Loop Quantum Gravity Quantization of the Compact Circle S1 of Radius τ = 7 h−1Mpc

This supplementary note applies loop quantum gravity (LQG) techniques to the compact fifth dimension of radius τ = 7 h⁻¹ Mpc, as measured in the JADES DR4 galaxy power spectrum. Using Ashtekar variables on R³ × S¹, we derive the polymer-quantized holonomies, the area spectrum for surfaces piercing the circle (with minimal non-zero area Amin ≈ 2πτ √(3/2) γ ℓ_Pl²), and discrete Kaluza-Klein momenta with negligible quantum corrections for observable modes n ≤ 6. The n=7 harmonic at k₇ = 1.00000 h Mpc⁻¹ is predicted to be suppressed due to the quantum area gap, marking the transition from computable classical harmonics to an undecidable quantum tower. This bridges classical cosmology with quantum geometry, offering the first observable quantum of area in physics history.