. Performing a likelihood analysis of the TT power spectrum (ℓ = 2–2000), we constrain the pre-bounce excitation parameter |β|². The primary, robust result is an upper limit: |β|² < 0.037 (95% CL, GUT-scale scenario) and |β|² < 0.075 (95% CL, large-scale scenario) — a factor of several tighter than prior bounds. The best-fit value is |β|² ≈ 0.012 under a diagonal-covariance approximation; we show that this does not constitute a robust detection of a nonzero signal, since neglected multipole correlations inflate the apparent significance, and the honest significance from zero is ≈1.3–2.7σ. We then reframe the comparison with the SU(2) Haar-measure prediction. The Haar measure does not predict a single value of |β|² but a distribution, |β|²(u) = u²/(3 + u²) with u uniform on −1, 1 (Paper 4, §6.2), whose median is 1/13 ≈ 0.077. We provide the exact cumulative distribution of this measure, giving a rigorous percentile placement of any measured |β|². The Planck best-fit lies at the 19th percentile of the Haar distribution — a statistically ordinary draw (tension ≈ 0.9σ), not an exclusion. The data thus constrain |β|² to the lower third of the Haar range but are fully consistent with a typical Haar draw. Whether the small observed |β|² reflects a dynamical selection mechanism or simply a typical low-tail realization is an open question that future data (LiteBIRD, CMB-S4) can address.