We consider a matching problem, which is meaningful in team competitions, as well as in information theory, recommender systems, and assignment problems. In the competitions which we study, each competitor in a team order plays a match with the corresponding opposing player. The team that wins more matches wins. We consider a problem where the input is the graph of probabilities that a team 1 player can win against the team 2 player, and the output is the optimal ordering of team 1 players given the fixed ordering of team 2. Our central result is a polynomial-time approximation scheme (PTAS) to compute a matching whose winning probability is at most epsilon less than the winning probability of the optimal matching. We also provide tractability results for several special cases of the problem, as well as an analytical bound on how far the winning probability of a maximum weight matching of the underlying graph is from the best achievable winning probability.

Transition metal dichalcogenide (TMD) moiré bilayers have realized a wide range of strongly correlated and topological phenomena. The physics in these materials is often sensitive to the interlayer stacking order. Polarization-resolved optical second harmonic generation (SHG) is the most used technique for stacking order characterization but unverified for most heterobilayers. Here we calibrate the optical SHG for angle-aligned MoTe2/WSe2 bilayers by the scanning transmission electron microscopy (STEM). We directly compare the transport and magnetic properties and the electronic phase diagram for two distinct stacking orders. With the calibrated stacking order assignment, we clarify the interpretation of earlier results, including the nature of the Chern insulator, mechanism of an electric-field-tuned metal-insulator transition at half band filling, and the Kondo lattice physics. Our work provides a consistent picture of the relation between the stacking order and the electronic properties of MoTe2/WSe2 moiré bilayers.

The stacking configuration of two-dimensional materials critically governs their optical and electronic responses. Monolayer transition-metal dichalcogenides (TMDC) lack inversion symmetry and exhibit exciton-enhanced second-harmonic generation (SHG). In TMDC bilayers, 60° (0°) stacking is conventionally expected to suppress (enhance) SHG owing to destructive (constructive) interference of the layer-resolved nonlinear polarizations. Here, we report an unconventional destructive SHG interference in nearly 0°-stacked (AA-stacked) MoTe2/WSe2 heterobilayers using two independent probes: atomic-resolution imaging and stacking-sensitive exciton hybridization measurements. Supported by ab initio GW and Bethe-Salpeter equation calculations, we show that distinct two-photon resonances associated with the WSe2 C exciton and the MoTe2 D exciton generate a nearly $π$ phase difference ($Δϕ$) in their second-order nonlinear susceptibilities $χ^{(2)}$, leading to the anomalous destructive interference. We further demonstrate that in small-angle twisted MoTe2/WSe2, the SHG polarization state is governed by the interplay between twist angle $α$ and phase difference $Δϕ$, and can be mapped onto trajectories on the Poincaré sphere. At excitation energies satisfying $Δϕ$ + 3$α$ = 180°, the SHG output becomes nearly circularly polarized (ellipticity ~ 0.91) and undergoes an abrupt 90° azimuthal rotation, corresponding to a geometric polarization singularity in the parameter space. Our findings open new routes for exciton-resonance engineered nonlinear photonics and stacking-resolved optical functionality in moiré materials.

We present a minimalist, symmetry-guided interaction for microscopic cluster models based on Wigner-$SU(4)$ symmetry. Retaining only an $SU(4)$-invariant two-body attraction and a local three-body repulsion, this framework is implemented via the generator coordinate method (GCM) to describe $α$--$α$ scattering phase shifts, the low-lying spectrum and transition properties of $^{12}\mathrm{C}$, and the cluster spectrum of $^{16}\mathrm{O}$. We show that the long-standing structural tension between the $^{12}\mathrm{C}$ and $^{16}\mathrm{O}$ ground states can be mitigated within this restricted $SU(4)$ operator space without introducing additional phenomenological complexity. These results indicate that Wigner-$SU(4)$ symmetry provides an effective organizing principle for $Nα$ clustering, offering a more fundamental baseline for understanding complex cluster structures.

Using one-dimensional simulations analyzed through the lens of open quantum systems, we study the photoelectron's strong-field dynamics from an atom surrounded by a scattering environment stochastically structured. We theoretically investigate high-order harmonic generation from this situation. We show that local dephasing of the photoelectron wavepacket induced by elastic scattering leads to global decoherence. This drives a transition from quantum to classical behavior, as witnessed by the photoelectron probability density localizing around specific trajectories of the classical analog system: unstable periodic orbits. This phenomenon mirrors quantum scars traditionally observed in the eigenfunctions of time-independent systems, such as quantum billiards. Here, it emerges in-situ within a time-dependent framework, manifesting directly in the real-time dynamics from the ground state rather than solely through spectral analysis.

We investigate a fifty-year-old conjecture of Erdős and Graham concerning whether the binomial coefficient ${n \choose k}$ with $1 \leq k \leq \frac{n}{2}$ must always have a divisor $\leq n$ that is ``close'' to $n$: that is, bigger than a constant times $n$. We show this is the case when $k$ is sufficiently large as a function of $n$. However, we show (under the Generalized Riemann Hypothesis) it is possible to find binomial coefficients ${n \choose k}$, where $k$ is small compared to $n$, such that ${n \choose k}$ does not have divisors $\leq n$ close to $n$. This settles the conjecture of Erdős and Graham, under GRH. This latter, more substantial argument involves a restricted covering problem with residue classes, sieve methods, and various exponential sum estimates.

Identifying network dynamics is a critical yet challenging task to to understand the mechanism of real-world social systems. There are two types of algorithms, and one requires the knowledge of self-dynamics function, interactive function, and interactive network to sparsely identify the network dynamics. Another one does not require any knowledge, but use simple functions to universally approximate complex functions. However, this type of algorithms lack interpretability, and the functional space is too extensive to search efficiently. Thus, to address this issue, this work proposes an Alternating Sparse Identification of Network Dynamics (ASIND) algorithm to sparsely identify the self-dynamics function, interactive function and interactive network alternatively. Extensive experiments are conducted to show the state-of-the-art identification and 100-steps prediction performance compared to the baseline. The experimental results also show the weak identifiability of interactive network, that means different networks can generate highly similar trajectories of network dynamics. The code is available at https://github.com/KMY-SEU/ASIND.

Quantum criticality is a resource for quantum-enhanced metrology, but existing schemes face intrinsic limitations. These arise because using criticality directly in the encoding dynamics restricts the accessible parameters to those explicitly supported by the critical Hamiltonian, and the requirement for critical conditions narrows the effective estimation range. To solve this, we introduce a general framework termed criticality-assisted noncommutative preparation (CANP). In this approach, critical evolution is employed as a state-preparation resource. We establish the underlying algebraic conditions and show that the intrinsic noncommutativity between the preparation and encoding operations leads to a genuine enhancement of the quantum Fisher information (QFI). Remarkably, this enhancement may be achieved at fixed total sensing time and energy cost. The effect is quantified by the Wigner-Yanase skew information, which measures noncommutativity and exhibits the same critical scaling as the QFI. We demonstrate effective use of CANP in the quantum Rabi and Lipkin-Meshkov-Glick models. Our results establish CANP as a robust technique to effectively implement criticality-enhanced quantum metrology.

In nuclear $β$-decay calculations, the axial-vector coupling constant $g_A \approx 1.27$ usually needs to be quenched phenomenologically by a factor $q~\approx$ 0.75 to reproduce {the Gamow-Teller (GT) transition strengths}. We propose a novel approach to quench the GT {strength} of $β$-decay within the microscopic random phase approximation (RPA) plus particle-vibration coupling (PVC) approach, incorporating the contributions of two-body currents (TBC) derived from chiral effective field theory ($χ$EFT). Self-consistent RPA+PVC calculations are performed in three doubly magic nuclei, $^{56}$Ni, $^{100}$Sn, and $^{132}$Sn, with various Skyrme energy density functionals, and the effect of TBC is evaluated by using the obtained many-body wavefunctions. A combined effects of the many-body correlations introduced by PVC and chiral TBC quench the GT strength and reproduce quantitatively experimental data without any additional adjustments. The extracted quenching factors $q$ by the present microscopic model lie in the range $\approx$ 0.73--0.80, which is quite close to the commonly adopted empirical value of $q \approx 0.75$.

We investigate a family of relational systems arising from interval partitions of $ω$, inspired by Vojtáš's characterization of the bounding and dominating numbers. By varying the underlying asymptotic quantifiers and interval constraints, we obtain several natural interval-type generalizations. We show that the universal variants are remarkably robust: in all the discrete, colored, restricted, bounded, and measure-theoretic settings considered here, the associated bounding and dominating numbers coincide with the classical invariants $\mathfrak b$ and $\mathfrak d$. In contrast, the existential variants systematically reverse these invariants, yielding that the bounding number coincides with $\mathfrak d$ and the dominating number coincides with $\mathfrak b$.

We present a strong lensing analysis of the system PJ011646 using high-resolution ($\sim$0.1 arcsec) Atacama Large Millimeter/submillimeter Array (ALMA) dust-continuum observations to test for the presence of dark matter substructures. The lens mass distribution is modelled with an elliptical power law and third- and fourth-order multipoles (PL+MP; $m=3,4$), plus external shear. The multipoles have amplitudes of $\simeq$1.5 per cent of the convergence, consistent with nearby early-type galaxies, and improve the fit by $Δ\ln Z = 52.1$ relative to a pure PL model. Using this best-fitting macromodel, we perform a grid-based subhalo search in the image plane, parametrising the perturber as a spherical NFW. A subhalo in two locations in the image plane improves the fit by $Δ\ln Z>10$. Both correspond to the same location in the source plane, so they are partially degenerate; follow-up analysis suggests that only one is physically real. This is a subhalo of mass $M_{200} = {2.78}_{-0.66}^{+0.43} \times 10^{10} \, M_\odot$ and concentration $c_{200} = 30_{-7}^{+5}$, detected at $\sim$5.8$σ$ significance (relative to the PL+MP). This concentration is consistent with that expected for a typical tidally stripped Navarro-Frenk-White subhalo. The enclosed projected mass is most tightly constrained within a radius of 2 kpc, where we infer $M_{\rm sub} = {3.57}_{-0.14}^{+0.16}\times 10^9 \, M_\odot$. From grid cells consistent with no detection ($Δ\ln Z < 10$), we derive limits on the minimum subhalo mass that could have been detected at $3σ$ significance, finding $M_{200} \approx 8 \times 10^{8} \, M_\odot$ in the most sensitive regions of the lensed arcs. This demonstrates that ALMA continuum imaging at sub-arcsecond resolution can probe dark matter substructure in a mass regime where cold and warm dark matter models predict different abundances of subhalos.

We use shear-mode Raman imaging to track ferroelectric switching in multilayer 3$R$-MoS$_2$. Within a single flake, mechanically segmented regions respond independently and follow distinct pathways. Partially polarized end states indicate that domain walls can reside between selected layer pairs, producing partial stacking transformations. The dwell time of intermediate states varies widely, indicating that pinning sites strongly influence the dynamics. Second-harmonic generation measurements further reveal three characteristic sample-boundary and domain-wall orientations, including a prevalent chiral direction near the zigzag-armchair bisector. These results provide a direct, noninvasive view of domain-wall-mediated switching in a prototypical sliding ferroelectric and identify pinning and exfoliation-created boundaries as key factors governing its dynamics.

We establish results for the first sensitivity analysis of the stochastic fluid models (SFMs). We derive expressions for the sensitivity analysis of the key stationary and transient (time-dependent) quantities of this class of models. We also construct numerical examples to demonstrate the application potential of our methodology in queueing systems, such as deteriorating systems and insurance risk processes. This work forms foundation for the sensitivity analysis of other Markovian modulated models, which are generalisations of the SFMs, and have widespread applications.

Photodetection is usually treated in the frame in which the detector is at rest relative to the optical apparatus. We show that uniform motion of an electric Glauber detector changes the single-click POVM realized on two counterpropagating single-photon modes. Motion Doppler-shifts the alternatives in the detector frame; finite bandwidth then converts propagation direction into a detection bias without decohering the photon. For a Lorentzian response near one Doppler branch, the readout crosses from phase-sensitive to direction-sensitive with a quality-factor-enhanced onset. Finite-time integration adds Doppler-beat visibility loss, separating passive covariance from measurement change.

This work investigates dynamically generated hidden-charm baryon resonances arising from meson-baryon interactions. Using the local hidden gauge formalism, we model the interaction via t-channel vector meson exchange and unitarize the amplitude using the Bethe-Salpeter equation. To address regularization issues, we propose a novel ``hybrid loop function'' scheme that eliminates the unphysical poles -- common artifacts in cutoff or dimensional regularization -- while keeping the predictions of physical states. Consequently, the model successfully reproduces six experimentally known hidden-charm pentaquarks as well as earlier theoretical results, and predicts new states in the S=-1, I=1 sector.

There is a growing consensus that large-scale, fault-tolerant quantum computing (FTQC) necessitates high-fidelity photonic interconnects to overcome the scaling limits of monolithic architectures. However, most current platforms were not originally designed for native photonic connectivity and require significant engineering overhead. To overcome these fundamental hardware limitations, we recently introduced a rationally designed organic molecule that serves as an ideal quantum node, featuring a robust qubit-photon interface (QPI) and a long-lived nuclear-spin register. In this work, we present PIQC (Photonic Integrated Quantum Circuits), a distributed architecture designed to scale these molecular nodes into a functional quantum computer. The PIQC framework integrates five mutually reinforcing innovations: (i) Designer molecular qubits, i.e. carbene molecules in an isosteric host that provide millisecond-coherence electron spins with high spectral stability and spin-dependent optical emission, (ii) deterministic nuclear registers made of synthetically placed $^{13}$C or $^{14}$N labels that enable fast ($\sim 1~μ$s), high-fidelity electron-nuclear gates, (iii) hybrid photonic integration, which allows molecular films to seamlessly integrate with existing mature fabrication technologies, e.g. thin-film lithium niobate (TFLN), (iv) heralded entanglement protocols that can tolerate up to 70% photon loss, and (v) stairway Floquetification, i.e. high-rate quantum low-density parity-check (qLDPC) codes that are converted into Floquet codes, reducing syndrome extraction to weight-two Bell-pair measurements that match PIQC's networked hardware. PIQC offers a hardware-efficient, commercially viable pathway toward a utility-scale quantum computer based on distributed FTQC.

As the need for more computing power grows, traditional methods are hitting limits. To boost performance, we're expanding Central Processing Unit (CPU) capabilities and using specialized hardware accelerators. For example, mobile devices usually have cameras, video encoding, and audio accelerators. To perform the different tasks, these accelerators execute microcode programs. These accelerators, however, take up space and often sit idle. Reconfigurable processors offer a solution. They have a normal core connected to several accelerator slots. These accelerator slots can be filled during runtime to accommodate the application running. Once one application finishes and another application is running, the accelerators can be switched. For example, playing music after using the camera. In this work, we introduce dynamic control-flow execution for the microcode of runtime reconfigurable processors, i.e., support for loops, conditional jumps, and exception handling. We benchmark using four different applications from four domains (object detection, ocean movement simulation, artificial intelligence and security) that all are compute-intensive and would require the dynamic control-flow when executed on reconfigurable processors. We show that the dynamic control-flow allows different applications to be executed with significant speedup in comparison with execution on general-purpose processors.

We investigate the blow-up analysis and quantitative behavior for a sequence of maps $\{u_n\}_{n=1}^\infty$ from degenerating tori $(T^2,g_n)$ or from degenerating cylinders $(S^1\times [0,π],g_n)$ with free boundary conditions $u_n(S^1\times \{0,π\})\subset K$ to a compact Riemannian manifold $(N,h)$ satisfying $$E(u_n)+\|τ(u_n,g_n)\|_{L^2}\leq Λ<\infty,$$ where $τ(u_n,g_n)$ is the tension field of $u_n$, $K\subset N$ is a smooth submanifold. We establish generalized energy identities and prove that away from bubbles, the asymptotic limit of the necks are either some geodesics on $N$ or some geodesic-like curves on $K$ where some length formulas are given. This partially confirms a conjecture by Ding-Li-Liu \cite{Ding-Li-Liu} in the sense of approximate sequence case. Moreover, we study an evolution system to seek minimal cylinders in a compact Riemannian manifold with free boundary and with arbitrary codimensions. By studying the convergence of the flow at infinity, we obtain some existence results of minimal cylinders with free boundary. Compared with the closed case in, an interesting new phenomenon here is that the neck may converges to a geodesic-like curve on $K$.

We consider the case of scattering by several obstacles in $\mathbb{R}^d$ for $d \geq 2$. We establish a relative trace formula for Neumann and transmission boundary conditions analogous to the one obtained in arXiv:2002.07291 for Dirichlet boundary conditions. In the case of $f(x) = x^{1/2}$ the trace has the interpretation of the Casimir energy of the obstacle configuration. In the one-dimensional case, we recover a rigorous version of the Lifshitz formula for the Casimir energy of parallel plates with frequency-independent electric permittivity and magnetic permeability. We thereby strengthen the mathematical foundations of the Casimir effect and demonstrate the flexibility of the rigorous approach established in arXiv:2104.09763 and arXiv:2002.07291.

In the context of the Equational Theories Project, Terence Tao posed the challenge of finding alternatives to a complicated 62-step proof found by the Vampire superposition prover. We introduce a proof minimization tool called Krympa. Using a combination of brute force and heuristics, and exploiting both Vampire and the Twee equational prover, the tool reduces the 62-step proof to 20 steps, each corresponding to a rewrite. In an empirical evaluation, it also performs well on 1431 equational problems originating from the same project, reducing in particular a 151-step proof to only 10 steps.

We prove ergodicity for random dynamics satisfying some expansion and irreducibility conditions. As a particular application, we show that if $R_1,R_2\in \mathrm{SO}(d+1)$, $d\ge 2$, generate a dense subgroup, then the random dynamics of $R_1$ and $R_2$ on $S^d$ is stably ergodic. Previously this was only known to hold in even dimensions. As a consequence, we deduce spectral gap and statistical limit theorems for such systems. In particular, our results apply in the presence of zero Lyapunov exponents.

Laplace approximations are a standard tool for computationally efficient inference in latent Gaussian models, but they fail for quantile regression with the asymmetric Laplace likelihood because the observed Hessian vanishes almost everywhere. We show that this obstacle can be overcome without smoothing the likelihood: the relevant local curvature is given not by the observed Hessian, but by the Fisher information when the model is correctly specified and by the population curvature of the expected loss under misspecification. On this basis, we develop a Laplace approximation framework for quantile regression with mixed-effects and Gaussian process models. We propose practical curvature estimators, including the triangular kernel curvature (TKC) estimator, that yield approximations for posterior distributions and marginal likelihoods, and we establish their asymptotic validity. Empirically, the proposed methods are scalable and numerically stable, and for latent Gaussian models, they achieve accuracy comparable to or better than MCMC and variational competitors at substantially lower computational costs. More broadly, the framework clarifies how Laplace approximations can be justified for non-smooth generalized posteriors through local quadratic behavior of the expected loss.

We investigate the effect due to grid resolution and subgrid-scale model on large-eddy simulations of low-level clouds using a novel framework that combines pseudo-spectral advection with the anisotropic minimum dissipation (AMD) subgrid-scale model. We use two field campaigns as reference, DYCOMS-II RF01 and ASTEX, which cover both non-precipitating and precipitating stratocumulus cloud regimes across different time scales. Our results demonstrate that the AMD model combined with pseudo-spectral advection produces robust and accurate predictions across varying grid resolutions without parameter tuning. We identify a recommended grid anisotropy where vertical spacing is approximately three times finer than horizontal spacing, balancing accuracy and computational efficiency. Finally, an error analysis based on cloud liquid water content and vertical velocity variance reveals good agreement with theoretical predictions for isotropic grids, while grid anisotropy effectively improves convergence rates.

Given a countable Turing ideal $\mathcal{I} \subseteq ω^ω$, we say that $x$ is a list (resp. weak list) of $\mathcal{I}$ if $\mathcal{I}=\{x^{[n]} : n \in ω\}$ (resp. if $\mathcal{I} \subseteq \{x^{[n]} :n \in ω\}$). We show that, for several natural ideals $\mathcal{I}$, $x$ computes a list of $\mathcal{I}$ if and only if it computes a function dominating all the functions in $\mathcal{I}$. On the other hand, we provide reals which are $\mathsf{HYP}$-strongly null engulfing (and hence $\mathsf{HYP}$-dominating, by results of Greenberg, Kuyper and Turetsky) but which cannot compute a weak list for $\mathsf{HYP}$, solving a problem left open in a recent paper by Greenberg and the second author. This result can be generalized to any countable ideal which is downward closed under $\leq_{\mathsf{HYP}}$. We also give a characterization of reals which compute a list of $\mathsf{HYP}$: $x$ computes a list of $\mathsf{HYP}$ if and only if $x$ is $\mathsf{HYP}$-dominating and $\mathcal{O}$ is $Σ^0_2(x)$.

We prove the sharp Gaussian isoperimetric inequality for conjugate heat-kernel measures along a Ricci flow via a monotonicity formula. As consequences, we obtain the exact Gaussian enlargement theorem and a Gaussian-quantile two-set concentration estimate. In particular, this recovers the exponential concentration estimate of Hein--Naber from a sharper isoperimetric profile. We also derive Gaussian rearrangement inequalities, recover the sharp Hein--Naber log-Sobolev inequality, and identify the universal Gaussian-model constants in Bamler's \(L^p\)-Poincaré inequalities. Further applications include Gaussian-profile localization near Bamler's \(H_n\)-centers, convex-order and moment estimates for logarithmic derivatives of the conjugate heat kernel, reverse hypercontractivity, entropy-regular profile stability, and a path-space Bobkov inequality.

Forecasting univariate time series in the financial market is a challenging endeavor. While numerous statistical and machine learning models have been introduced to address this challenge, they typically concentrate solely on analyzing temporal patterns within the time series data. In this research, we study the statistical significance of the inclusion of geometric patterns in enhancing forecasting accuracy within the context of time series analysis. We introduce the Time-Geometric model, a combination of models designed to exploit both geometric and temporal patterns. The contribution of this research lies in advancing the domain of univariate time series prediction,as demonstrated through extensive empirical evaluations. Our findings underscore that leveraging geometric patterns, captured through Graph Neural Networks, yields statistically significant improvements in forecasting accuracy.

While vorticity is the classical tool for analyzing rotational fluid kinematics, it inherently focuses on local, differential spin. This paper introduces a complementary framework based on the angular momentum density field, $\mathbf{L} = \mathbf{r} \times \mathbf{u}$, deriving generalized transport equations that explicitly balance macroscopic torque and rotational momentum. This $\mathbf{L}$ perspective offers several distinct theoretical advantages over traditional velocity/vorticity formulations. Specifically, this approach: (i) provides a novel decomposition of the viscous torque into a diffusive component and a local spin dissipative term; (ii) shows the mechanism by which lift is generated in viscous boundary layers by vorticity acting as a source of angular momentum; it also explains stall (iii) reformulates the hydrodynamic impulse to yield a remarkably clean separation of terms into dilatational, volumetric, and rotational flux components; The $\mathbf{L}$ formalism provides the kinematic closure necessary to unify non-circulatory added mass and circulatory lift within a single, dimensionally consistent budget. (iv) enables the direct calculation of the viscous added mass force, accounting for the inertial resistance of boundary layers and separated wakes; (v) simplifies geophysical fluid dynamics by absorbing the planet's rotation, traditionally treated as an artificial virtual vorticity term which directly gets absorbed into the conserved axial angular momentum $m$, revealing the fundamental physics of global circulation through explicit torque balances; (vi) identifies the rotlet as a fundamental Green's function for the $\mathbf{L}$ transport equation in the Stokes regime; and (vii) demonstrates that both oblique shocks and vortex sheets act as singular sources of $\mathbf{L}$ that turn the macroscopic flow.

We construct a generic extension of $L$ satisfying Martin's Axiom, $2^{\aleph_0}=\aleph_3$, a lightface $Δ^1_3$ wellorder of the reals, and $Σ^1_n$-uniformization for every $n\geq 2$ simultaneously.

We study privacy guarantees in the framework of pointwise maximal leakage (PML) that satisfy two requirements: they are robust under post-processing and upper bound the failure probability, i.e., the probability that the information leakage exceeds a given threshold. We first examine two candidate definitions inspired by (approximate) differential privacy and show that neither one satisfies both requirements simultaneously. We then introduce the notion of the PML envelope, which quantifies the largest amount of information leakage about a secret after arbitrary post-processing of a mechanism's output. By construction, the PML envelope satisfies both requirements. We discuss basic structural properties of the envelope, such as monotonicity, and derive general upper and lower bounds. We further analyze the envelope for two widely used privacy mechanisms: the PML-extremal mechanisms in the high-privacy regime and randomized response. Overall, this work establishes the PML envelope as a natural and operationally meaningful definition for providing privacy guarantees that are preserved under arbitrary downstream transformations.

It is well known that the graph of a total $\mathbfΣ^1_n$-function is $\mathbfΠ^1_n$. We prove the consistency of the dual assertion at the third projective level: there is a model of $\ZFC$ in which the graph of every total $\mathbfΠ^1_3$-function is $\mathbfΣ^1_3$. This principle is incompatible with $\mathbfΠ^1_3$-uniformization and hence with the usual projective-determinacy picture. The construction also repairs the final step of the failure-of-uniformization argument from~\cite{HOFFELNER2023103292}.