Miguel A. Sabogal, Luis A. Escamilla, Davide Pedrotti, Edvig Selimaj, Sunny Vagnozzi

The standard "no-go theorem" against late-time solutions to the Hubble tension is essentially a normalization wall, since Baryon Acoustic Oscillation (BAO) measurements constrain the product $H_0r_d$, with $r_d$ the sound horizon at baryon drag. However, late-time solutions (which keep $r_d$ fixed) are tightly constrained not only by the BAO normalization $H_0r_d$, but also by the shape of the expansion history, i.e. the dimensionless expansion rate $E(z) \equiv H(z)/H_0$. We show that, if $r_d$ and the acoustic angular scale $θ_s$ are fixed, an increase in $H_0$ needs to be matched by an equal fractional increase in the dimensionless distance integral $I \equiv \int dz/E(z)$: $δH_0/H_0 \simeq δI/I$. We use this to quantify the "shape wall" set by relative distance constraints on $E(z)$, which we reconstruct nonparametrically, using Gaussian Processes and the latest unanchored Type Ia Supernovae (SNeIa) and BAO data. In our most conservative analysis using PantheonPlus SNeIa and DESI DR2 BAO data, we find a maximum fractional increase in $H_0$ of $\lesssim 2\%$, falling well short of the $\gtrsim 8\%$ required to solve the tension. This shape wall holds even in the presence of early-time new physics, and limits the maximum increase in $H_0$ which can be contributed by late-time modifications to $E(z)$ at fixed $θ_s$. Therefore, late-time modifications, whether invoked alone or alongside early-time new physics, face not only the well-known normalization wall, but also a stringent percent-level shape wall.