Time-Delay Lensing and the Price of a Mass Sheet: One Degeneracy Carries the Whole Answer

Strong-lens time delays measure an absolute distance — the time-delay distance, inversely proportional to the Hubble constant — from the arrival-time differences of a variable quasar's multiple images, with no ladder and no ruler. The H0LiCOW programme's six-lens result, H0 = 73.3 (+1.7/−1.8) km/s/Mpc, is accordingly quoted as independent confirmation of the local ladder. This paper prices the assumption on which that independence rests: the mass-sheet degeneracy, an exact rescaling of the lens model that leaves every imaging observable unchanged while scaling the inferred Hubble constant by a free factor lambda. Reconciling the lensing value with the Planck value requires lambda = 0.919 — a mass sheet equivalent to an external convergence of 8.1 percent, or an internal profile rescaling of the same size. When the successor programme TDCOSMO broke the degeneracy with stellar kinematics alone, the value moved to 74.5 with errors near 6; when it additionally imported the SLACS lens population hierarchically, the value moved to 67.4 (+4.1/−3.2) — on top of Planck (0.01 sigma) and 1.49 sigma below SH0ES. The spread among these three analyses of substantially the same sky, from 67.4 to 74.5, spans the entire Hubble tension, and it is generated not by data but by how the degeneracy is broken. The statistical road ahead is short — the six-lens sample implies 5.9 percent per lens, so 9 lenses reach 2 percent and 35 reach 1 percent statistically, and forty lenses would reach 0.93 percent — but the mass-sheet term does not average down with lens count; it is broken by kinematics and population assumptions or it is not broken at all. Time-delay lensing is therefore currently a measurement of the degeneracy-breaking assumptions with a lensing likelihood attached, and it becomes an arbiter of the tension exactly when those assumptions are independently secured. This study is computed from published values; it is not a measurement.