arXiv subjects

Honghuai Fang

Publications and source records attributed to Honghuai Fang.

7 recordsLinked to original sources

arXiv2026

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 probability and in $L^p$ for every $1\le p<\infty$. Quantitatively, it lies within $O(D^{-1/8}(\log D)^{1/2})$ of this limit outside a set of Haar measure at most $\exp\{-\Omega(D^{7/4}\log D)\}$. For each fixed $0 \pi/\sqrt3$, its complement has measure at most $e^{-c_xD^2}$ for some $c_x>0$. As $x\uparrow\pi/\sqrt3$, the lower and upper logarithmic rates are both asymptotic to $(\pi^2/3-x^2)^2/(16\zeta(3))$. The small-ball upper bound follows from a comparison of Jacobi determinants, obtained by rescaling the linearized geodesic equations and applying Kato transport. Weyl integration reduces the remaining integral to an Abel-regularized logarithmic-energy estimate on the circle. A centered principal logarithm and concentration of the circular unitary ensemble eigenangle second moment give the distance upper bound.

math.PR↗

arXiv2026

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 $0 0$ yields a Haar-typical lower bound of order $D^{4/3}M^{-2/3}$ for the corresponding infinite-cliff Carnot--Carath\'eodory distance, outside an $e^{-\Omega(D^2)}$ exceptional set. The infinite-cliff diameter therefore has exponential lower rate at least $4/3$, disproving Brown's exponent-one conjecture. The same estimate gives a fixed-error no-ancilla two-qubit circuit lower bound of the same order.

quant-ph↗

arXiv2026

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 $H_k^{(n)}(K;q)\in\mathbb{Z}[q^{\pm1}]$ such that \[ J_r^{SU(n)}(K;q)=\sum_{k=0}^{r} \left(\prod_{i=0}^{k-1}\{r-i\}\{r+n+i\}\right) H_k^{(n)}(K;q), \] where $\{m\}=q^m-q^{-m}$. The finite dual interpolation formula of Beliakova--Gorsky gives an integral one-sided factorial expansion. After identifying their reduced scalar with the Habiro--L\^e convention, we restrict the completed center to one-row colors. Completed Harish--Chandra reflection then yields inversion symmetry in the variable $z$, and integral descent through $X=z+z^{-1}$ converts the one-sided expansion into the two-sided Newton basis. Cyclotomic-local interpolation and a UFD denominator-removal argument prove Laurent integrality of the Newton coefficients. We also determine a natural coefficient ring for a rank-uniform expansion. For every zero-framed knot there are unique Laurent differential coefficients $G_k(K;A,q)\in\mathbb{Z}[A^{\pm1},q^{\pm1}]$. The associated Newton coefficients are Laurent polynomials in $A$ over $\mathbb{Q}(q)$ whose values at every geometric node $A=q^n$, $n\ge2$, lie in $\mathbb{Z}[q^{\pm1}]$. They define a two-variable Newton inverse-limit element whose positive-rank specializations recover all symmetric-color HOMFLY--PT polynomials. The completion is taken in the Newton kernels rather than coefficientwise at roots of unity. After a positive rank and a color have been fixed, the series is finite and may be evaluated at a root of unity.

math.GT↗

arXiv2026

Bloch-regulator Principal Parts of Cyclotomic Iwasawa Pseudomeasures

Let $p$ be an odd prime and let $K/\mathbb{Q}_p$ be a finite unramified extension. From a finite presentation by roots of unity of order prime to $p$, we construct a localized Iwasawa pseudomeasure on $\mathbb{Z}_p^\times$. Although the pseudomeasure depends on the chosen presentation, its image modulo bounded measures depends only on the associated Bloch class: it is the Frobenius-depleted Coleman regulator of that class multiplied by a universal half-shifted zeta principal part. Consequently, every nonexceptional weight component is bounded, while the exceptional component has at most a simple pole with explicitly determined residue. For $p>3$ and $\mathbb{Z}_p$-valued coefficients, vanishing of the principal part is equivalent to vanishing of the corresponding class in $K_3(K;\mathbb{Z}_p)$. Cyclotomic refinements preserve the Bloch class and act on the associated pseudomeasures by explicit Iwasawa multipliers. Normalized finite linear combinations of refinements interpolate arbitrary finite jets of the bounded weight-space data, subject only to the normalization at the exceptional point. We also establish a half-shifted complex Mellin factorization, compare the construction with the GSWZ germ family, and derive, at simple degree-one places above primes $p>3$, a finite-polylogarithm criterion for the local $K_3$-class of the knot $5_2$.

math.NT↗

arXiv2024

An edge-type state integral over local field and A-polynomials

To each local field, Garoufalidis and Kashaev recently associate a quantum dilogarithm that satisfies a pentagon identity and some symmetries. By employing an angled version of these quantum dilogarithms, they developed three generalized TQFTs, one given by a face state-integral and others by edge state integrals. These TQFTs produce distributional invariants for one-cusped three-manifolds, which are believed to be related to counting points on the A-polynomial curve. In this paper, we will calculate partition functions of an edge-type generalized TQFT over a local field for several examples and prove the appearance of A-polynomial in these new invariants.

math.QA↗

arXiv2023

$SO(3)$-Berezin-Toeplitz quantization and the AJ conjecture

We investigate the Berezin-Toeplitz operators that operate on the geometric quantized space corresponding to the $SO(3)$-Witten-Chern-Simons theory. We conjecture that the $SO(3)$-Berezin-Toeplitz operators quantized from the A-polynomial annihilate the corresponding $SO(3)$-knot states.

math.QA↗

arXiv2022

SO(3)-Knot States and the Volume Conjecture

We study the alternating subspace of holomorphic sections of a special prequantum line bundle over SU(2)-character variety of torus, and show that it is isomorphic to the projective representation of mapping class group of peripheral torus given by the SO(3) Witten-Chern-Simons theory. We conjecture that the large r asymptotics of $L^2$-norm of SO(3)-knot states via geometric quantization capture the simplicial volume of knot complements.

math-ph↗