The observable-specific weak-mixing conjecture for phantom gain discrepancies

Let $T$ be the odd-to-odd Syracuse map, let $n_0$ be odd, and let $h_K$ be the phantom gain observable at depth $K$, with reference mean $R(K)$. Define

$$\eta_K(n_0)=\limsup_{N\to\infty}\left|\frac1N\sum_{i=1}^N h_K\bigl(T^i(n_0)\bmod 2^K\bigr)-R(K)\right|.$$

Observable-specific weak-mixing conjecture. For every odd starting value $n_0$,

$$\sum_{K\ge3}\eta_K(n_0)<\varepsilon-R\approx0.326.$$

This is a weaker, observable-specific target than full total-variation equidistribution and is intended to supply the quantitative error budget needed by the phantom-gain reduction. Its status is open in the paper.