Testing a Transient Extended Cuscuton as a Solution to the Hubble Tension

We test whether a genuinely two-degree-of-freedom Extended Cuscuton can raise the late-time Hubble constant by transiently modifying the pre-recombination expansion history without spoiling the cosmic microwave background (CMB). Using CLASS 3.4.0, we first show that a two-window expansion history can match H0=72kms−1Mpc−1 and the acoustic angle at the background level but already leaves several-percent TT/EE residuals. We then reconstruct this target within the luminal Extended-Cuscuton subclass (v4=MPl2/2, transient v3(ϕ)), implementing the nondynamical scalar sector as a radiation-inclusive algebraic D=0 constraint rather than as an extra evolution equation, and we show analytically that the original, minimal Cuscuton could never have supplied the required transient in the first place. For a single recombination-centered braiding window, the perturbation sector amplifies rather than compensates for the background mismatch, to a combined TT/EE screening residual of ∼100% (vs. ∼8% background-only) for both signs of the braiding amplitude. Widening this window, or replacing it with two windows aligned to the background target’s own bumps, brings the residual down to ∼81% in the best completed case—still far from the background-only value—before the scan is stopped by a pole in the algebraically solved perturbation coefficients, traced to one super-horizon mode and shown not to be an integrator artifact; cases beyond this point remain unresolved rather than excluded. We conclude that the transient Extended-Cuscuton constructions tested here, minimal and extended alike, do not provide a viable resolution of the Hubble tension.