Author

Hong-Huai Fang

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Preprint Sep 2026

Variable-Cliff Nielsen Geometry and an Exponent -4/3 Lower Bound for the Infinite-Cliff Diameter

Let $D=2^n$ and $M=3n+9\binom n2$. We study the right-invariant one-step-cliff metric $d_Q$ on $\operatorname{PU}(D)$, with unit penalty on Pauli weights one and two and penalty $Q$ on all higher weights. If $M/Q_D\to0$ and $MQ_D^{3/4}/D^2\to0$, then, for every fixed $0<x<\pi/\sqrt3$, \[ \mu_D\bigl(B_{Q_D}([I],x\sqrt{Q...

Hong-Huai Fang · 0 citations

Preprint Sep 2026

Sharp Typical Distance and Exponential Small-Ball Bounds in One-Step-Cliff Nielsen Geometry

Let $D=2^n$ and equip $\operatorname{PU}(D)$ with the one-step-cliff Nielsen metric, with quadratic metric coefficients one in Pauli directions of weights one and two and $D^2$ in all higher weights. We prove that the distance from the identity of a Haar-random element, normalized by $D$, converges to $\pi/\sqrt3$ in p...

Hong-Huai Fang · 1 citation

Preprint Aug 2026

Cyclotomic Newton Expansions and a Rank-Uniform Integer-Valued Newton Completion

Let $J_r^{SU(n)}(K;q)$ denote the reduced $SU(n)$ quantum invariant of a zero-framed knot $K$, colored by the $r$th symmetric power of the defining representation and normalized to be $1$ for the unknot. For every fixed $n\ge2$ we prove the Chen--Liu--Zhu cyclotomic expansion conjecture: there are unique coefficients $...

Hong-Huai Fang, Tian Zhou · 2 citations