Flat Irrational Torus Topology: Simultaneous Resolution of Low-ℓ Anomaly and Hubble Tension

We test the Flat Irrational Torus model IT3—a compact flat topology with irrational aspect ratios Ly/Lx= √2 and Lz/Lx= √3—against Planck PR4 temperature power spectrum data. Using Bayesian MCMC inference with the CLASS Boltzmann code and emcee sampler, we find that the topology scale is well-constrained: Lx= 28.57+0.73 −0.87 Gpc, consistent with the geometric lower bound Lx ≥ 2χrec ≈ 28.0 Gpc. The model yields H0= 67.55 ± 1.77 km/s/Mpc, reducing the tension with SH0ES 2024 local measurements from 3.04σ to 2.68σ. We discover a positive correlation (r= 0.258) between Lx and H0, revealing a geometric mechanism where compact topologies suppress low-ℓ power while simultaneously allowing higher expansion rates. The model improves the fit to the low-quadrupole anomaly with ∆χ2= −5.33 (> 2σ improvement), while the Bayesian Information Criterion gives ∆BIC ≈+2.49 (using N= 2500 effective multipoles), placing the result in the "inconclusive to positive" regime. Furthermore, importance-sampling validation against DESI 2024 BAO and Pantheon+ supernova data yields an effective sample size of Neff= 17,633 (∼ 27% survival rate), demonstrating that the IT3 topology remains statistically compatible with late-time geometric probes. We conclude that IT3 is a statistically viable alternative to infinite ΛCDM, offering a purely geometric explanation for two major cosmological tensions without introducing new physics beyond general relativity.