In this paper we establish optimal Lorentz estimates for the Riesz potentials acting on closed or co-closed $k$-forms of finite mass on a smooth, compact Riemannian manifold of dimension $n$: For $α\in (0,n)$ and $k=1,\ldots,n-1$, there exists a constant $C>0$ such that \begin{align*} \| \mathcal{I}_{α,k} F \|_{L^{n/(n-α),1}(Λ^k)} \leq C \| F\|_{L^1(Λ^k)} \end{align*} for all $k$-forms $F \in L^1(Λ^k)$ orthogonal to the space of harmonic $k$-forms and satisfying $\mathrm{d} F=0$ or $\mathrm{d}^* F=0$. We show how this inequality implies analogous Lorentz bounds for solutions of the $k$-form Poisson equation and for the Hodge system with data having finite mass. These results include as a special case the div--curl system on the $3$-dimensional torus, where we answer an open question originally posed by J. Bourgain and H. Brezis.

Pinching antenna (PA) systems have recently emerged as a promising flexible-antenna technology, which can reconstruct the wireless propagation environment by dynamically adjusting the positions of pinching elements along dielectric waveguides, thereby providing new spatial degrees of freedom (DoFs) for enhancing wireless system performance. This paper investigates a multi-waveguide PA-based multi-cell communication system, focusing on the joint optimization of precoding matrices, waveguide power allocation, and antenna placement to maximize the weighted sum rate (WSR). In multi-cell scenarios, inter-cell interference typically leads to a highly coupled and nonconvex WSR maximization problem. To address this challenge, an efficient alternating optimization framework is adopted to optimize each variable in an iterative way. Specifically, fractional programming is first employed to reformulate the original problem by introducing auxiliary variables that decouple the signal and interference terms. Based on this reformulation, block coordinate descent is then applied to optimize the precoding matrices and power allocation, leading to closed-form or semi-closed-form updates. For the high-dimensional and nonconvex PA placement problem, particle swarm optimization (PSO) is utilized to perform an efficient search and improve scalability. Numerical results demonstrate that, under various system configurations, the proposed scheme significantly outperforms baseline methods, including average power allocation, fixed antenna placement, conventional multiple-input multiple-output (MIMO), and massive MIMO. These results highlight the strong potential of PA systems for large-scale multi-cell wireless communications.

We study a variant of the $k$-in-a-row game in which players alternatively claim positions until a $k$-in-a-row is created among all claimed positions. This leads to the constraint near $k$-in-a-row avoiding on configurations and the associated problem of determining their extremal densities of such configurations. We investigate this problem on two types of boards: the grid $\mathbb{Z}^2$ and hypercubes $[k]^d$. For the grid $\mathbb{Z}^2$, we establish nearly tight bounds on the maximum density $D(k,\mathbb{Z}^2)$, showing that $D(k,\mathbb{Z}^2)=1-\frac{2}{k}$ whenever $3\nmid k$, and determine both $D(3,\mathbb{Z}^2)$ and $d(3,\mathbb{Z}^2)$ exactly. We also bound the minimum density $d(k,\mathbb{Z}^2)$ up to a gap of $(8+o(1))k^{-1}$. For hypercubes $[k]^d$, we derive asymptotic bounds on $D(k,[k]^d)$ up to order $k^{-2}$ and obtain the exact value of $d(k,[k]^d)$. Our results contrast with the classical no-$(k+1)$-in-line problem, a similar problem imposing different constraint, where the trivial upper bound is conjectured to be attainable.

We study continuous-time reinforcement learning for the optimal switching problem, in which a decision-maker controls a diffusion process by switching among finitely many regimes, incurring both running and transition costs. To enable exploration, we relax the classical deterministic switching control to a randomized framework, where the switching decisions are governed by a continuous-time Markov chain with state-dependent generator, and augment the cost functional with a KL-divergence regularization weighted by a temperature parameter $λ$. Under mild assumptions on the coefficients, we establish that the regularized value function is the unique smooth solution of an elliptic Hamilton--Jacobi--Bellman system, and derive an explicit optimal Gibbs policy given by an exponential transformation of the value function differences across modes. We further prove that the regularized value function approximates the classical optimal value function with error of order $O\left(λ\log \frac{1}λ\right)$, which is consistent with analogous bounds established in other entropy-regularized control problems and is believed to be sharp. To solve the regularized problem numerically, we introduce a mirror descent flow in the dual logarithmic policy space, prove its well-posedness and the monotonic decrease of the value function along the flow, and establish quantitative error bound to the classical optimal value function. For a constant temperature scheduler, the convergence rate is of order $O\left(\frac{1}{e^{λs} - 1}+λ\log\frac1λ\right)$, while under the annealing scheduler $λ_s = \frac{1}{\sqrt{1+s}}$, we obtain the rate $O\left(\frac{\log s}{\sqrt{s}}\right)$, which decays to zero as the flow time $s \to \infty$.

In this paper, we propose a new approach, called maximum utility split (MUS) scheme, which is built on random utility split (RUS) scheme but with a notable difference: one lottery is designed with two fixed outcomes but with varying probability, and the other has a deterministic outcome specifically chosen at the point where the range between the largest and smallest possible utility values is maximized. Consequently, the probability of random lottery is set such that the range of the ambiguity set of utility functions is reduced by half at the point. Under moderate conditions, we show that MUS can successively generate a sequence of such questionnaires and effectively reduce the ambiguity set, eventually converging to the true utility function as the number of questionnaires increases. The main challenge is to effectively identify the point with the largest utility range for a given ambiguity set constructed from preference information. Based on the structure of the ambiguity set, we propose an interval-based algorithm which identifies each certain-outcome lottery by solving a sequence of linear programs. Moreover, to deal with the case where elicitation terminates before the ambiguity set reduces to a singleton, we demonstrate how to figure out a nominal utility function by solving optimization programs. These identify the smallest and largest utility functions under the Kantorovich metric within the ambiguity set, after which we identify a nominal utility function located in the middle of them. Finally, numerical results demonstrate the efficiency of the MUS method and the performance of a robo-advisor system based on MUS-type queries and the nominal utility elicitation. While the main discussions focus on concave utility functions, we also demonstrate how the MUS approach can be extended to accommodate general non-concave utility functions, particularly S-shaped ones.

We introduce certain families of AF algebras associated to Bratteli diagrams arising from numerical semigroup theory, a branch of combinatorics. Curry-Schoenberg B-splines, staples of computer-aided design, provide insight into the statistical properties of these algebras. This permits us to consider certain ensembles of "rapidly growing" AF algebras from a probabilistic viewpoint.

The aim of this paper is to study $m$-partial isometries on Hilbert spaces, a natural extension of partial isometries and $m$-isometries. We establish structural and spectral results, characterize the $m$-partial isometric weighted shifts, and investigate similarity to $m$-isometries and $m$-partial isometries.

We introduce a partition-selection method to generalize the flow, chromatic, and Tutte polynomials of a graph by restricting the standard edge subgraph expansions to subgraphs given by prescribed connected vertex partitions. We establish similar deletion-contraction formulas and specialization relations for these polynomials, recovering all classical polynomial invariants when the selection is the set of all partitions. Next we study a relation between Jaeger et al.'s nonhomogeneous flows and a special class of partition-selected flow polynomials (called affine flow polynomials). Specifically, we give a geometric realization of nowhere-zero nonhomogeneous flows by restricting the edge-coordinate arrangement to affine flow spaces. The resulting characteristic polynomials coincide with Kochol's admissible assigning polynomials and with affine flow polynomials, which enumerate nowhere-zero nonhomogeneous flows over finite fields. To see the key role of the partition-selection framework, we further introduce boundary arrangements determined by the bond structure of a graph. Using the intersection posets of boundary arrangements, we obtain the classification of all restricted arrangements mentioned above, the comparison of unsigned coefficients of affine flow polynomials, and the decomposition formulas for affine flow polynomials.

We compute the capacity region of a sum of broadcast channels whose components are degraded, less-noisy, more-capable, deterministic, or semi-deterministic. We achieve this by showing that an auxiliary-receiver outer bound, previously introduced by some of the authors, matches Marton's inner bound. This result generalizes a previously known result for the sum of two reversely degraded broadcast channels due to El Gamal (1980). Moreover, we define a class of primary broadcast channels and show an analogous result for the sum of primary broadcast channels.

Spherical Interpolation of orDER $n$ (SIDER-$n$) is a recursive high-order interpolation method for data on the unit sphere $\mathbb{S}^2$, built from repeated spherical linear interpolation (SLERP). This paper develops a quaternion--Lie algebra framework for proving the local consistency of SIDER for smooth spherical curves sampled at equally spaced parameter values. Points on $\mathbb{S}$ are represented as pure unit quaternions, and interpolation errors are measured in fixed-base quaternion logarithmic coordinates. In this setting, each SLERP operation admits an exact Baker--Campbell--Hausdorff (BCH) representation, which converts the geometric interpolation problem into an algebraic problem involving filtered Lie-polynomial expansions. The BCH expansion shows that SLERP is affine to leading order, has no quadratic correction, and has a first nonlinear correction that is cubic and commutator-valued. Using this structure, we prove that SIDER2 has a third-order divided-error form with the same leading nodal factor as ordinary quadratic interpolation. We then show that the recursive SIDER step raises the order by one: the affine part gives the Neville-type finite-difference cancellation, while the nonlinear BCH remainder preserves the sharp filtered degree structure after the nodal factor is removed. Consequently, for every fixed $n\geq2$, $d_{\mathbb{S}^2}\bigl(γ(θh),P_i^{[n]}(θ;h)\bigr) = O(h^{n+1}) $under the stated smoothness and small-stencil assumptions. The proof also identifies the shift-invariance of the leading divided-error coefficient as the algebraic compatibility condition underlying the SIDER recurrence.

Let $T_{1}, T_{2}, \dots, T_{k}$ be $k$ spanning trees of a graph $G$. For any pair of vertices $u$ and $v$, if the $u$--$v$ paths in the $k$ spanning trees are pairwise openly disjoint, then the spanning trees are called completely independent spanning trees (CISTs) of $G$. In this paper, we first prove that every 3-connected 2-outerplanar triangulated disc has two completely independent spanning trees. Next, for a 3-connected 3-outerplanar triangulated disc $G$, we provide sufficient conditions for $G$ to have two completely independent spanning trees. We provide an example of a 3-connected 4-outerplanar triangulation that does not have two completely independent spanning trees.

This paper investigates a class of nonautonomous multiscale McKean-Vlasov stochastic systems. By leveraging the nonautonomous Poisson equation, we rigorously establish both strong and weak averaging principles, accompanied by explicit convergence rates. Notably, the coefficients of the averaging equations derived in the general case retain dependence on the scaling parameter $\varepsilon$. However, under the additional assumptions that the fast-scale coefficients are either asymptotically convergent or time-periodic, we demonstrate that the slow component converges, in the strong or weak sense, to averaging equations with coefficients independent of $\varepsilon$.

We associate to each positroid variety in the Grassmannian $\mathrm{Gr}(k,n)$ a polyhedral complex, which we call a fence complex. Fence complexes consist of unions of faces of the Gelfand-Tsetlin polytope $P_{k,n}$ associated to a fundamental weight $ω_k$. We show that these fence complexes are homeomorphic to closed balls. Furthermore, they endow the Gelfand-Tsetlin polytope with the structure of a regular CW complex, giving a polyhedral complex presentation of the regular CW complex structure on $\mathrm{Gr}(k,n)_{\geq 0}$. We also show that the Ehrhart polynomial of a fence complex equals the Hilbert polynomial of the associated positroid variety. We prove that under the Sturmfels-Gonciulea-Lakshmibai degeneration of $\mathrm{Gr}(k,n)$ to the toric variety of the Gelfand-Tsetlin polytope, positroid varieties degenerate to the reduced union of toric varieties corresponding to their fence complexes. As an application, we classify when positroid varieties contained inside hook Schubert varieties are arithmetically Gorenstein. We also derive a recursive character formula for cyclic Demazure modules, which we show is equivalent to a formula of Almousa, Gao and Huang.

We study the splitting behavior of parametrized families of polynomials over finite fields through a geometric and Galois-theoretic approach. While the underlying techniques are widely considered folklore in arithmetic geometry, they have rarely been written down explicitly. To maximize accessibility, we develop a framework based on classical Galois theory and the Chebotarev Density Theorem over an affine normal variety, avoiding the heavy machinery of Grothendieck's étale topology. The primary goal is to extend and conceptually explain a recent result by Slavov, which established the condition for square values of several polynomials over a finite field to be independent. In the case where $q\equiv 1 \pmod n$, we generalize this phenomenon to $n$-th power residues, and reframe this independence condition as the natural condition on Kummer extensions to be mutually linearly disjoint. Finally, we briefly mention how these results can be translated into the modern language of étale fundamental groups, generalizing the base to geometrically integral, normal schemes of finite type over $\mathbb{F}_q$.

We study eigenvalues and eigenfunctions of the ``div-grad type" sub-Laplacian with respect to Popp's volume on a compact equiregular sub-Riemannian manifold $M$. Since Popp's volume is canonically determined by the sub-Riemannian structure of $M$, the spetra of the sub-Laplacian carry geometric meanings. In this paper, we first embed $M$ into the Hilbert space of square-summable sequences using eigenfunctions and then define a spectral distance between two compact equiregular sub-Riemannian manifolds. Our result is a sub-Riemannian analogue of Berard-Besson-Gallot's classical work in the Riemannian case.

Spherical Interpolation of orDER $n$ (SIDER-$n$) is a recursive high-order interpolation construction for data on the unit sphere $\mathbb{S}^2$, built from repeated spherical linear interpolation (SLERP). This paper gives a local consistency analysis of SIDER for smooth spherical curves sampled at equally spaced parameter values. The analysis is carried out in geodesic normal coordinates, which allows the SIDER recursion to be compared with classical Neville interpolation while retaining the curvature-dependent corrections introduced by SLERP. We first derive local expansions of SLERP and show that SIDER2 has third-order accuracy; its leading error has the same shifted nodal structure as Euclidean quadratic interpolation. We then prove that the adjacent SIDER2 errors entering SIDER3 have a common leading coefficient, so that the SIDER3 recurrence cancels the cubic term and yields fourth-order accuracy. Carrying the expansion one order further gives the corresponding coefficient compatibility for SIDER3 and proves fifth-order accuracy of SIDER4. Finally, we introduce a degree-filtered formal expansion framework for the general SIDER recursion. This framework proves that, for each fixed $n$, SIDER-$n$ preserves the required polynomial degree structure in the normalized stencil variable. Together with the interpolation conditions at the $n+1$ nodes, this yields the local consistency estimate $d_{\mathbb{S}^2}\bigl(γ(θh),P_i^{[n]}(θ;h)\bigr)=O(h^{n+1})$ under the stated smoothness and small-stencil assumptions.

Digital elevation models (DEMs) have reached resolutions and sizes that only parallel computaters can efficiently process. One important application of DEMs is predicting how much water flows where, the so-called ``flow routing problem'' (a variation of which is the problem of determining the drainage area upstream of a point in a DEM). The traditional algorithm for flow routing is sequential, and attempts to parallelize this method have so far only been moderately successful. Herein, we build on earlier work in Richardson et al. (2014) and propose an algorithm and several variations that can efficiently solve the flow routing problem on very large models with very large numbers of parallel processes. For the largest model we use, with 1.88 billion points, the best algorithm herein can route water in 4.0 seconds on 12,288 processes of a computer cluster.

In this paper, we present a second-order linear scheme based on the variable-step Alikhanov formula and central difference discretization for the time-fractional Allen-Cahn equation. The nonlinear potential is treated explicitly via a second-order extrapolation with preprocessing, which enables the discrete maximum-bound principle (MBP) to be preserved through an appropriate stabilization technique. Moreover, by developing a discrete fractional Grönwall inequality together with the uniform boundedness of numerical solutions guaranteed by the MBP, we establish an $α$-robust and optimal second-order maximum-norm error estimate under initial weak singularity assumption. In addition, energy stability is proved in the sense that the discrete original energy is uniformly bounded by the initial energy plus a high-order spatiotemporal correction term. Finally, extensive numerical experiments are presented to demonstrate the effectiveness of the proposed scheme.

This paper focuses on investigating a class of suspension bridge models coupled with the Cattaneo heat conduction law, with special attention paid to two distinct scenarios: systems with viscoelastic memory and those without. Operator semigroup theory is adopted as the core mathematical tool to conduct a comprehensive analysis of the models' dynamic behaviors. For the memoryless suspension bridge system ($δ= 0$), we first establish its well-posedness (i.e., the existence and uniqueness of solutions). Further stability analysis reveals that this system exhibits polynomial decay behavior, where the decay rate depends on the structural parameter relationship of the bridge deck: a decay rate of $t^{-1/2}$ is achieved when $\frac{ρ_1}κ = \frac{ρ_2}{b}$, and the rate slows down to $t^{-1/4}$ when $\frac{ρ_1}κ \neq \frac{ρ_2}{b}$. Moreover, exponential stability is proven to be unachievable under specific coefficient configurations (i.e., $χ_0 \neq 0$, or $χ_0 = 0$ and $χ_1 = 0$, with $χ_0, χ_1$ being parameter-dependent coefficients). For the counterpart system incorporating viscoelastic memory ($δ= 1$), the well-posedness of solutions is also verified, and more importantly, the system is shown to achieve exponential decay. This indicates that viscoelastic memory significantly enhances system stability and accelerates energy dissipation compared to the memoryless case. By systematically exploring the regulatory role of viscoelastic memory in the stability of suspension bridge systems under the framework of Cattaneo thermal conduction, this study enriches and extends the existing research on suspension bridge models with viscoelastic memory.

We study abstract dynamic programs on partially ordered spaces, pairing the order-theoretic approach to dynamic programming with topological and metric foundations. We show that readily verifiable forms of topological stability, such as global stability and contractivity of the policy operators, deliver the fundamental optimality properties of dynamic programming together with convergence of value function iteration, Howard policy iteration, and optimistic policy iteration. We also prove that stationary policies dominate nonstationary policy plans under very weak assumptions. Applications include Markov decision processes, structural estimation problems in which maximization and integration are interchanged, optimal stopping without discounting, and Bayesian sequential analysis. For the last two, our results weaken existing assumptions and extend algorithmic guarantees for foundational problems.

This paper presents an AI proof that there is no algorithm deciding whether a polynomial equation over the Gaussian integers in $18$ unknowns has a solution. The proof improves the $20$-unknown theorem of Matiyasevich and Sun. It follows their rationality criterion and integer test, but saves two variables: the clearing-denominator variable is avoided by imposing two integer conditions, and the remaining nonzero condition is absorbed into the relation-combining lemma by the one-variable gadget $(2R+1)(3R+1)$.

Indirect signal production is often treated as a higher-order variant of classical Keller-Segel chemotaxis, but its critical structure depends strongly on how the signal cascade is closed. This paper separates two asymptotic regimes of a two-stage signalling mechanism. In the parabolic--elliptic--elliptic limit, the chemoattractant is generated by the self-adjoint fourth-order operator $K_τ=(I-τΔ)^{-1}(I-Δ)^{-1}$. We prove its spectral positivity, entropy-dissipation structure, fourth-order principal scaling, and logarithmic kernel singularity in four dimensions. Consequently, the correct critical space is $L^{N/4}$, and $N=4$ is the mass-critical dimension. A concentration calculation identifies the natural threshold candidate $M_* = 64π^2τ/χ$, while the sharp threshold theorem is formulated as an Adams/logarithmic-HLS open problem. In contrast, the mixed elliptic--parabolic cascade cannot be reduced to a static fourth-order kernel. Its eliminated signal is a Volterra memory operator whose near-diagonal multiplier has the same order as the classical Keller-Segel drift. Thus its critical theory must be based on mixed space--time estimates, not static elliptic scaling. Numerical experiments support our operator-level distinction.

We study Prandtl's classical problem on minimising the induced drag for a finite wing with fixed span. The induced drag is given by a singular quadratic functional of the circulation, with admissible functions satisfying the prescribed lift and second-moment conditions. We formulate the problem in the fractional Sobolev space \(H^{1/2}\), which is the natural energy space for the functional, prove existence and uniqueness of minimisers by variational methods, and derive the corresponding Euler--Lagrange equation. % Passing to a periodic formulation on the one-dimensional torus, we identify the drag functional with the quadratic form of the half-Laplacian and solve the resulting singular integral equation explicitly and recover Prandtl's bell-shaped circulation profile.

We formulate a conditional scale-critical defect-cascade reduction for the local regularity problem of the three-dimensional incompressible Navier--Stokes equations. The theorem concerns a potential singular point for which no sufficiently small dyadic scale enters the Caffarelli--Kohn--Nirenberg smallness regime. Under the structural hypotheses of the framework, such a point cannot be explained by an undifferentiated concentration of energy or pressure. It must lead either to non-effective moving-window observability or to an NS-realizable, cleaned, scale-critical defect cascade invisible to the combined active-pressure, flux, energy, and adjoint-trace tests. The reduction is built from dyadic rescaling, coarse graining, active/harmonic pressure splitting, Reynolds covariance positivity, pressure compatibility, and local energy-flux identities. Finite-window observability reduces possible invisible directions to explicitly defined residual kernels, while budget compatibility and sign coherence convert visible pressure--flux activity into depletion. Consequently, within a controlled window class where dyadic defect extraction, observable depletion, and moving-window growth control hold, effective observability together with exclusion of NS-realizable combined-invisible cascades yields a CKN scale and hence local regularity. The final component interprets the remaining obstruction as a critical recurrence problem and proposes diagnostics for vortex stretching, active pressure work, and interscale flux. Spatially harmonic pressure is retained as a physical local pressure component; only purely time-dependent pressure functions are treated as gauge.

We examine upper bounds on Nordhaus-Gaddum type problems for parameters related to graph connectivity. Our main result is that for a graph $G$ on $n$ vertices where both $G$ and its complement $G^c$ are connected, then the sum of the algebraic connectivity of $G$ and the algebraic connectivity of $G^c$ cannot exceed $n-3$ (with finitely many exceptions with a small number of vertices). We obtain similar results for the isoperimetric number of a graph, and explore similar Nordhaus-Gaddum type questions for the Cheeger constant and the second eigenvalue of the normalized Laplacian matrix.

We give a detailed, corrected presentation of some fundamentals of the constructive theory of locally convex spaces that appear without proofs in <cite>BVtech</cite>. This suffices for some important functional analytic theorems that are stated in our final section.

We prove a large-diameter fundamental-gap lower bound for compact horoconvex domains in real hyperbolic space of curvature \(-1\). The geometric part reduces large horoconvex domains to a fixed-width radial-height problem in all dimensions. The analytic part proves the needed radial-height theorem by comparing the low-energy Dirichlet form with a limiting angular operator on the sphere, while the radial complement is separated by a one-dimensional branch gap and endpoint Green estimates. The result gives the polynomial \(D^{-3}\) scale matching the Nguyen--Stancu--Wei large-diameter upper bound.

We study the relationship between almost toric base diagrams, perfect exceptional classes, and optimal ellipsoid embeddings for $H_b=\mathbb{CP}^2_1 \# \overline{\mathbb{CP}\!}\,{}^2_b$ and $P_b=S^2_1\times S^2_b$. Starting from a quadrilateral almost toric base diagram with one Delzant corner, we encode the three non-Delzant corners by a recursive triple. We show that every quadrilateral obtained via a well-defined sequence of mutations from the initial diagrams is encoded by a recursive triple in the same way. Moreover, geometric mutation of these diagrams corresponds to algebraic mutation of the associated triples. These algebraic mutations are the recursive operations used to generate the $(p,q)$-perfect classes for $H$. We apply this dictionary to realize every $(p,q)$-perfect class for $H$ by an explicit sequence of almost toric mutations for suitable values of $b$. We also prove the analogous realization result for triples of quasi-perfect classes for $P$, showing that these classes are in fact $(p,q)$-perfect. Finally, we apply these results to ellipsoid embedding problems, including visible embeddings, visible obstructions, and ATF-visible staircases.

We formulate and prove linear counterparts of results on partial matchings between finite subsets in abelian groups. In the group setting, there are necessary and sufficient criteria for the existence of partial matchings under suitable hypotheses; our aim is to obtain parallel statements in a linear framework. In particular, for a field extension $K\subsetneq L$ we introduce a notion of partial matching between finite-dimensional $K$-subspaces $A,B\subseteq L$, and we prove existence theorems mirroring known results for subsets of abelian groups. Along the way, we recover and extend various parts of this area of matching theory, emphasizing the close analogy between the group-theoretic and linear settings. Our approach blends classical linear-algebraic techniques with tools from matroidal transversal theory, and utilizes a linearized version of a method originating in additive number theory.

We study Cauchy singular integral operators on planar wedges whose opening angle is a rational multiple of $π$. For $θ=pπ/q$, the covering $w=ζ^q$ yields an exact finite-sheeted factorization of the wedge Cauchy transform into $2q$ interval Cauchy transforms with explicit algebraic recombination coefficients. The factorization is formulated on weighted conormal Hölder spaces. We prove that the lifting operator preserves conormal order, lowers the Hölder exponent from $β$ to $β/q$, and has sharp $\ell^1$ sheet norm $q$. Combining this operator factorization with a Mellin model for interval Cauchy transforms, we derive a mode-by-mode propagation rule for polyhomogeneous endpoint expansions. Nonresonant powers preserve their logarithmic order, while integer exponents raise it by one. The results also give a local singular decomposition for Cauchy operators on piecewise analytic curves with rational corner angles.