Papers

In this paper, when the quiver Hecke algebra R is symmetric, we present a new construction of quantum monoidal seeds $ \mathscr{S}_{w,v}$ for $\mathcal{C}_{wv}$ using the reflection functors $\mathcal{F}_i$ and the newly introduced operators $\mathcal{K}_i$. The monoidal seed $ \mathscr{S}_{w,v}$ is obtained as a subseed of the monoidal seed of $\mathcal{C}_w$ constructed by applying $\mathcal{K}_i$ and $\mathcal{F}_i$ along the special KF sequence determined by a reduced expression of $w$ and $v$. We further prove that the monoidal seed $\mathscr{S}_{w,v}$ coincides with the set of all prime factors of the determinantial modules $M(w_{\le k } \Lambda_{i_k}, v_{\le k} \Lambda_{i_k} )$. We prove that the Grothendieck ring $K(\mathcal{C}_{wv}) $ lies between the cluster algebra and the upper cluster algebra.

In the previous paper \cite{Jin26}, the author introduced the notions of an adapted current $T$ and an adapted Hermitian-Einstein metric to establish the Kobayashi-Hitchin correspondence for a nef and big class $\alpha$. As a continuation of the previous work, this paper studies the solvability and the convergence of the Yang-Mills flow for a nef and big class $\alpha$ on a holomorphic vector bundle $E$ over a compact Kähler manifold $X$. In particular, we show that the limit of the Yang-Mills flow at infinity is determined by the holomorphic structure of $E$ and the nef and big class $\alpha$. More precisely, if we fix an integrable unitary connection $A_0$ on $E$, we show that the $T$-Yang-Mills flow on $E$ with initial condition $A_0$ is solvable for all time and it converges to a $T$-Yang-Mills connection $A_{\infty}$ in the sense of Uhlenbeck limit. Furthermore, we also show that, on the ample locus of $\alpha$, $A_{\infty}$ is complex-gauge equivalent to the direct sum of the Chern connections of the $T$-adapted Hermitian-Einstein metrics on the factors of the graded sheaf associated with the $\alpha^{n-1}$-Harder-Narasimhan-Seshadri filtration of $E$.

Let $G$ be an $r$-partite graph such that the edge density between any two parts is at least $\alpha$. We consider the problem of determining how large $\alpha$ must be in order to guarantee that $G$ has a Hamiltonian traversal (an $r$-cycle subgraph containing exactly one vertex from each part), and show that this critical density tends to $\frac 1 2$ as $r$ increases. This resolves a conjecture of Badakhshian, Falgas-Ravry, and Sharifzadeh. We also study the critical densities necessary to guarantee the existence of other spanning structures in traversals, particularly subgraph factors, and obtain asymptotically the critical densities for traversal $F$-factor subgraphs for several classes of graphs $F$. The proofs of our results involve the absorption method.

Let $d\geq 2$ and let $\bf{f}_1,\ldots,\bf{f}_m\in\mathbb C^d$. We prove that if $m\leq 2d-1$, then the intensity measurement map \[ \bf{x}\longmapsto \bigl( |\langle \bf{x},\bf{f}_1\rangle|^2, \ldots, |\langle \bf{x},\bf{f}_m\rangle|^2 \bigr) \] fails to recover almost every signal in $\mathbb C^d$ uniquely up to a global phase factor. Combined with the known generic sufficiency of $2d$ measurements, our result establishes that the minimum number of measurements required for almost-everywhere phase retrieval in $\mathbb C^d$ is exactly $2d$. This resolves an open problem in phase retrieval by determining the exact measurement threshold for almost-everywhere phase retrieval in ${\mathbb C}^d$.

We address the question of which common differences can arise in arithmetic progressions contained in fractal sets. For a compact set $C\subset\mathbb R$, we investigate not only whether arithmetic progressions occur in $C$, but the full collection of their common differences. More generally, for a finite pattern $P$, we study the set of scales at which affine copies of $P$ appear in $C$. For affine self-similar sets satisfying strong separation, we obtain explicit restrictions on admissible common differences. Specializing to middle-$\varepsilon$ Cantor sets, we prove that the longest arithmetic progression has length four whenever $3-2\sqrt2<\varepsilon\le 1/3$, showing that the maximal progression length drops immediately from six at the critical parameter $\varepsilon=3-2\sqrt2$. We further develop recursive bounds for the sets of admissible common differences and derive explicit blackout intervals, namely ranges of scales for which arithmetic progressions cannot occur. On the positive side, sufficiently thick Cantor sets exhibit the opposite behavior. Combining a refinement of the Hunt-Kan-Yorke construction with the Newhouse Gap Lemma, we prove that every sufficiently small common difference occurs in a three-term arithmetic progression. In particular, if the largest bounded gap of a Cantor set is at most $0.067 diam(C)$ and its thickness is at least $6.96268\ldots$, then every common difference in $(0,0.435 diam(C)]$ occurs in a three-term arithmetic progression contained in $C$. Analogous interval results are obtained for four-term arithmetic progressions and asymmetric three-point patterns.

We study the global dynamics for the 3D Zakharov system with radial initial data of energy slightly above the ground state energy, proving that the initial data set splits into nine nonempty, pairwise disjoint regions in which the solutions have distinct behaviors: growup, scattering, or trapped by the ground state, as time goes in the forward or backward directions. It is a classification similar to those for the nonlinear Klein-Gordon equation and for the nonlinear Schrödinger equation, extending the previous results on the Zakharov system below the ground state energy. The proof relies on the normal form technique to handle the quadratic frequency interactions and careful modulational analysis around the ground state. One novelty is a new family of localized virial estimates with monotonicity. These virial estimates are crucial for us to study the dynamics away from the ground state and may be of independent interest for other purposes.

We prove an analogue of a theorem of Berget and Rhoades about extending the parking space $\mathrm{Park}_n$ to an $S_{n+1}$-module $\mathrm{Slim}_n$. Specifically, we show that the bipartite parking space $\mathrm{Park}_{K_{n,m}}$, which naturally comes with an $S_{n-1} \times S_m$ action, extends to an $S_n \times S_m$-representation $\mathrm{Slim}_{n,m}$. We then formulate a conjecture generalizing this statement to any simple graph.

Sylvester's denumerant $d(t; \boldsymbol{A})$ counts the number of nonnegative integer solutions to $\sum_{i=1}^{N} a_i x_i = t$, where $\boldsymbol{A} = (a_1, \dots, a_N)$ is a sequence of positive integers with $\gcd(\boldsymbol{A}) = 1$. In 2025, Xin and Zhang gave a polynomial time algorithm in $N$ for computing $d(t; \boldsymbol{A})$ when the entries of $\boldsymbol{A}$ are bounded by a constant. In this paper, we extend this algorithm by incorporating Barvinok's algorithm, enabling it to handle the case where a fixed number of entries of $\boldsymbol{A}$ are allowed to be unbounded.

We study the structure of the Birkhoff center and the recurrent behavior of differentially positive systems whose linearizations along trajectories preserve a homogeneous cone field on a homogeneous space. Such cone fields arise naturally from general relativity and Lie theory. We establish an order-structural dichotomy for the Birkhoff center: every connected component of the Birkhoff center, as well as the support of any invariant measure, is either strongly ordered or unordered. This yields a comprehensive characterization of recurrent dynamics for differentially positive systems on homogeneous spaces, interpreted through the lens of the underlying order relation.

A scuba diver holding constant depth balances on an unstable equilibrium: the gas carried in the suit and buoyancy compensator compresses with depth, so the buoyant force falls as the diver sinks and rises as the diver ascends. We represent the diver as a proportional-derivative controller that regulates this compressible-buoyancy saddle after a finite reaction delay, and we derive the governing delay differential equation from the vertical force balance and the isothermal gas law, reducing it to a damping ratio, two control gains, and a dimensionless delay. The characteristic spectrum, obtained by pseudospectral collocation of the semigroup generator and checked against a direct Newton solution of the characteristic equation, locates the Hopf boundary that separates stable hovering from sustained porpoising; for the baseline diver the critical reaction delay is 3.36 s and the onset period is 28.8 s. The bifurcation is supercritical, and because the saturating force is the quadratic hydrodynamic drag, the limit-cycle amplitude grows in proportion to the delay excess rather than as its square root. The safe-operating envelope shows that runaway ascent is triggered by saturation of the compensator, not by loss of linear stability, so a stable and an unstable diver can share the same escape threshold. As onset is approached, the lag-one autocorrelation and variance rise while the fitted recovery rate falls and matches the spectral abscissa, giving an eigenvalue-exact early warning of the transition.

Let $G=F\Gamma$ be a factorization of a finite group, with neither factor assumed normal and with $F\cap\Gamma$ allowed to be nontrivial. We prove that $\exp(G)$ divides $\operatorname{lcm}(|F|,|\Gamma|)$, or equivalently that $\gcd([G:F],[G:\Gamma])\exp(G)$ divides $|G|$. This answers a cohomological divisibility question posed by Natale. Combining the group-theoretic divisibility with Natale's exponent bound and a lifting argument, we prove Kashina's exponent conjecture, in the arbitrary-field formulation of Etingof and Gelaki, for every finite-dimensional semisimple and cosemisimple Hopf algebra $H$ over a field $k$ for which $\operatorname{Rep}(H\otimes_k\overline{k})$ is group-theoretical. The same argument proves the corresponding degree-three cohomological divisibility for coefficients in an arbitrary $G$-module. For complex group-theoretical categories, we also establish Frobenius-Schur exponent divisibility under a cohomological factorization hypothesis, without assuming a fiber functor. We derive applications to low-dimensional Hopf algebras and abelian extensions.

In this paper we present a discrete version of Bertrand's paradox, where each sample space is a finite set of chords and a chord is chosen uniformly at random.

Periodically driven quantum systems are commonly modeled using master equations constructed in the eigenbasis of an undriven Hamiltonian, implicitly assuming that environmental dissipation couples to static energy transitions even under strong time-periodic driving. The validity of this approximation beyond weak or near-resonant driving remains poorly understood. To address the need for a more self-consistent quantum theory approach, we formulate a nonsecular Floquet–Markov generalized master equation (F-GME) in the quasienergy basis, treating interaction-induced (internal) and drive-induced (external) nonperturbative dressing on an equal footing. We subsequently investigate dissipation in two minimal driven open quantum systems—a harmonically driven two-level system and a harmonically driven coupled-two-level-system—each weakly coupled to a Markovian bath. Comparing the F-GME to a time-independent dressed-basis master equation, we show that even for a flat-bath spectral density and weak dissipation, the two approaches can yield qualitatively different steady-state populations and emission spectra. We resolve dissipation into drive-assisted sideband processes decaying via Floquet extended-space quasienergy channels, and show these channels can hybridize through nonsecular couplings into collective Floquet–Liouville modes governing observable spectral resonances. This analysis demonstrates that time-independent dissipative descriptions can incorrectly weight multiphoton Floquet transitions by collapsing quasienergy-resolved decay pathways into static energy gaps. The F-GME framework provides a systematic diagnostic for identifying regimes where Floquet-consistent dissipation is essential and clarifies the physical origin of discrepancies between commonly used master-equation approaches.

We extend the classical Taibleson-Weiss molecular theory to arbitrary radii, reaching the critical decay |x|^{-n/p}, with the resulting embedding shown sharp. This molecular theory governs the largest class of functions certifying membership in h^p(R^n), 0 < p <= 1, through Goldberg's convolution splitting: a critical Hölder modulus and an \ell^p-Dini decay majorant, sharp on the regularity, decay, and cancellation axes.

Noise modulation (NoiseMod) encodes information in the variance of a transmitted noise-like waveform rather than in its amplitude, phase, or frequency, and is attractive for ultra-low-power and covert links. Its main weakness is that, unlike classical modulation, it exhibits no diversity under Rayleigh fading: its bit error probability (BEP) decays only as $1/(N_s\delta)$, where $N_s$ is the number of noise samples per bit and $\delta$ the useful-to-thermal noise variance ratio. Independently, fluid antenna systems (FAS) have been shown to recover substantial selection diversity from a single radiating element that switches among $N_p$ closely spaced ports, without extra radio-frequency chains. This paper combines the two: we equip a NoiseMod receiver with a fluid antenna and derive its average BEP. For idealized, mutually independent ports, we obtain an exact closed-form BEP via order statistics of the port envelopes. For the physically accurate, spatially correlated case governed by Jake's model, we build a semi-analytical BEP using the two-stage channel approximation of Khammassi et al. We validate both regimes against full signal-level Monte Carlo simulation and show that (i) FAS restores a diversity order that grows with the number of ports $N_p$ when ports are weakly correlated, (ii) this gain saturates once the fluid-antenna aperture $W\lambda$ is fixed and $N_p$ grows, mirroring the outage-probability saturation reported for FAS, now observed for BEP, and (iii) an intrinsic, correlation-independent (and $\delta$-independent) BEP floor set only by $N_s$ and the variance ratio $\alpha$ persists regardless of the antenna diversity order.

Widely used in mechanics and mathematical physics, physical components remove the inherent coordinate-dependent scaling in curvilinear coordinates, yielding components of consistent physical dimension. In orthogonal coordinates, they are constructed by normalizing the coordinate frame and coframe; in general coordinates, however, their construction requires additional, non-trivial choices. In this paper, we extract physical components of arbitrary tensors on arbitrary Riemannian manifolds by orthonormalization of the coordinate frame. We further formulate a general normalization framework distinguishing three requirements: dimensional consistency, dual frame-coframe compatibility, and unit normalization. We show that dimensional consistency alone leaves independent general linear gauge freedoms for the contravariant and covariant components. Further requiring dual compatibility locks these into a single general linear gauge; a dual-compatible frame and coframe are both of unit length if and only if they are orthonormal. Thus, the choice of orthonormal transformations emerges as the only physical-components framework that satisfies all three requirements. We apply this framework to nonlinear elasticity and inelasticity, referring broadly to constitutive responses involving internal distortions, of which we study anelasticity, viscoelasticity, and visco-anelasticity. We examine the deformation gradient, inelastic distortions, strain measures, and stress tensors. We conclude by arguing that physical components remain neither intrinsic nor unique.

Naor proved that $L_p$ satisfies the metric $X_p$ inequality with the sharp scaling parameter and asked for the optimal dependence of its constant on $p$. The best previously known upper bound was $O(p^4/\log p)$, while first-degree chaos gives a lower bound of order $p/\log p$. We prove the matching upper bound. More generally, if $h$ is a mean-zero scalar function on the Hamming cube, $p\geq2$, and $S$ is uniformly distributed over the $k$-subsets of $[n]$, then \[ \bigl(\E_S\|\E_{[n]\setminus S}h\|_p^p\bigr)^{1/p} \lesssim \frac p{\log p} \left[\frac{k}{n}\sum_{j=1}^n\|\partial_jh\|_p^p+ \left(\frac{k}{n}\right)^{p/2}\|h\|_p^p\right]^{1/p}. \] The proof combines a sharp fixed-cardinality Rosenthal comparison with the exact reconstruction $h=\sum_jD_j\Delta^{-1}h$. A dimension-free inverse-gradient square-function estimate follows by integrating a pointwise reverse Poincaré inequality for the Walsh heat semigroup. Naor's transference argument then yields the optimal metric bound $O(p/\log p)$.

Banach asked whether a normed space must be Hilbert if, for one fixed dimension greater than one, all subspaces of that dimension are linearly isometric. We prove the complex case. By the codimension-one reduction, the essential finite-dimensional problem is to characterize a balanced convex body in a complex $(n+1)$-space whose complex hyperplane sections are all complex-linearly equivalent. We show that such a body is a Hermitian ellipsoid. The proof adapts the recent bundle–degree method of Lu and Yang for the real problem, with two features specific to the complex setting. After covariance normalization, write $G=\operatorname{Aut}{\mathbb C}(S)$ for the complex-linear symmetry group of a model section $S$. The exact section maps form a principal $G$-bundle over $S^{2n+1}$; reduction to $G^{\circ}$ places its obstruction in $\pi{2n}(G^{\circ})$, which is finite. Pulling back by a suitable positive-degree self-map trivializes this bundle and yields a global Lipschitz family of exact complex-linear section maps. Brouwer degree and a signed degree formula then imply that $p_S^{2n+2}$ and $p_S^{2n+4}$ are real homogeneous polynomials, where $p_S$ is the norm of a model section. Unique factorization forces $p_S^2$ to be quadratic, and phase invariance makes the resulting quadratic form Hermitian. The parallelogram identity then yields the general complex Banach-space statement.

Three general plane conics admit $184$ complex tangent circles, and it had been conjectured that at most $136$ of them could be real. We construct an explicit strongly general triple of smooth conics over $\mathbb{Q}$ with exactly $160$ real tangent circles; the same count therefore occurs on a nonempty Euclidean chamber. The construction combines a fourfold splitting theorem for two flagged double-line degenerations with exact Sturm–Tarski, elimination, and interval certificates. We also show that the Grothendieck–Witt-valued count is $92\mathbb{H}$ and express its real local signs, up to a fixed orientation convention, in terms of curvature differences, residual intersection divisors, and contact normals. The exact-arithmetic code and certificate data are archived in the accompanying repository.

Buoyancy points straight upward, yet in the absence of external forces, a rising air bubble may trace a spiral through the water. This phenomenon was already observed by Leonardo da Vinci. We show that the gravity-free, three-dimensional two-phase incompressible Euler equations with surface tension already admit such a motion. For every nonnegative inner-to-outer density ratio and every sufficiently small positive axial Weber number, we construct smooth near-spherical bubbles and drops with compact interfaces and phasewise irrotational flow. They are stationary in a frame translating along and rotating about a fixed axis, while their centroids lie off that axis and therefore trace genuine circular helices in laboratory coordinates. These helical relative equilibria form a symmetry-breaking branch that issues from the axisymmetric family of rectilinearly translating solutions and is generated by the spherical harmonic mode of degree two and azimuthal number one. The mathematical construction exploits an overdetermined free-boundary formulation, its variational structure, and an $\mathrm{SO}(2)$-equivariant Lyapunov-Schmidt reduction. Particular difficulties in the form of resonances arise at a discrete set of density ratios, including the frequently studied vacuum case with vanishing inner density. Complementary to our existence result, in the absence of surface tension and if the inner density is not larger than the outer density we prove that the model admits neither nontrivial translating relative equilibria nor relative equilibria with helical centroid trajectories. As a byproduct, we show that every regular finite-energy stationary vortex sheet of spherical topology for the homogeneous three-dimensional Euler equations, with irrotational flow on both sides, is trivial.

We study the spectral behavior and approximation properties of linearized shallow networks with analytic activation functions on the unit sphere $\mathbb S^d$. Motivated by earlier spherical approximation results for analytic zonal functions \cite{mhaskar1999approximation}, we first derive an asymptotic formula, expressed explicitly in terms of the residues, for the ultraspherical coefficients of a real-analytic function whose relevant singularities are a conjugate pair of simple poles on the imaginary axis. In particular, the formula gives a matching lower bound along the relevant high-degree subsequences and identifies the exponential decay scale and sign oscillation. It applies directly to $\tanh$, the logistic sigmoid, and the rational activations considered here, while the corresponding estimate for $\arctan$ is obtained through differentiation. For such activation functions and quasi-uniform points $\{\theta_j^\ast\}_{j=1}^n \subset \mathbb S^d$, functions $f \in \mathcal H^r(\mathbb S^d)$ satisfying the appropriate spectral compatibility condition can be approximated by a linear combination $$f_n(\eta) = \sum_{j=1}^n a_j \sigma(\theta_j^\ast \cdot \eta),$$ with the sharp Sobolev rate $$\|f - f_n\|_{\mathcal H^s(\mathbb S^d)} \lesssim n^{-\frac{r-s}{d}} \|f\|_{\mathcal H^r(\mathbb S^d)}.$$ The approximation theorem places the $\mathcal H^s$ norm on the left-hand side and simultaneously provides an explicit normalized $\ell^2$ bound on the coefficients. For spectrally compatible analytic target functions, the same construction further gives the stated analytic convergence rate. Thus, the analysis connects spectral asymptotics and lower bounds at the degrees where the ultraspherical coefficients do not vanish with Sobolev-norm approximation and coefficient control for fixed analytic activation functions.

We introduce the category $\mathbf{SSS}$ of sample-smooth spaces over a mixed site. The test objects are the products $\Omega_n := \mathbb{R}^n \times \Omega$ of a Cartesian space with the universal Hilbert cube $\Omega$ carrying all universally measurable sets, and a space is a set with a family of admissible plots $\Omega_n \to \mathcal{X}$ closed under precomposition. Smoothness and measurability are then not two structures glued along an axiom, but one structure over one site. The site has finite non-empty products, because $\Omega$ absorbs its own square; its Karoubi envelope contains every $\mathbb{R}^n$; and it has mixed morphisms $\omega \mapsto (W(\omega),\Phi(\omega))$, which turn measurability of a smooth family from an axiom into a consequence. $\mathbf{SSS}$ is a concrete quasitopos: complete, cocomplete, cartesian closed and locally cartesian closed, with a classifier for embeddings. Morphisms of Cartesian spaces are exactly the $C^\infty$ maps and manifolds embed full and faithfully, both without Boman's theorem. Every object has tangent and cotangent spaces, every morphism a differential. The modalities sit in an adjoint string $\Pi \dashv \flat \dashv \natural \dashv \sharp \dashv \Lambda$, making $\mathbf{SSS}$ cohesive over quasi-universal spaces. The point is the probability monad. Defining the plots of $\mathsf{P}(\mathcal{X})$ as push-forwards of $\mathcal{X}$-plots at every test object, $\mathsf{P}$ is an unconditional strong commutative affine monad on all of $\mathbf{SSS}$ – functor, unit, product of kernels, multiplication and the monad laws are each one line of seed splitting – and its Kleisli category, of differentiable simulators, is a Markov category. The reparametrisation trick holds by construction: every Kleisli morphism is plot-wise a sampler, stably under composition. A reflection theorem locates the whole gain in a single plot family.

An admissible configuration $\Lambda\subset\Hh^m\simeq\Rr^{4m}$ defines a compact intersection of quadrics $Z_{\Hh}(\Lambda)$ and a free diagonal $\Sp(1)$ quotient $N_{\Hh}(\Lambda)$. We realize this quotient as the leaf space of a real dilation action on an open subset of quaternionic projective space. The total space is a quaternionic moment-angle polyhedral product. Every quotient is $2$-connected; when there are no indispensable coordinates, it is $3$-connected and its fourth integral cohomology is generated by the Euler class $e$ of the associated oriented rank-four bundle. We compute $p_1(TN_{\Hh})=2(n-2)e$ and prove that no LVMQ manifold admits a quaternionic Kähler metric. An explicit infinite family consists of products of spheres admitting both homogeneous positive Einstein metrics and quaternionic toric structures in the sense of Gentili–Gori–Sarfatti. We also describe the trace foliations of the real action in the Poincaré domain. For the separate rank-one distribution $X\Hh$, an invertible quaternionic-linear field is integrable only in the real scalar case. A nonlinear field with positive real scalar derivative at the origin and involutive $X\Hh$ has exactly the standard quaternionic Hopf foliation on every sufficiently small centered sphere.

Spacecraft attitude control becomes challenging when light-sensitive instruments must avoid bright objects. This work proposes a path-planning-based spacecraft attitude reorientation maneuver considering these constraints to reach the desired attitude. Constraint-compliant attitude trajectories are computed using a fuzzy enhanced potential field approach in the attitude space. Based on the path-planning output, the control input is determined by considering the dynamics and kinematics of the spacecraft to satisfy dynamic constraints in the time domain. Numerical simulation results demonstrate successful reorientation maneuvers without violating the constraints.

Can the characteristic monodromy of string scattering be derived from S-matrix consistency alone, without assuming a worldsheet? We show that finite-particle consistency fixes it to a sharply defined extent. In a massless doubly ordered identity model, two exact six-point residues generate a scalar pentagon and the odd Fay identity, forcing every analytic four-point null kernel to satisfy $q''=\kappa q$ and leaving only the linear, trigonometric, and hyperbolic branches. This converts finite-multiplicity factorization into an all-orders constraint normally associated with worldsheet geometry. For a crossing-symmetric target in the opposite KLT module, positive unsubtracted dispersion with nonzero heavy spectral weight selects the trigonometric branch, and moment determinacy fixes the Veneziano germ up to scale. A two-scale counterexample proves that module entrance is an independent input. Our result therefore identifies both a finite-to-infinite rigidity mechanism for string scattering and the precise missing principle needed for a first-principles derivation.

Bradač, Sudakov, and Wigderson characterized $p$-locally dense graphons by a quadratic inequality for all bounded nonnegative functions. Their proof uses Reiher's finite lemma and graphon approximation, and they asked for a direct proof. We give one by rounding simple functions. Divide each level set into $m$ equal-measure pieces and retain each piece independently with probability equal to its level. Off-diagonal terms agree in expectation; the diagonal error is at most $\norminf{W-p}/(4m)$. Letting $m\to\infty$ and then approximating in $L^1$ proves the inequality. A three-atom example shows that atomlessness is necessary if arbitrary probability spaces are allowed.

Let $F$ be a field, let $0\le \ell\le k-2$, and suppose that $n\ge k-1$. We determine the minimum degree of a polynomial in $F[x_1,\ldots,x_n]$ that vanishes to order at least $k$ at every nonzero vertex of the Boolean cube and to order exactly $\ell$ at the origin. The answer is \[ n+2k-2-\rho_F(k-\ell), \] where $\rho_F(s)$ is the least number of positive integers summing to $s-1$ whose corresponding Catalan numbers are nonzero in $F$. The proof gives an explicit basis of the reduced vanishing space. In this basis the top-degree map is diagonal, with Catalan numbers on the diagonal; a refinement using block coordinates shows that the basis is compatible with polynomial degree. In odd characteristic the degree is either $n+2k-3$ or $n+2k-4$, according as $C_{k-\ell-2}$ is nonzero or zero. In characteristic $2$ it is \[ n+2k-2-s_2(k-\ell-1), \] where $s_2$ denotes binary digit sum. Thus the first characteristic drop, and all later drops, are determined exactly.

Pairwise relationships among mental-health symptoms are routinely summarised asscalar edge weights, which cannot express whether two symptoms carry overlapping information about a third or information that appears only in combination. Partial information decomposition (PID) addresses this gap but is computationally intractable beyond a few sources. We introduce embedding-based PID (ePID), a scalable pipeline that compresses all nonfocal symptoms into a low-cardinality discrete embedding and computes a tractable two-source PID, yielding source-unique, remainder-unique, redundant, and synergistic components for each ordered source-target pair. We benchmarked 13 candidate embeddings on synthetic Bayesian networks calibrated to PHQ-9 and on 83 real-world datasets across five PID measures. A supervised Agglomerative Conditional Information Bottleneck (ACIB) embedding recovered the reference decomposition most accurately of the 13 embeddings tested, and did so for every PID measure yielding non-negative atoms once four or more symptoms were compressed (synergy recovery r = 0.92). The two instruments then diverged sharply. In PHQ-9 networks (UK Biobank,N = 154,291; Xinxiang student sample, N = 24,292) the surrounding symptom context carried most pairwise dependence through redundant and remainder-unique channels; synergy contributed 6 to 9%, and no directed edge was synergy-dominated in either cohort. In the 28-item Interpersonal Reactivity Index, 45% of source pairs were. The identical pipeline, applied without parameter changes, therefore returned opposite profiles for the two instruments, each consistent with how that instrument was constructed. By separating overlapping from interaction-dependent information, ePID provides a scalable, model-agnostic complement to standard symptom-network methodology to distinguish redundant and synergistic contributions to observed correlations.

Training datasets for upcoming LLMs would include a significant amount of AI text/image data generated from current LLMs. In such a scenario, it is important to understand how this affects batch decompositions and thereby, the performance of the resultant new LLM. In this paper, we consider AI generated data as anomalies “linked" to main data points and study decomposition and undersampling properties of the overall random dataset. We use redundancy graphs and iteration techniques to obtain bounds for the minimum size of a strongly dissimilar (SD) decomposition and demonstrate a phase transition phenomena, wherein the minimum size is essentially determined by the main data points when the number of anomalies is small and is “taken" over by the anomalies above a certain threshold. We also establish a size criticality result for the strong similarity of a randomly undersampled dataset and illustrate our results with examples involving categorical datasets, whose overall space size is much larger than the size of the dataset.

Algorithmic Information Dynamics (AID) studies systems by perturbing them and measuring changes in algorithmic complexity, but its usual estimator, the Block Decomposition Method, is piecewise constant, restricting the calculus to finite differences. We use $K^{\mathrm{CDM}}_{\mathrm{s}F}$, a certified, differentiable estimator, to bring the calculus into learning dynamics: grokking, where a complexity order parameter is known but has not been made to act. As a transient loss kick, the estimator becomes a controller that accelerates grokking in Levin's description-length–versus-time sense, within a data-dependent Occam boundary whose finite-size trend, $f_c\sim\ln p/p$, is consistent with a coupon-collector interpretation. Ablations show that a complexity gate matches a train-loss gate in rescuing failing seeds with $27\%$ less intervention; among the tested signals, only map complexity marks the transition's completion; the certified prior and the per-parameter $\nabla K$ attribution are both fungible (a uniform-prior sensor makes bit-identical gate decisions, and random supports match $\nabla K$-selected ones above a sparsity threshold); and direct field perturbation shows a nucleation-like response to the Occam field (no linear regime is resolved over the probed amplitudes, so these measurements do not justify a fluctuation–dissipation surrogate), with a finite-field response growing by orders of magnitude toward the phase-transition. These measurements account for the empirically tuned staircase: bang–bang pulses, stall-fired and released on yield, whose iteration plausibly builds the response it exploits. The kick transfers to sparse parity and to a transformer; a sustained weight-space loss fails. The algorithmic estimator's distinct contribution is timing (when to fire and when to release), not attribution.