Papers

Ultrafunctions are generalized functions defined on a non-Archimedean field extending R. They provide a framework for treating problems that fall outside the classical theory of distributions. In this paper we introduce an improved notion of ultrafunctions which refines the previous constructions. This new approach allows a more flexible functional framework than the one used in other papers and yields a more flexible functional setting. In particular, this approach allows us to prove the existence of ultrafunction solutions for several classes of ill-posed evolution problems arising in partial differential equations.

This paper derives a multiscale Bakry-Émery criterion for an upper bound on the Araki relative entropy of perturbations of an equilibrium state in relativistic QFT. Using stochastic positivity, affiliated perturbations of the relativistic bosonic field can be put in correspondence with perturbations of the massive Euclidean free field. This gives a Feynman-Kac formula, which in turns is used to derive a probabilistic expression for the Araki relative entropy. The description of the massive Euclidean free field as an abstract Wiener space provides the infinite-dimensional framework to derive a Polchinski equation. Extending the methods of Bauerschmidt, Bodineau, and Dagallier to this context yields a probabilistic bound on the Araki relative entropy, which can be formulated entirely in operator-algebraic terms. Using the criterion, it is shown that the Lorentzian massive Sine-Gordon model in the ultraviolet finite regime satisfies the relative entropy bound.

Quantiles are among the most fundamental concepts in Probability and Statistics, from descriptive to inferential. However, quantile functions are well-defined and well-understood in the context of one-dimensional probability distributions, where their definition as the inverse of distribution functions—a definition which is intimately related to the canonical ordering of the real line. Starting with dimension $d=2$, such a canonical ordering is no longer available in~${\mathbb R}^d$; nor is it available in nonlinear manifolds such as hyperspheres, tori, or polyspheres (vectors of directional variables). This results in the absence of an obvious and widely accepted definition of quantiles. The need to extend the concept of quantile beyond the classical univariate context nevertheless has sparked a large body of literature, giving rise to a variety of more or less satisfactory quantile concepts; the recent years have been particularly active in this respect. This short review is an attempt to complement and update Serfling's 25-year-old survey (Serfling 2008) by summarizing and unifying some of these concepts in light of recent contributions.

Fix $k\ge 3$, and let $f_k(N)$ be the largest harmonic sum of a subset of $[N]$ containing no $k$ distinct integers with a common pairwise least common multiple. We prove that $f_k(N)=(\log N)^{\gamma_k+o(1)}$ for a well-defined exponent $\gamma_k\in(0,1]$. Following the weighted-pressure idea of Chojecki, we give a self-contained proof of variational formulas for $\gamma_k$ in terms of weighted sunflower-free families. We then eliminate the continuous weight: if $M_k(n,r)$ is the largest size of an $r$-uniform $k$-cosunflower-free family on $[n]$, then \[ \gamma_k=\sup_{n\ge1,\ 1\le r\le n}\frac{r}{en}M_k(n,r)^{1/r}. \] This finite-block formula yields a direct transfer principle from uniform set-system constructions, recovers the Tang–Zhang bounds, and gives $0.438899\ldots<\gamma_3\le0.889881\ldots$.

We study disorder-free localization in a spin-coupled quantum walk on the hypercube within the quantum Bernoulli noises framework. We derive a general criterion for disorder-free localization: it occurs if and only if some spectral subspace of a reduced unitary evolution contains a vector whose position marginal probability is non-uniform. For the Grover walk with distance-dependent coupling \(\phi_\sigma=|\sigma|\pi/(n+1)\) in dimension \(d=4\), we explicitly construct a non-uniform fixed point in the all \(+1\) spin configuration. This implies that disorder-free localization occurs in our model.

We study external difference families arising from cyclotomic classes in finite fields from the viewpoint of a fixed number of blocks. For families consisting of even-indexed cyclotomic classes, the EDF condition can be expressed in terms of relations among cyclotomic numbers. We first study the two-block case and recover a classical characterization in terms of quadratic forms. Our main result shows that, for a prime $p=12k+1$, the family $\{C_0^6,C_2^6,C_4^6\}$ forms an EDF in $\mathbb{F}_p$ if and only if $k$ is a square. The proofs combine symmetry relations of cyclotomic numbers with their explicit evaluations.

This paper presents some results useful in identifying or ruling out fold and cusp bifurcations in mass action networks, and hence in studying multistationarity in these networks. The main result is that the occurrence of fold and cusp bifurcations in mass action networks can be confirmed or ruled out using alternative systems of equations, obtained using principles of Gale duality, rather than the original mass action equations. This can, for certain classes of networks, greatly simplify the calculations involved, avoiding the need for explicit centre manifold reduction or Lyapunov-Schmidt reduction when checking bifurcation, nondegeneracy and transversality conditions. Moreover, the approach leads to immediate corollaries on the non-occurrence of bifurcations in classes of networks with certain combinatorial properties. We present a number of examples illustrating the results and corollaries.

We study the existence of maps of nonzero Brouwer degree between two orientable, equal-dimensional generalized Dold manifolds (GDMs) fibered by complex partial flag manifolds over real projective spaces. The GDMs considered here are obtained as orbit spaces of the diagonal involution on the product of a sphere and a complex partial flag manifold, acting antipodally on the sphere and by complex conjugation on the flag manifold. We show that no map of nonzero degree exists between two distinct GDMs when their base real projective spaces are different, except in one exceptional case. Furthermore, when the base real projective spaces coincide but fibers are distinct, we prove the nonexistence of maps of nonzero degree in most cases, including those in which at least one of the fibers is not a Grassmannian. As an application of our study, we establish cohomological rigidity for the class of products of a sphere with a complex or quaternionic partial flag manifold.

Given $\alpha\in \mathbb R$ and $ B_1,\ldots,B_q\in M_{m,n}$, consider the map \[\Phi^{\pm}_{\alpha,\mathfrak{B}}(X)=\alpha \mbox{tr}(X) I_m\pm \sum_{s=1}^qB_sXB_s^*\ \mbox{for all}\ X\in M_n.\] In this study, necessary and sufficient conditions on $\alpha $ and $B_1,\ldots,B_q$ are given that guarantee $\Phi_{\alpha,\mathfrak{B}}^\pm$ is $k$-positive, $k$-copositive, and $k$-PPT, respectively. As application, known results about the map $\alpha \mbox{tr}(X)I_n\pm X$ are recovered. Furthermore, singular value inequalities associated with these positivity constraints are considered.

We study positive entire solutions of the flat constant $Q/R$-curvature equation in dimension $n\geq5$. We prove that the weak superharmonicity condition $-\Delta u\geq0$ already forces every nonconstant solution to be admissible. With this automatic admissibility, we classify all such solutions as positive constants or standard bubbles. We also show that the weak superharmonicity condition is structurally necessary. Once it is removed, there is a one-parameter family of positive radial entire solutions near every positive constant. These solutions converge to a positive constant at infinity, while both $-\Delta u$ and $-\Delta(u^{(n-2)/(n-4)})$ change sign infinitely many times.

Mathematical programming provides a common language for many decisions embedded in modern machine-learning (ML) and artificial-intelligence (AI) systems: selecting retrieval context, routing tokens, allocating inference compute, fitting structured predictors, protecting against distribution shift, and balancing competing objectives. However, the relevant literature is fragmented across optimization, information retrieval, recommendation, natural-language processing, computer vision, and learning theory. This paper organizes various applications under common mathematical programming paradigms: linear, quadratic, binary and mixed-integer, conic, bilevel, multi-objective, inverse, distributionally robust, submodular, and min–max optimization. We normalize the models with a mostly unified notation and, for every application, identify inputs, decision variables, a principal formulation, structural properties, solution strategies, and limitations. Across paradigms, we compare tractability, relaxation quality, decomposition, approximation guarantees, and scalability bottlenecks. The paper shows that mathematical programming is most useful not as a claim that all learning is LP or MIP, but as a disciplined interface between predictions and constrained decisions.

We study connected Riemannian manifolds on which either the sectional curvature or the Ricci tensor is, at each point, strictly positive, strictly negative, or zero, and ask whether the two signs can coexist under pointwise pinching. A general support-rigidity theorem for positive semidefinite divergence-free symmetric tensors is the common analytic mechanism. For sectional curvature in dimension $n\ge 3$, any locally uniform positive lower bound for the absolute pointwise pinching ratio rules out a change of sign. With a fixed pinching constant $\delta_0$, the flat set has no $C^1$ hypersurface piece; it is empty when $\delta_0>1/2$, and is locally porous when $\delta_0=1/2$. These conclusions are sharp in several senses: there are smooth conformally flat local metrics whose sectional curvature changes sign across a flat hypersurface when the pinching degenerates, and there are closed exactly $1/q$-pinched metrics on spheres with isolated flat points. For Ricci curvature, the sharp threshold is \[ \delta_c=\frac1{n-1}. \] A locally uniform gap $\delta_{\mathrm{Ric}}>\delta_c$ forces one Ricci sign globally. Conversely, for every $0<\delta<\delta_c$ there are local sign-changing metrics with exact pointwise pinching $\delta_{\mathrm{Ric}}\equiv\delta$, and exact local examples also exist at $\delta=\delta_c$. Pointwise strictness $\delta_{\mathrm{Ric}}>\delta_c$ is insufficient if the gap collapses at a Ricci-flat interface. For every subcritical $\delta$ we give explicit closed metrics on $\mathbb{S}^1\times\mathbb{S}^{n-1}$, and complete periodic lifts to $\mathbb{R}\times\mathbb{S}^{n-1}$, whose optimal global lower pinching constant is exactly $\delta$. Finally, we prove two ansatz-specific obstructions to compactifying the exact local constructions. The main content of the proof is generated by ChatGPT 5.6 sol and verified by the authors.

Adaptive Conformal Inference (ACI) extends conformal prediction to non-exchangeable settings by adjusting the nominal miscoverage level online in response to recent coverage errors. When forecasts are issued with horizon $\tau$, however, the outcome needed to evaluate a prediction is observed only $\tau$ steps later, so these adaptive updates must rely on delayed feedback. We study this setting and ask how the effect of delay depends on the persistence of the residual process. First, we show that the $\tau$-delayed ACI recursion can be decomposed into $\tau$ interleaved ACI-like sequences. This representation yields a finite-sample bound on long-run empirical coverage with explicit dependence on $\tau$. We also derive an approximate marginal coverage bound that relates coverage deviation to changes in the underlying environment across the forecast horizon and to the adaptation rate $\gamma$. We then introduce the delay-to-memory ratio $r=\tau/L$, where $L$ is the time scale over which the temporal signal driving non-exchangeability decays. Simulation results show that the usefulness of this ratio depends on the form of temporal dependence: it strongly organizes performance under AR(1) dependence, is less predictive of overall performance under GARCH(1,1) and Markov switching, but more clearly characterizes when scale normalization remains useful in those settings. Abrupt mean- and variance-shift experiments further show that the preferred adaptation rate depends on the residual dynamics. Overall, the results show that the effect of forecast delay is best understood relative to the time scale over which past residual information remains relevant.

We establish five families of integer-parameter extensions of Ramanujan's partial theta identities. The families follow from a common two-parameter specialization of Andrews' transformation, for which we give an independent proof based on a $q$-difference recurrence and a boundary estimate. Specializations of the integer parameter recover six identities from Ramanujan's lost notebook. As an application, a residue argument applied to the fifth family gives an integer-parameter extension of Lovejoy's residual identity, from which we construct a corresponding family of conjugate Bailey pairs.

Controller design for physical systems directly from data is of tremendous interest in both the industry and academia. However, the research on direct data-driven control has thus far mainly focused on discrete-time linear time-invariant systems. To make the highly non-trivial step to continuous-time nonlinear systems, this paper explores an intermediate route through the framework of linear parameter-varying (LPV) systems. LPV systems can be used as a convenient surrogate for nonlinear systems to achieve systematic analysis and controller design. In this work, we generalize the LPV Fundamental Lemma for discrete-time systems towards a class of continuous-time LPV systems, achieving data-driven representations that can be used to design continuous-time LPV controllers.

Here we investigate the stability of the Kolmogorov–Arnold representation theorem (KART) under adversarial reparameterisations of the hidden layer for multivariate discontinuous and unbounded functions. Our results provide a rigorous mathematical foundation for the structural robustness of modern deep learning architectures, such as Kolmogorov–Arnold Networks (KANs), under adversarial configurations.

We determine the first correction to the leading Vandermonde fusion law for the correlation functions of the Sine$*\beta$ process in the critical and and subcritical regimes $m\beta\leq1$. When $m\beta<1$, the normalized correction is of order $|\varepsilon|^{1+m\beta}$, with a strictly negative coefficient given by an explicit gamma-function ratio times an absolutely convergent arithmetic-geometric mean deficit integral. At $m\beta=1$, it is $-\frac{\sum*{i<j}(a_i-a_j)^2}{8m^2(2m+1)}\varepsilon^2\log(1/|\varepsilon|)+O(\varepsilon^2)$. For two merging points, the subcritical coefficient reduces to a gamma-function expression involving $\sec(\pi\beta)-1$, and the critical logarithmic coefficient is $-1/160$. The argument starts from a geometric interpolation of circular-Jacobi weights. Selecting one particle in the interpolation derivative increases the fused charge by $\beta$ and leaves a strict positive-power moment margin in the remaining stochastic-zeta expectation. This yields an absolutely convergent one-particle identity and uniform control at the collision scale. The results resolve the critical and subcritical conjecture in the author's earlier preprint and complement its supercritical second-order expansion.

Let $a(n)$ be the sequence A158690 in the On-Line Encyclopedia of Integer Sequences (OEIS), defined by the exponential generating function $\sum_{n\ge0} a(n)t^n/n! = 1+\sum_{m\ge1}\prod_{j=1}^m(1-e^{-(2j-1)t})$. We prove two congruence conjectures of Peter Bala. The first states that, for every integer $k\ge1$, the sequence $a(n)$ modulo $k$ is eventually periodic with period dividing $\varphi(k)$. We prove the stronger statement that the Carmichael function $\lambda(k)$ is an eventual period. The second conjecture asserts the shifted Gauss congruences $a(np^r+i)\equiv a(np^{r-1}+i)\pmod{p^r}$ for every $i\ge0$, every prime $p$, and all $n,r\ge1$. Both results follow from a general theorem for exponential generating functions of the form $G(e^t-1)$ with $G\in\mathbb Z[[y]]$, together with the standard power-sum formula for Stirling numbers of the second kind.

The restricted augmented Zarankiewicz number \(z_L(m,n)\) yields core combinatorial lower bounds for the maximal SOS rank of biquadratic forms. All previously known infinite admissible graph families rely on \(K_{4t}\) incidence graphs, attaining an asymptotic relative gap limit of \(1/4\). This work develops a new infinite family built from \(K_{5t}\) incidence bipartite graphs with \(\mathbb{Z}_5\) cyclic labeling for block partitions. We construct valid nondegenerate intra-block and inter-block 2-edges, derive a quadratic closed-form lower bound of \(z_L\), and prove its relative gap converges asymptotically to \(2/5\). Full enumeration for \(t=1\) verifies the exact value \(z_L(10,5)=23\). Under nondegenerate and generalized \(C_4\)-free constraints, the ratio \(2/5\) is shown to be the maximal asymptotic ratio attainable under this block framework. Our results expand the library of extremal bipartite graphs and sharpen lower bounds for biquadratic SOS rank, with further open problems for general \(K_{kt}\) constructions outlined in closing.

In this article, we study the relations between algebraic models for the MinRank problem, which is a well-known NP-hard problem. We establish connections between the minors modeling, the Kipnis-Shamir modeling, and the support minors modeling, providing insights into their equivalences and differences. Our results contribute to a deeper understanding of the algebraic structures underlying the MinRank problem and may inform future research on efficient algorithms for solving it.

We study finite-rank multiplicative deformations of rotationally invariant non-Hermitian random matrices. More precisely, we consider models of the form $\mathbf{A}(\mathbf{I}+\mathbf{T})$, where $\mathbf{A}$ is a large rotationally invariant non-Hermitian random matrix, $\mathbf{T}$ is a finite-rank normal perturbation, and $\mathbf{I}$ denotes the identity matrix. We characterize the emergence of outlier eigenvalues, their fluctuations, and the associated eigenvector overlaps. Our results provide a multiplicative non-Hermitian counterpart to the classical Baik–Ben Arous–Péché framework.

Analytic continuation of imaginary-time quantum Monte Carlo data to real-frequency spectra requires the inversion of a severely ill-posed two-sided Laplace transform and arises naturally in quantum many-body calculations of dynamic properties. In this work, we distinguish the intrinsic ill-posedness of the continuous inverse two-sided Laplace problem from the conditioning of its finite-dimensional discretization. For equidistant sampling and reconstruction grids, we express the discrete problem via a diagonally scaled monomial Vandermonde matrix with exponentially distributed nodes. Imposing the physical detailed-balance symmetry transforms the discretization into a diagonally scaled Chebyshev-Vandermonde system. Exploiting these structures, we derive explicit lower and upper bounds on the condition numbers in terms of the physical and discretization parameters. For the unconstrained discretization, our bounds reveal super-exponential growth of the condition number with the reconstruction dimension, which cannot be removed by increasing the number of imaginary-time samples alone. Detailed-balance substantially improves the conditioning, especially in the practically relevant pre-asymptotic regime, although the asymptotic super-exponential dependence remains. In both cases, our bounds identify a low-dimensional regime in which the super-exponential contribution is suppressed and stable reconstruction remains feasible. These results provide a mathematical explanation for the effectiveness of low-dimensional spectral representations and detailed-balance in analytic continuation. While they do not remove the fundamental ill-posedness, they substantially mitigate the ill-conditioning of its finite-dimensional discretization, delay the onset of its catastrophic super-exponential growth and thereby enlarge the range of numerically accessible reconstructions.

The notion of admissible $g$-twisted $V$-comodules is investigated for a graded vertex operator coalgebra $V$ and a finite order automorphism $g\in \Aut V$. We prove that $\mathcal {M}$ is an admissible $g$-twisted $V$-comodule if and only if its graded dual $\mathcal {M}'$ is an admissible $g^{-1}$-twisted $V'$-module for vertex operator algebra $V'$. Then we establish a dual theory of twisted Zhu's theory, i.e., there is a coassociative coalgebra $C_g(V)$ for a graded vertex operator coalgebra $V$, any admissible $g$-twisted $V$-comodule gives a $C_g(V)$-comodule, and vice versa. We also prove that $V$ is $g$-corational, which means every admissible $g$-twisted $V$-comodule is completely reducible, if and only if its dual vertex operator algebra $V'$ is $g$-rational.

In this paper, we consider the computation of $M$-eigenpairs of fourth order partially symmetric tensors arising from elasticity theory. First, a lifted fourth order tensor is constructed, and the original $M$-eigenvalue problem is reformulated as a parameterized generalized tensor eigenvalue problem under the $\mathbf B_{\alpha,\beta}$-normalization. Then, an exact correspondence between the two eigenvalue problems is established, which provides a procedure for computing all real $M$-eigenpairs through the proposed reformulation. Furthermore, perturbation bounds for the largest $M$-eigenvalue are derived, and the lifted reformulation is shown to preserve these bounds without introducing any additional relaxation. Finally, numerical experiments are reported to show the effectiveness of the proposed method.

Priors in Bayesian inverse problems are often approximated through discretization, hyperparameter estimation, or generative modeling. Understanding how prior approximation errors propagate to the posterior and subsequent predictions is therefore important. In this work, we study the stability of the prior-to-posterior map where both prior and posterior perturbations are measured in the same Wasserstein metric $W_p$, $p\geq1$. We identify verifiable conditions on likelihood regularity and admissible prior classes that ensure uniform, Hölder, and Lipschitz stability. For bounded likelihoods that are uniformly continuous on bounded sets, uniform stability holds over prior classes with uniformly integrable $p$-th moments and a common positive evidence lower bound. With global Hölder regularity of the likelihood and uniform bounds on higher prior moments, a coupling argument leads to a Hölder estimate with a sharp exponent. For $p>1$, a Lipschitz likelihood need not give Lipschitz stability, even for priors with bounded support. We establish Lipschitz stability through an interpolation argument under uniform Poincaré bounds and a globally Lipschitz potential with uniformly bounded essential oscillation under the priors. For Gaussian priors with additive Gaussian noise and bounded Lipschitz forward models, these estimates give posterior $W_2$ bounds even for mutually singular prior perturbations. Numerical experiments for a Darcy inverse problem illustrate the predicted Hölder and Lipschitz rates and the resulting control of errors in the posterior mean and standard deviation of a Lipschitz quantity of interest.

Let $(x_n)_{n\geq1}$ be a sequence of distinct points on the unit circle. An $r$-span is the total length of $r$ consecutive gaps determined by the inserted points. Write $M_n^{(r)}$ and $m_n^{(r)}$ for the largest and smallest $r$-spans after the first $n$ insertions. We prove that there is an absolute constant $c>0$ such that, for every sufficiently large $r$, \[ \limsup_{n\to\infty}\bigl(nM_n^{(r)}-r\bigr) \geq c\sqrt{\log r}, \qquad \limsup_{n\to\infty}\bigl(r-nm_n^{(r)}\bigr) \geq c\sqrt{\log r}, \] and \[ \limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}} \geq 1+\frac{\log r}{100r}. \] Thus all three asymptotic conjectures made by de Bruijn and Erdős in 1949 are resolved. The ratio bound matches the upper bound of Clément and Steinerberger up to an absolute constant and answers a question of Brethouwer. The proofs compare interval counts at nearby times. Pointwise control leads to a one-dimensional sequence-discrepancy argument for the ratio, while averaged control and Halász's planar $L^1$ discrepancy theorem give the two one-sided conclusions.

We study the existence of nontrivial solutions to nonlocal semilinear diffusion equations of the form \[ a_0(x)u(x)-\int_\Omega k(x,y)\big(u(y)-u(x)\big)\,d\mu(y)=F(x,u), \quad u\geq 0, \] on a general measure space $(\Omega,M,\mu)$, where the kernel $k$ is nonnegative and symmetric. In this framework, a major challenge is establishing the strict positivity of solutions. To this end, we provide a novel complete characterization of the strong maximum principle in terms of a connectivity structure induced by the kernel, which leads to a decomposition of the domain into independent components. We also develop the method of sub- and supersolutions in this setting, obtaining existence, uniqueness, and nonexistence results, as well as a regularity result that ensures the continuity of solutions under suitable assumptions. The main interest of working within an abstract measure space lies in its rich applicability, as it unifies and extends a wide variety of frameworks, including discrete, continuous, and hybrid models, or systems interacting across different dimensions. Finally, numerical experiments are presented to illustrate the potential of our theoretical results.

We present a data-driven approach to identifying linear index-1 differential-algebraic pH systems (pH-DAEs) based on input-output measurements. In comparison to the identification of port-Hamiltonian (pH) systems, the algebraic constraint and the index condition pose additional challenges. First, we establish a structure-preserving formulation of the considered pH-DAE class and derive an implicit midpoint discretization that preserves the algebraic constraints and discrete dissipation inequality. We formulate the identification problem as a regularized least-squares minimization problem subject to the pH-DAE dynamics. Exploiting the index-1 structure, we reduce the constrained problem to an unconstrained optimization problem over the system parameters while preserving the port-Hamiltonian structure. Next, we derive an adjoint-based formulation to efficiently evaluate the gradient of the resulting reduced cost functional. This enables us to use gradient-based optimization methods for parameter estimation. Under suitable assumptions on the admissible parameter set, the existence of a minimizer is established. Numerical experiments demonstrate that the proposed approach can identify surrogate pH-DAE systems that accurately reproduce the input-output behavior of reference systems. Further investigations show the approach's potential for identifying reduced-order surrogate models. Cross-validation with independent input signals confirms the predictive capability of the identified models.

We present tensor network representations for discrete maximum entropy distributions under expectation constraints. To this end, we introduce Computation-Activation Networks (CompActNets), a tensor network architecture that subsumes exponential families. By leveraging the geometry of the convex polytope of realizable expectation vectors, we represent any maximum entropy distribution in the same architecture. We exploit the fact that proper faces of this polytope correspond to the boundary closure of exponential families, which restricts the distribution's support. We then derive explicit representations for the support within the CompActNet architecture. The proposed framework suggests tensor network ranks as complexity measures for faces. Finally, a case study on Boolean statistics links the geometry of 0/1-polytopes directly to propositional formulas.

Let $k$ be an algebraically closed field of positive characteristic $p$, and let $C_p$ act linearly on a finite-dimensional $k$-vector space $V$, with indecomposable Jordan summands $V_{d_i}$ of dimension $d_i$. We study projective (crepant) resolutions of $V/C_p$ by constructing a model using the invariant weighted blowup. For $V=V_2^{\oplus n}$, our model is smooth and is the normalization of a blowup of $V/C_p$. It has a unique exceptional divisor of discrepancy $n-p$, such that our model is a crepant resolution when $n=p$. In almost all other cases when the quotient is canonical but not terminal, the invariant weighted blowup model does not produce crepant resolutions, but gives the unique nontrivial projective crepant birational model of the quotient. Consequently, up to trivial summands, we classify the $p$-cyclic linear quotients admitting a projective crepant resolution.