pNGB Higgs Naturalness at a Tipping Point

In scenarios where the Higgs is viewed as a pseudo Nambu-Goldstone boson (pNGB) the question of naturalness finds itself, from a phenomenological perspective, at a tipping point between direct searches and precision. If, by the end of the High-Luminosity LHC operation, all experimental results were to remain consistent with the Standard Model, precision Higgs coupling measurements will begin to drive the naturalness tension. To illustrate this from a fresh perspective we construct a maximally natural `Kitchen Sink' model, throwing into the mix three approaches to symmetry-based naturalness: Supersymmetry, Twin Higgs, and pNGB Higgs models with a Gegenbauer potential. In other words, we build a `Supersymmetric Gegenbauer's Twin' model. This model not only maximises naturalness, at least from a technical perspective, but can also interpolate between all three ingredients smoothly, revealing the interplay between direct exploration and precision. Implications for FCC-ee and FCC-hh are discussed.