Papers

We study the problem of finding the norm of the point evaluation operator in the Paley–Wiener space $PW^{1}(\r^d)$, consisting of $d$-variable functions of spherical exponential type that are integrable on $\r^d$. The extremal functions can be taken radial, which naturally leads us to consider a related extremal problem in a weighted Paley–Wiener space of single-variable functions. We establish that the radial extremal function must satisfy a third-order linear ODE with polynomial coefficients for every $d \geq 1$, extending Gorbachev's recent odd-dimensional result. Along the way, we prove interpolation and reciprocal formulas involving the zeros of the extremizer.

Let $p$ be a complex polynomial of degree $n\ge2$ whose zeros lie in the closed unit disk. Sendov's conjecture asserts that every zero of $p$ lies within distance one of a zero of $p'$. Mazur's recent proof, and Tao's streamlined exposition of it, reduce a hypothetical counterexample to a scalar lower bound \[ 1\le F(\eta,\alpha,n) \] together with two upper bounds for the endpoint parameter $\eta$ and the uniform restriction $0<\alpha\le 17$. We complete this scalar reduction by a two-stage optimization argument. First, $F$ is nondecreasing in $\eta$, so $\eta$ may be replaced by a piecewise polar envelope $\eta^{\ast}(\alpha)$. Writing $s=(n-1)/2$ converts the degree to a half-integer variable and gives \[ F(\eta^{\ast}(\alpha),\alpha,n) = E(\alpha,s) + \kappa(\alpha,s) \int_0^1 t^3 \widehat{\beta}(t;\alpha,s)^{\,s-\frac{3}{2}} \,dt. \] On each half-unit $\alpha$-slab, $E$ and $\kappa$ decrease with $\alpha$, whereas $\widehat\beta$ increases. Convexity in $t$ reduces all but the first mesh interval to eight explicit nodal terms. The first interval satisfies a uniform bound $3/200$. Each nodal upper term is a strictly log-concave function of the half-integer $s$, so its global discrete maximum is certified by two adjacent ratio evaluations. A fixed finite certificate over the $34$ half-unit slabs gives \[ F(\eta^{\ast}(\alpha),\alpha,n)<\frac{97}{100}<1, \] contradicting the scalar lower bound. The main contribution is a substantial reduction of the computer assistance required after the Mazur–Tao scalar reduction: all continuous optimization and the unbounded degree parameter are handled analytically, leaving only a small fixed collection of explicit one-variable inequalities.

For $n\geq 0$, Stirling-Ramanujan constants $S_n$ are the Ramanujan summation of the divergent series $\sum_{k\geq 1} k^n\log k$. These constants are exponential periods over the exponential base field $\mathbb{E}=\mathbb{Q}(t,e^{-t})$. We generalize this result to a broad class of General Stirling-Ramanujan constants. Given polynomials $P$ and $Q$, with $\Re Q(s)>0$ for $\Re s >0$, the constant $S(P,Q)$ is the Ramanujan summation of the series $\sum_{k\geq 1} P(k)\log Q(k)$. They are exponential periods over $\mathbb{K}_P((\omega)) (t,e^{-t}, (e^{-\omega t}))$ where $\mathbb{K}_P$ is the field of definition of $P$ and $(\omega)$ are the zeros of $Q$. These constants are related to the derivatives $\zeta'_H(-n,w)$ of Hurwitz zeta function at negative integers, which we prove are exponential periods over $\mathbb{E} \left ( w,e^{-wt}\right )$. They can be expressed using Bendersky Gamma functions $\hat \Gamma_n$ and the values $\log \hat \Gamma_n(w)$ for $\Re w >0$ are exponential periods over $\mathbb{E} \left ( w,e^{-wt}\right )$. The logarithms of the determinants of the Laplacian on spheres and lens spaces are exponential periods over $\mathbb{E}$. The values $\zeta(2n+1)/\pi^{2n}$ are exponential periods over $\mathbb{E}$. For a periodic function $\chi:\mathbb{Z}\to \mathbb{C}$, we extend Ramanujan summation to the twisted series $\sum_{k\geq 1} \chi(k) P(k)\log Q(k)$ and define General Twisted Stirling-Ramanujan constants $S_{\chi}(P,Q)$. We derive integral formulas proving that they are exponential periods over $\mathbb{E}(\chi )=\mathbb{Q}(\chi)(t,e^{-t})$. For $n\geq 0$, $L_\chi'(-n)$ and $L_\chi(n+1)/\pi^n$ are given in terms of these constants and are exponential periods over $\mathbb{E}(\chi )$.

In 2002, Lins Neto introduced three remarkable one-parameter families of holomorphic foliations by curves on the complex projective plane, of degrees two, three, and four, whose properties established that the general form of the Poincaré Problem had no solution. These foliations are known to be birationally equivalent to certain quotients of linear foliations on abelian surfaces. We give explicit formulas for these birational equivalences, obtaining, in particular, parametrizations of the leaves of the foliations. For the families of degrees three and four, we determine explicit generators for the groups of birational transformations of the projective plane preserving them. For foliations in these families admitting a rational first integral, that is, for those whose parameter is an Eisenstein rational, we give a complete description of the nature and position of the singular points of a generic integral curve, and we present an algorithm that computes the rational first integral explicitly. Lins Neto's foliations can also be defined over algebraically closed fields of positive characteristic, and, in this setting, we characterize those that are algebraically integrable. Finally, we study the integrability of the reductions to fields of positive characteristic of some of the non-integrable foliations in the complex family.

For $\mu>0$, let \[ M(\mu)=\left\{(z,w)\in\mathbb{C}^2 : |z|^2+|w|^{2/\mu}<1\right\} \] be the Thullen domain, and let $g_{\mathrm{CY}}$ denote its complete Cheng–Yau Kähler–Einstein metric, normalized by \[ \operatorname{Ric}(g_{\mathrm{CY}})=-3g_{\mathrm{CY}}. \] We prove that, for every $6/7\leq\mu<1$, a suitable rescaling of $g_{\mathrm{CY}}$ admits a global holomorphic isometric immersion into the infinite-dimensional complex hyperbolic space $\mathbb{CH}^{\infty}$. To the best of our knowledge, these are the first examples of complete nonhomogeneous Kähler–Einstein manifolds admitting such an immersion. By \cite[Lemma~6]{DSIL2012}, the same metrics also admit Kähler immersions into the flat Hilbert space $\ell^2(\mathbb{C})$. Since the Thullen domains considered here are nonhomogeneous, these examples are neither totally geodesic complex hyperbolic spaces nor products of rescaled complex hyperbolic spaces. Consequently, they provide counterexamples to \cite[Conjecture~4.1]{LoiZedda2018}, for both the complex-hyperbolic and flat Hilbert-space alternatives, and also disprove the earlier flat-Hilbert rigidity conjecture formulated in \cite[Remark~10]{LoiZedda2011}. As a further consequence, the flat realization yields complete nonhomogeneous $\eta$-Einstein Sasakian manifolds admitting global Sasakian immersions into the infinite-dimensional Heisenberg space form.

We prove $C^{k/2,\alpha/2}$ -regularity up to a non-umbilic elliptic complex point for the Bishop family of holomorphic discs with boundary in a $C^{k,\alpha}$ regular real surface. Furthermore, we prove existence and regularity of holomorphic discs near certain complex points of index $\ge 2$. The proof employs a novel blow-up of the real surface which resolves the complex point to a pair of totally real surfaces and leads to a $\mathbb{Z}_2$ -equivariant Riemann-Hilbert problem for holomorphic annuli. The index is computed to be 1 and the problem is shown to be Fredholm-regular.

A compact complex manifold has the bounded mass property if, for one (equivalently, every) Hermitian form $\omega$, the masses $\int_X(\omega+\mathrm{dd}^{\mathrm{c}}\varphi)^n$ are uniformly bounded over all smooth $\varphi$ with $\omega+\mathrm{dd}^{\mathrm{c}}\varphi>0$. We prove that this property fails on the Hopf threefold $(\mathbb C^3\setminus\{0\})/\langle z\mapsto\mathrm{e}^{-1}z\rangle$, answering a question of Boucksom–Guedj–Lu.

Let $f:(X,\Delta)\rightarrow Y$ be a surjective holomorphic map between two normal Kahler varieties, where (X,\Delta) is a log canonical pair, -(KX+\Delta) is nef and Y is Q-Gorenstein. In Fu-Guo-Song-Wang [18], it is proved that -KY is pseudo-effective if f is a projective morphism. In this paper, prove that -KY is pseudo-effective without assuming that f is projective

In this paper, we establish several higher-dimensional generalizations of refined Bohr-type inequalities for bounded holomorphic functions mapping into the unit polydisk $\mathbb{P}\Delta(0;1_n)$ in $\mathbb{C}^n$. First, we formulate multidimensional analogues of sharp Bohr-type inequalities originally established by Liu et al. [Bull. Sci. Math. 173 (2021) 103054], incorporating both squared coefficient terms and area functional components. Second, we provide improved inequalities for a recent multidimensional extension by Ahamed et al. [Complex Anal. Oper. Theory 20(6) (2026), 142] by introducing an analogous term corresponding to the area functional. Finally, we extend a refined Bohr-type inequality involving the term $\vert{}f(z)-a_0\vert{}$ to the setting of several complex variables. All the results are shown to be sharp.

We investigate the asymptotic behavior of Jacobi-Piñeiro polynomials of degree $2n$ orthogonal on $[0,1]$ with respect to weights $w_j(x) = x^{\alpha_j}(1-x)^{\beta}$, $j=1,2$ where $\alpha_1,\alpha_2, \beta>-1$, and $\alpha_1-\alpha_2 \notin \mathbb{Z}$. These polynomials are characterized by a Riemann-Hilbert problem for a $3 \times 3$ matrix-valued function. We use the Deift-Zhou steepest descent method for Riemann-Hilbert problems to obtain strong uniform asymptotics in the complex plane. The local parametrix around the origin is constructed using Meijer G-functions. We match the local parametrix around the origin with the global parametrix with a double matching, a technique that was recently introduced.

For a countable family of hyperplanes $H_j\subset \mathbb{P}^m$, $j\in\mathbb{N}$, in general position and a linearly nondegenerate holomorphic curve $f\colon \mathbb{C}\to \mathbb{P}^m$ of finite lower order, we prove that the Nevanlinna defects $\delta_f(H_j)$ satisfy $$ \sum_{j=1}^{\infty}\delta_f(H_j)^{1/3}<\infty. $$ This resolves a long-standing open problem in Nevanlinna theory and extends Weitsman's celebrated scalar endpoint theorem (the case $m=1$) as well as Krutin's results for exponents strictly greater than $1/3$. The same uniform finite-family estimate yields the corresponding endpoint theorem for divisors cut out on a projective variety by ambient hypersurfaces of uniformly bounded degree, assuming that the divisors are in general position with respect to the variety and that the curve is not contained in the support of any divisor.

We characterize the positive Borel measures $\mu$ on the unit disc $\mathbb{D}$ for which the Möbius-invariant space $Q_K$ embeds continuously or compactly into $L^2(\mu)$. The characterization is given in terms of a discrete dyadic capacity $C^{(b)}_{K,\mathcal{R}}(\mu)$ built from a polar dyadic resolution of $\mathbb{D}$, and of an equivalent capacity $D^{(b)}_{K,\mathcal{R}}(\mu)$ expressed as a semidefinite program. The equivalence of the two capacities is established through conic duality and the complex Grothendieck inequality. As an application, we characterize the boundedness and compactness of the Volterra integral operator $T_g$ on $Q_K$, bridging and completely resolving the gap between the sufficient and necessary conditions established by Li and Wulan (2010). We also obtain a complete non-testing characterization of the pointwise multipliers $\mathcal{M}(Q_K)$ on $Q_K$, thereby answering an open problem posed in the survey of Bao and Wulan (2021).

We study the generalized Hilbert operator \[ \mathcal{H}_b f(z)=\int_0^1 f(t)\,\frac{(1-t)^b}{(1-tz)^{b+1}}\,dt, \qquad b>0, \] acting on the Hardy spaces $H^p$ for $1\leq p\leq \infty$. We establish the precise operator norm \[ \|\mathcal{H}_b\|_{H^p\to H^p}=B\!\left(\frac1p,b+1-\frac1p\right) \] for every $1<p<\infty$ and, by continuous extension to $b=0$, recover the classical norm $\pi/\sin(\pi/p)$. We also prove that $\mathcal{H}_b$ is bounded on $H^1$ for every $b>0$, in contrast with the classical Hilbert operator, and we obtain the sharp restricted norm estimate \[ \|\mathcal{H}_b\|_{H^1_0\to H^1}=B(1,b). \] We also determine the exact norm \[ \|\mathcal{H}_b\|_{H^\infty\to \mathcal B}=\frac{1}{b+1}+2. \]

We prove uniform a priori estimates for solutions to degenerate complex Monge–Ampère equations in big cohomology classes, using both auxiliary-function technique developed by Guo, Phong and Tong [On $L^\infty$-estimates for complex Monge-Ampère equations, Ann. of Math. (2) 198 (2023), no.1, 393-418], and quasi-psh envelope approach developed by Guedj and Lu [Quasi-plurisubharmonic envelopes 1: Uniform estimates on Kähler manifolds, J. Eur. Math. Soc. (JEMS) 27 (2025), no. 3, 1185-1208.]. As an application, we apply our method to prove the Moser-Trudinger and Brezis-Merle-type inequalities for complex Monge-Ampère equations.

We study the extension of the conformal structure of a Riemann surface, obtained as a submanifold of $\mathbf R^n$, across an isolated singular point, under hypotheses that are metric rather than analytic. The main result (1989) is that an isolated singularity is $\textbf{conformally point-like}$ (conformal to a punctured disc) whenever it is $\textbf{$M$-regular}$: the pair (surface minus the point, the point) satisfies a Whitney condition, and the length of the spherical slice $S(0,r)\cap E$ decreases at most linearly in $r$, a condition described as “metric decay”. This is proved via a modulus-of-rings (extremal-length) argument, generalizing the classical planar technique to submanifolds of $\mathbf R^n$. Two classes of examples are treated: surfaces of revolution generated by a single curve, and subanalytic surfaces, for which $M$-regularity is verified directly from Hironaka's structure theory, giving as a corollary that every isolated singularity of an orientable subanalytic surface is conformally point-like. A further original result is a $C^\infty$ counterexample showing that a strict Whitney condition alone does $\textbf{not}$ imply the linear length bound: the two hypotheses in the definition of $M$-regularity are independent.

The titular object of this paper is an analogue of the Mandelbrot set over the 3-dimensional real algebra whose multiplication is the bilinear extension of the rock-paper-scissors operation. In order to study the dynamics of the mappings $x\mapsto x^2+c$ in this setting, a notion of holomorphy for functions on general finite-dimensional real algebras is introduced. Under a mild assumption it is shown that for such algebras holomorphy is always equivalent to solving a finite system of linear first-order PDEs generalizing the Cauchy-Riemann equations. Code is provided for generating animations of these fractals and for exploring them in a video game format.

We construct a polarized smooth projective fivefold and prove that it is K-polystable but does not admit a constant scalar curvature Kähler metric. This disproves the Yau–Tian–Donaldson conjecture for constant scalar curvature metrics. The main result of this paper was obtained using generative AI, particularly GPT-5.6-sol, Fable 5, and the Danus system. A detailed report on the use of generative AI in this paper is enclosed in the appendix, joint with Bin Dong and Guoxiong Gao.

Let $\Phi$ be the classical Jacobi-theta kernel in the Fourier representation of the Riemann $\Xi$-function, set $s(t)=\Phi(\sqrt t)$, and define the first Laguerre expression $f(t)=s'(t)^2-s(t)s''(t)$. Csordas and Dimitrov (2000) conjectured that $\log f$ is strictly concave on $(0,\infty)$; Csordas (2015) later restated the assertion as Open Problem~4.14. We prove the conjecture by two complementary methods, sharing only a local certificate near the modular fixed point. The first proof is a direct theta-series argument: a directed-rounding Taylor certificate near $t=0$ is joined to a dominant-first-summand estimate with rigorous theta-tail bounds. The second proof uses Jacobi's nonlinear third-order differential equation for $\theta_3$ to obtain a three-dimensional autonomous phase space, a sharp elliptic monotonicity theorem, and an exact quartic reduction of the target inequality. The quartic boundary is a polynomial shear of the quadratic cone $XY=Z^2$. A directed interval certificate proves that every possible cone contact on the only remaining compact interval points strictly into the desired region; beyond that interval a pointwise monotonicity theorem closes the argument. The finite certificates and exact symbolic checks are supplied as reproducible scripts. The result implies the associated double Turán inequalities through the theorem of Csordas–Dimitrov, but no assertion of the Riemann Hypothesis is made.

We describe certain finite-time singularities of the Kähler-Ricci flow arising in the analytic minimal model program. Assuming that convergence to an asymptotically conical Kähler-Ricci shrinker is realized by holomorphic maps, we prove that, in a fixed holomorphic gauge, the nearby flow is modeled on the shrinker at the level of Kähler potentials. Consequently, every noncollapsed Kähler-Ricci flow through singularities in complex dimension two is modeled on a shrinker-cone-expander transition, confirming a strong form of Song's conjectural picture. We also show analogous results in higher dimensions under the Calabi ansatz, and improve known results in the compact shrinker case. These give the first compact Ricci flows through conical singularities whose small-scale behavior is fully described.

Let $g$ be analytic in the unit disc and consider the generalized Hilbert operator $$ \mathcal{H}_g(f)(z)=\int_0^1 f(t)g'(tz)\, dt. $$ The boundedness of $\mathcal H_g$ on $H^p$ is characterized by the mean Lipschitz condition $g\in\Lambda\left(p,\frac{1}{p}\right)$ when $1<p\leq2$, while the problem remains open for $2<p<\infty$. It has been recently proved that the condition $g\in\Lambda\left(p,\frac{1}{p}\right)$ does not imply the boundedness of $\mathcal H_g$ on $H^p$, $2<p<\infty$ \cite{GuoTang2026}. We show that this condition is far from sufficient in the latter range: for every $2<p<\infty$, there exists a function $g\in\Lambda\left(p,\frac{1}{p}\right)$ such that $\mathcal H_g$ is not bounded even from $H^p$ into $H^1$. The main ingredient is an exact characterization of the boundedness of $\mathcal{H}_g:H^p\to H^2$ for all $1\leq p\leq\infty$. In particular, when $2<p<\infty$, this mapping is bounded if and only if $g'$ belongs to a certain mixed-norm space. For lacunary symbols, the same mixed-norm condition also characterizes the boundedness of $\mathcal H_g$ on $H^p$, and hence gives a complete solution of the open problem within this class of symbols. We also show that, for $1\leq q\leq\infty$, boundedness of $\mathcal H_g:H^1\to H^q$ is characterized by the condition $g'\in H^q$. We also characterize compactness of $\mathcal H_g$ in the aforementioned cases.

This paper relates the minimal Harder–Narasimhan slope to the surjectivity of real-linear Cauchy–Riemann operators. We establish a conformally invariant $L^2$ criterion and an asymptotic slope criterion, which yield higher-rank automatic transversality criteria for pseudoholomorphic curves beyond the classical rank-one framework. Applications to pseudoholomorphic spheres in $S^6$ provide quantitative $L^2$ obstructions to the integrability of almost complex structures.

Let $g:\mathbb{D}\to\mathbb{D}$ be a hyperbolic holomorphic self-map of the unit disk with Denjoy–Wolff point $\tau\in\partial\mathbb{D}$ and angular derivative $\alpha\in(0,1)$. We say that $g$ has an extremal hyperbolic rate if its forward iterates satisfy the sharp metric asymptotic \begin{equation*} \rho_{\mathbb{D}}(g^{\circ n}(z),w) = n\log\frac{1}{\alpha} + O(1) \quad\text{as }n\to\infty, \end{equation*} for every $z,w\in\mathbb{D}$. Using the Herglotz–Nevanlinna representation, we prove that $g$ has an extremal hyperbolic rate if and only if the associated boundary measure $\sigma$ satisfies \begin{equation*} \int_{\partial\mathbb{D}\setminus\{\tau\}} \log\frac{1}{|\zeta-\tau|} \,d\sigma(\zeta) < \infty. \end{equation*} We further show that this condition is equivalent to a non-degenerate angular asymptotic of the Koenigs linearization: for any conformal map $\phi_\tau:\mathbb{D}\to\mathbb{H}$ with $\phi_\tau(\tau)=\infty$, \begin{equation*} 0< \left| \angle\lim_{z\to\tau} \frac{h(z)}{\phi_\tau(z)} \right| < \infty, \end{equation*} where $h$ is a Koenigs function. We then extend the extremal-rate theory beyond the unit disk. For finitely connected hyperbolic planar domains, the disk characterizations transfer to ordinary boundary points of the associated deck transformation group. For holomorphic self-maps of the unit ball $\mathbb{B}^n$ and for $K$-quasiconformal self-maps, where no comparable Herglotz representation is available; we establish sufficient boundary regularity conditions that guarantee the extremal hyperbolic rate. The quasiconformal result is further extended to finitely connected planar domains at ordinary boundary points. These results show that, across the settings considered here, extremal hyperbolic growth is governed by the non-degeneracy of the boundary linearization at the Denjoy–Wolff point.

In the present paper it is discussed to which extend conformal mappings of special circular triangles, which are thoroughly examined by M. Bonk using analytic and classical special functions methods, can be defined and investigated geometrically. This is in analogy to the elliptic modular function, which admits an analytic and also a geometric definition.

By constructing explicit entire maps, we prove that every exceptional Morimoto–Nagano model $M_t^3$, $t>1$, can be realized as a compact real-analytic hypersurface in $\mathbb C^3$. This extends Isaev's realization results from restricted parameter ranges to all $t>1$ and provides a complete solution to a long-standing open problem in the Morimoto–Nagano classification of compact, simply connected, real-analytic hypersurfaces in complex Euclidean spaces that are homogeneous under the action of a connected Lie group of CR automorphisms.

In a recent article of the authors, we proved a result called the Isomorphism Theorem for holomorphic maps from an irreducible Shimura varieties of rank $\ge 2$. The proof of the Isomorphism Theorem uses in essential ways Kähler geometry, function theory of several complex variables, harmonic analysis and ergodic theory. Here we will focus on a slight variation of the Isomorphism Theorem where the target is uniformized by a simply connected complete Kähler-Einstein manifold $(M,h_M)$ which is moreover assumed to be Carathéodory hyperbolic (i.e., the infinitesimal complex Finsler pseudometric $\kappa_M$ induced from the space of bounded holomorphic maps into the Poincaré disk is a complex Finsler metric) and $\Gamma' \subset {\rm Aut}(M)$ is a torsion-free discrete subgroup such that the quotient manifold $Y_{\Gamma'} := M/\Gamma'$ is of finite volume with respect to the quotient Kähler-Einstein metric. In this setting, we will explain the essential roles played by Kähler geometry, harmonic analysis and ergodic theory in the proof of the Isomorphism Theorem.

Let $[p]$ be the principal submodule generated by a polynomial in $H^2(\mathbb D^2)$. For homogeneous $p$, the homogeneous slices of $[p]$ admit a weighted OPUC model in which the two wandering vectors are an orthonormal polynomial and its reversal. We show that the associated Verblunsky coefficients determine the singular values of the wandering-projection product and the restricted cross-commutator, as well as the non-zero spectrum of the core operator. Toeplitz-determinant and Mahler-measure identities yield exact Fredholm determinants and Schatten estimates, while $[(z-w)^N]$ rules out a uniform Hilbert–Schmidt bound. The same model gives explicit singular values of $[S_z^*,S_w]$ on the homogeneous quotient $H^2(\mathbb D^2)\ominus[p]$; for $p=(z-w)^N$, its squared Hilbert–Schmidt norm is asymptotic to $N$. For arbitrary polynomial generators, we construct a weighted bivariate model with a doubly Toeplitz, block-banded moment matrix and prove $C_p^2|_{\mathscr E_z}=\Gamma_p^*\Gamma_p$, relating the core spectrum to the cross-Gram operator between the two edge spaces. We also discuss cyclic-factor obstructions, represent higher numerical invariants by alternating CMV products, and give a quadratic counterexample to their proposed monotonicity.

We extend the Donaldson-Sun theory of metric tangent cones to non-collapsing Gromov-Hausdorff limits of conical Kahler-Einstein pairs whose boundary coefficients lie in a fixed finite subset of Q, and prove uniqueness of the log metric tangent cone. Furthermore, under a mild lc compatibility condition, we relate this with the Li-Xu and Li-Liu-Xu stable degeneration machinery. We also construct polarized smooth Kaehler metrics on CP^2 with a uniform lower Ricci bound and volume non-collapsing, whose limit has nonunique tangent cones, showing that the Kahler-Einstein assumption is crucial for rigidity.

For $k\geq1$, let $M_k=[(z-w)^k]\subset H^2(\mathbb D^2)$. We first determine the banded Toeplitz matrices associated with the homogeneous components of $M_k$, together with explicit formulas for their determinants and the relevant algebraic cofactors. These formulas lead to a complete description of the spectrum of the core operator: \[ \sigma(C_{M_k}) = \{0,1\} \cup \left\{ \pm\frac{k}{n+k}:n\geq1 \right\}. \] In particular, the spectral data determine the parameter $k$. The determinant and cofactor formulas further yield a unified finite-sum representation for $\alpha_{n,j}^{(k)} =\langle w^j\phi_n,z^j\psi_n\rangle$, and hence for Yang's higher numerical invariants. We derive an adjacent relation connecting $\alpha_{n,j}^{(k)}$ and $\alpha_{n,j+1}^{(k)}$ by means of an explicit telescoping certificate, and show that the corresponding finite-section transformations are strict contractions. Combining these finite-dimensional estimates with the asymptotic behavior of $\alpha_{n,j}^{(k)}$, we prove the strict monotonicity \[ \Sigma_0(M_k)> \Sigma_1(M_k)> \Sigma_2(M_k)> \cdots . \] The cases $k\geq3$ constitute the new part of the analysis, while the previously known cases $k=1,2$ are recovered within the same framework. Consequently, Yang's monotonicity conjecture holds in strict form for the entire family $\{[(z-w)^k]:k\geq1\}$.

For a bounded operator \(T\) between Banach spaces, we introduce the fixed-copy exponent sets \[ \begin{aligned} \operatorname{Fix}_{\ell}(T) &:= \{r\ge1:T\text{ fixes a copy of }\ell^r\},\\ \operatorname{Fix}_{L}(T) &:= \{r\ge1:T\text{ fixes a copy of }L^r(0,1)\}. \end{aligned} \] For \(1\le p,q<\infty\), we completely determine both sets for every bounded composition operator \(C_\varphi:H^p\to H^q\). In particular, \(C_\varphi\) is strictly singular if and only if \(\operatorname{Fix}_{\ell}(C_\varphi)=\varnothing\). The classification also gives complete characterizations of the \(\ell^r\)-singular and \(L^r(0,1)\)-singular subclasses for every \(r\ge1\). As a further consequence, it completely resolves Problems~4.3(1) and~4.3(2) posed by Laitila, Nieminen, Saksman, and Tylli. The proofs introduce a new localization method for producing fixed copies from boundary lower estimates. Its main ingredient is a localization theorem independent of composition operators: for every measurable \(E\subset\mathbb T\) with \(m(E)>0\) and \(1\le p<r\le2\), it constructs a single copy of \(L^r(0,1)\) in \(H^p\) on which lower \(L^s(E)\) estimates hold simultaneously for all \(1\le s\le p\), with constants depending on \(E\) only through \(m(E)\). The classification combines this method with pullback measure criteria, known fixed-copy results, and classical subspace restrictions. For \(r<2\), the localization theorem is proved using stable integrals and analytic lifting; the endpoint \(r=2\) is handled by an \(E\)-adapted lacunary construction.

Let $B$ denote Bloch's constant. We prove \[ B \ge \frac{\sqrt{3}}{4}+0.0153 = 0.448312701\ldots, \] improving the lower bounds by Chen–Gauthier ($\sqrt{3}/4+2\cdot10^{-4}$) and Xiong ($\sqrt{3}/4+3\cdot10^{-4}$). The proof refines Bonk's method through computer-assisted estimates rigorously verified using interval arithmetic.