Supermassive black holes (SMBHs) powering quasars at $z\gtrsim 6$ are difficult to grow from stellar-mass remnants, motivating seeds from primordial black holes (PBHs) with masses ${10}^5–{10}^7{M}_{\odot }$. This range is constrained by the Cosmic Background Explorer’s Far Infrared Absolute Spectrophotometer (FIRAS) bound on the cosmic microwave background $\mu$-distortion, which limits the small-scale curvature variance to ${\sigma }_{\zeta }^2\lesssim {10}^{-4}$. For Gaussian perturbations, the variance fixes the far tail of the one-point probability distribution function (PDF), making the PBH abundance negligible. We call this the Gaussian barrier. The barrier can be evaded only if the variance probed by the distortion is decoupled from the tail probability controlling collapse. We implement this idea in the nonperturbative $\delta N$ formalism and relate asymptotic PDF tails to the global shape of the $\delta N$ map. Four Gaussian-cored families are analyzed: generalized-normal, stretched-exponential, power-law, and log-normal tails. After standardizing each family to unit variance, we impose the FIRAS cap and compute the distortion-limited PBH abundance in the tail-shape parameter space. The ordinary exponential tail produced by standard single-field nonattractor dynamics is still too light to reopen the seed window. Algebraic tails from fractional-potential dynamics, and sufficiently heavy log-normal tails treated as a phenomenological proxy for multiplicative dynamics, can supply seed-relevant abundances while respecting the distortion bound.

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• Cosmic microwave background
• Inflation
• Massive compact halo objects