The Hubble Tension in Plain Arithmetic: What Each Published Value Demands of the Sound Horizon

The disagreement between early-universe and late-universe determinations of the Hubble constant is usually reported as a single significance figure. This paper instead works the arithmetic probe by probe, using only the published central values and uncertainties of Planck 2018, SH0ES 2022, the tip-of-the-red-giant-branch calibration, the megamaser distance scale, time-delay lensing, and the GW170817 standard siren, and converts each late-time value into the sound horizon it requires at fixed acoustic angle. The Planck–SH0ES difference is 5.68 km/s/Mpc, a 4.85-sigma discrepancy under independent Gaussian errors; combining SH0ES, the megamasers, and the lenses gives 73.17 ± 0.86 km/s/Mpc and raises the formal tension to 5.74 sigma. At fixed acoustic scale, the SH0ES value requires a sound horizon of 135.65 Mpc against the Planck value of 147.09 Mpc, a reduction of 7.78 percent (11.44 Mpc); the lensing value requires 135.17 Mpc, a reduction of 8.10 percent; the red-giant-branch value requires only 141.95 Mpc, a reduction of 3.50 percent, and stands just 1.25 sigma from Planck. Mimicking the SH0ES excess with a local underdensity would require a density contrast of about −0.51 on the ladder's volume, far outside what galaxy counts permit. The computation is simulated arithmetic on published values, not a measurement.