Stability under scaling in the local phases of multiplicative functions

We introduce a strategy to tackle some known obstructions of current approaches to the Fourier uniformity conjecture. Assuming GRH, we then show the conjecture holds for intervals of length at least $(\log X{)}^{\psi (X)}$, with $\psi (X)\to \mathrm{\infty }$ an arbitrarily slowly growing function of $X$. We expect the methods should adapt to nilsequences, thus also showing that the Generalised Riemann Hypothesis implies close to exponential growth in the sign patterns of the Liouville function.