Papers

We extend the algebro-geometric construction of bifiltered affine schemes that encode both the Higgs branch and the Macdonald index of 4d $\mathcal{N}=2$ superconformal field theories to theories with non-trivial Higgs branches. For the Argyres–Douglas theories $\mathcal{D}_p(SU(N))$, we find that a universal potential, given by a function in the Casimir invariants of $\mathfrak{sl}(N)$, determines the scheme. The scheme is given by a `fattened' nilpotent orbit closure, encoding the OPE decoupling structure of the 4d theory. A procedure we call syzygy maximization fixes the precise potential in a broad range of examples, and the resulting schemes reproduce the 4d data. The underlying graded ring gives conjectural presentations of Zhu's $C_2$-algebra of the associated vertex operator algebra, while the bifiltration refines Arakawa's associated scheme by retaining the R-symmetry information. We find that Higgs branch renormalization group flows are realized by pulling back the same universal potential to the corresponding Slodowy slices, including flows for which the infrared Higgs branch is a point. In particular, our construction provides a systematic method for determining the Macdonald indices of the $\mathcal{D}_p(SU(N), O)$ theories, for which no general expressions were previously known.

We describe the implementation and use of colour_tensors: a Mathematica package built for the concrete construction of explicit colour tensors involving arbitrarily charged particles in any gauge theory. These can be used to directly determine linear relations among colour tensors and compute a number of otherwise difficult, representation-theoretic quantities relevant to scattering amplitudes such as colour-interference between partial amplitudes.

We prove nondegeneracy, modulo horizontal translation, of the normalized Sadovskii vortex patch. Building on the free-boundary and root-map framework developed by Huang and Tong, we first obtain a precise quantitative approximation of an exact Sadovskii profile, trapping its free boundary between two explicit barriers constructed around a rational approximation. This localization is then used to control the natural weighted quadratic form at the touching configuration. In the odd class, a ground-state identity shows that horizontal translation is the only zero mode. In the even class, dilation gives a negative direction, while a finite-rank comparison on a compact part of the boundary together with analytic contact-tail estimates yields a positive gap on a codimension-one subspace. Consequently, the Sadovskii dipole is locally rigid at the linearized level once translation is fixed. The computer-assisted part is confined to rigorous interval verification of a finite collection of explicit inequalities; the continuum, contact and operator reductions are proved analytically.

A cycle system for a matroid $M$ is a collection of cycles (unions of circuits) whose intersection properties mimic the cut sets of a graph. Cycle systems were introduced by Corry, the first author, McClain, Perkinson, and Yi, who showed that the $h$-vector of any matroid that admits a cycle system is a pure $O$-sequence, confirming a conjecture of Stanley for this class. Those authors also conjectured that any matroid admitting a cycle system must be binary. Here we answer this conjecture in the affirmative, and prove the stronger result that any such matroid must in fact be regular (representable over any field). From this, we conclude that if $M$ is connected, every cycle system for $M$ is a basis for its circuit space of. We establish other properties of cycle systems along the way, which may be of independent interest.

We show that Medvedev's logic of finite problems, a well-known superintuitionistic logic, is undecidable. The key method is a reduction from the periodic tiling problem to non-theoremhood in Medvedev's logic. This settles a longstanding open problem. Using similar techniques, but reducing instead to the ordinary tiling problem, we likewise obtain undecidability of Skvortsov's logic of infinite problems, and the fact that the two logics are distinct – in fact, they are separated by any aperiodic tiling of the plane. Due to the fact that Medvedev's logic figures in so many different areas, these results have implications for several fields – for example, the study of schematic fragments of logics such as propositional dependence logic, or the study of internal logics of toposes. The core idea and technical work of the undecidability proof were obtained using ChatGPT Sol 5.6, and formally verified in Lean by Claude Opus 5. A detailed methodology section outlines how such results were obtained.

We formulate and prove existence of pure quasifree Hadamard states for the Klein–Gordon field with Dirichlet boundary condition on a globally hyperbolic spacetime with timelike boundary. The microlocal condition is imposed on the one-particle Hilbert-space-valued distribution $p\circ G_D$ rather than on the two-point function. This is expressed by a $b$-wave-front set condition on the compressed future cone. The analysis on the manifold with corners $\bar M\times\bar M$ is then almost completely avoided. We prove that our version of the Hadamard condition is universal in the class of quasi-free Hadamard states. We also prove a propagation theorem for positive-energy microlocal splittings which uses only a microlocal splitting in the small $b$-calculus and well-posedness of the Dirichlet problem, and in particular requires no propagation theorem for reflected or gliding rays; combined with an explicit computation in the ultrastatic case and a Fulling–Narcowich–Wald deformation argument this yields the construction of a pure quasi-free Dirichlet–Hadamard state.

We prove that the capacity of the binary deletion channel satisfies $C(d)\le (1-d)/4$ for every $13/20\le d<1$. The proof describes the output from right to left, using a six-bit context to assign a description length. We bound the increase in expected description length minus output entropy when one input bit is added. A relative-entropy identity reduces this bound to finitely many linear inequalities. A potential on input windows of length 26 makes the inequalities telescope, giving a bound for every input word. Deletion composition extends the result from $d=13/20$ to all larger deletion probabilities. We also obtain finite-block bounds on mutual information and decoding error, with explicit $O(\log n/n)$ corrections. The finite certificate is checked using exact integer arithmetic, and the proof is formalized end to end in Lean.

For every finite set $A\subset\mathbb{Z}$, we construct a bijection $p:\mathbb{N}_0\to\mathbb{Z}$ beginning with $0,1$ that contains no four-term arithmetic progression, in occurrence order, whose first value belongs to $A$. More generally, for each $n\ge2$ the enumeration can begin with $0,1,3,\ldots,2^{n-1}-1$ and simultaneously avoid every such progression starting in $A$ or among these first $n$ entries. The construction uses finitely branching prerequisite relations and an explicit bounded integer potential. This proves finite prerequisite closure and gives an exhaustive enumeration, rather than merely a total order. We also give an extension theorem for finite prefixes with compatible binary-tail constraints. These results do not determine whether every enumeration of the integers contains an ordered four-term arithmetic progression.

We study Open Problem 3 of Kolokoltsov's survey "Quantum filtering and propagation of chaos for open quantum systems" (2026) on $H=\mathbb{C}^d$ ($2\le d<\infty$ fixed): a dynamic central limit theorem for $N$ mean-field coupled quantum particles under continuous diffusive (homodyne) measurement. The tagged conditional state $\Gamma^{(1)}_t$ is compared with the one-particle filter $\gamma_t$ driven by its own innovation, the mean-field Hamiltonian evaluated at the deterministic Hartree-Lindblad mean $\eta_t$. The fluctuation $F^N_t=\sqrt{N}(\Gamma^{(1)}_t-\gamma_t)$ is not autonomous in the limit: it is driven through the linearised filter by its own innovation, by an idiosyncratic Gaussian martingale carried by the connected correlations with the other particles, and by the empirical field $\hat G^N_t=\sqrt{N}(\Gamma_t-\eta_t)$, which itself satisfies an autonomous linear Gaussian equation, so that $(F,\hat G)$ solves a closed system. Unconditionally, $(F^N,\hat G^N)$ is tight, every limit point solves this system driven by a Brownian motion orthogonal to a square-integrable martingale pair, and the mean covariance of the idiosyncratic martingale is identified as the limit of the rescaled pair correlations. The remaining clauses, concerning signs rather than amplitudes, reduce to a single decorrelation statement fixing the law of the driving pair. It survives in a conditional form, with the common response to the empirical field and the tagged-slot component projected out; without those projections it is false for generic data, which is what fixes them. The conditional form is proved at leading order on a shrinking horizon and from below at its exact orders on every horizon; it is needed only for the concentration of the driving covariances, the Gaussianity of the pair, and uniqueness, and what remains open is one-sided and above the leading order.

Stochastic gradient descent (SGD) admits diffusion approximations that replace the complicated randomness of stochastic gradients by Gaussian noise, providing a powerful tool for understanding its dynamics and long-time behavior. We investigate whether an analogous approximation principle holds for optimization over probability measures, where the objective is a functional defined on the Wasserstein space P2. The nonlinear geometry and infinite-dimensional nature of P2 prevent a direct extension of the classical Euclidean theory. Using Lions differentiability, we lift the problem to a linear Hilbert space, where higher-order differential calculus becomes available. We then construct a Gaussian random-field approximation whose velocity field matches the mean and covariance of the original stochastic gradient. By exploiting this moment matching through higher-order Taylor expansions, we show that the Gaussian approximation captures the SGD dynamics with second-order weak accuracy. Our result provides a rigorous foundation for replacing sample-driven randomness by analytically tractable Gaussian fluctuations in stochastic optimization over probability measures.

We improve the upper and lower bounds for the smallest Lipschitz constant K for which there exists a K-Lipschitz retraction from the unit ball to the unit sphere of the infinite-dimensional Banach space X, for several classical spaces X.

We introduce GrassTop, a genome representation that integrates Grassmann manifolds and algebraic topology for viral classification and phylogenetic analysis. The framework begins by constructing multiscale topological and spectral descriptors of (k)-mer positional patterns. It then extracts a low-rank subspace that summarizes variation across the filtration and compares genomes using a Grassmannian distance. Although the reported implementation uses the chordal distance, the framework is not restricted to this particular choice. We evaluate GrassTop on four families of viral classification datasets, four phylogenetic clustering datasets, and a sequence perturbation experiment. Under the reported 5-nearest-neighbor protocol, GrassTop achieves higher scores than five published alignment-free reference methods across all reported classification metrics and datasets. Its UPGMA (unweighted pair-group method using arithmetic averages) trees achieve an average label purity of 1.0 on every phylogenetic dataset. The perturbation experiment provides a more nuanced result: the subspace representation differs most clearly from direct comparison of the unprojected feature matrices for SARS-CoV-2, whereas the differences are smaller or non-monotonic for the other datasets. Overall, these results support GrassTop as an effective topological-geometric representation for viral classification and phylogenetic analysis.

In fashion domain, recommending complementary clothing items that match selected pieces is a crucial cross-selling technique that improves customer satisfaction. Nevertheless, fashion matching presents significant challenges, as recommendations must not only align with individual user fashion preferences but also ensure compatibility between garments. These challenges are twofold. First, existing models often assume an overly simplified decoupled relationship between product compatibility and personalized user preferences, overlooking the natural complexity between the two. Second, existing data-driven approaches are not optimized for real-world fashion data, which is typically sparse and characterized by noisy interactions. To address these challenges, we propose Personalized Compatibility Graph Network, a multi-objective graph learning framework that organically unifies the modeling of product compatibility and personal preferences. PCGNet uses contrastive mutual information maximization to extract and align shared and view-specific patterns, thereby capturing the complex interplay between compatibility and personal preferences. Moreover, we introduce a correlation-aware neighbor sampling and a learnable global graph augmentation, which enhance the model by incorporating self-supervised signals mined directly from the graph, ensuring more stable and informative representations. Finally, PCGNet generates recommendation scores through the joint optimization of BPR ranking loss and multi-view mutual information losses. Experimental validation on two benchmark datasets demonstrates that PCGNet significantly outperforming state-of-the-art methods across all four evaluation metrics.

Restricted forms of Lagrange's four-square theorem, in which the representing variables are required to satisfy additional arithmetic conditions, have been studied from several points of view. Among the refinements proposed by Zhi-Wei Sun is the conjecture that every nonnegative integer has a representation \[ N=x^2+y^2+z^2+w^2,\qquad x,y,z,w\in\mathbb Z_{\ge0}, \] for which \[ x\ \text{and}\ x+24y\ \text{are perfect squares}. \] We prove this conjecture for all sufficiently large integers. The proof starts from an elementary parametrization of the two square conditions and combines a semi-linear two-squares sieve with Poisson summation, finite-field cancellation, and switching.

We compute all stringy invariants, encoding orbifold cohomology numbers, of (the normalization of) the identity component of $G$-character varieties of free abelian groups and of moduli spaces of $G$-Higgs bundles on abelian varieties, for all complex connected reductive groups $G$. As an application, we provide a direct proof of a topological mirror symmetry statement: the equality of the stringy invariants for Langlands dual groups. In the framework of Ginzburg–Kaledin local resolutions, we discuss the meaning of these stringy invariants, showing that symplectic resolutions of these moduli spaces can only exist in the cases of groups of Dynkin type $A$, $B$, or $C$. When such resolutions exist, the computed stringy Hodge numbers agree with the actual Hodge numbers of the resolution.

Clustering systems provide a natural way to encode structural information contained in phylogenetic networks. In this note, we study the clustering systems of normal and tree-child networks through an overlap-based property of set systems, called not-overlap-covered (NOC). We show that the NOC property is equivalent to inclusion-visibility, a memberwise formulation of the strict-compatibility condition previously used for tree-child clustering systems. We characterize normal networks as precisely the semi-regular networks whose clustering systems satisfy NOC. Consequently, a clustering system is realized by a normal network if and only if it satisfies NOC, or equivalently, if every one of its clusters is inclusion-visible. In this case, the Hasse diagram provides a canonical normal realization. These are exactly the clustering systems realized by tree-child networks. The NOC formulation yields a sharp quadratic upper bound on the number of distinct clusters of tree-child and normal networks and a direct polynomial-time recognition algorithm. Finally, we explore several consequences of the NOC perspective beyond the phylogenetic setting. These include connections to the enumeration of normal networks, an order-theoretic interpretation of inclusion-visibility, structural properties of NOC set systems, and a tractable special case of Minimum Set Cover, which is NP-hard in general.

Let $P_n$ be the matrix of a random permutation of $n$ symbols and let $M_n=\log\max_{|z|=1}|\det(I-zP_n)|$. Cook and Zeitouni proved that $M_n/\log n$ converges in probability to a constant $x_0$ for a uniform permutation. We show that the $\sqrt{\log n}$ fluctuations of $M_n$ are carried entirely by the number of cycles $K_n$. Write $\lambda(s)=\log\{\Gamma(1+s)/\Gamma(1+s/2)^2\}$, let $s_\kappa$ minimize $(1+\kappa\lambda(s))/s$ on $(0,\infty)$, and put $v(\kappa)=\kappa\lambda'(s_\kappa)$ and $a_\theta=\lambda(s_\theta)/s_\theta$. Under the Ewens measure with any fixed parameter $\theta>0$ we prove $M_n=v(\theta)\log n+a_\theta(K_n-\theta\log n)+O_P(\log\log n)$, so that the standardized pair $(K_n,M_n)$ converges jointly to $(G,G)$ with $G$ standard normal: the maximum and the cycle count are asymptotically perfectly aligned. This is deduced from a statement about the exact conditional law, which does not depend on $\theta$: for every compact $[\kappa_-,\kappa_+]\subset(0,\infty)$ there is a finite $C$ such that $P(|M_n-v(k/\log n)\log n|>C\log\log n \mid K_n=k)$ tends to $0$ uniformly over integers $k$ with $\kappa_-\log n\le k\le\kappa_+\log n$, that is, over exact and possibly atypical cycle counts. The proof keeps the size and the cycle count simultaneously in a two-variable coefficient extraction. Cycles longer than $n/(\log n)^4$ are reserved as an analytic factor whose coefficients are flat under every size shift produced by the shorter cycles; positivity then converts a scalar coefficient asymptotic into a relative comparison of the entire path-constrained measure, with an error that does not degrade with the number of constraints or with the rarity of the event. The constrained lower bound comes from pointwise saddle estimates for killed convolutions along a dyadic chain of endpoint boxes.

Practical OFDM-based integrated sensing and communication (ISAC) signals contain deterministic pilots and random data payloads, yet how these two types of resources jointly affect matched-filter sensing performance remains insufficiently understood. This paper establishes an analytical and optimization framework for pilot-data (P-D) OFDM sensing. We first derive a closed-form mean-square periodic autocorrelation function (MS P-ACF), which characterizes the respective contributions of the average subcarrier power spectrum and data-symbol randomness to sensing sidelobes. Based on this result, we reveal that pilot placement has no impact on normalized sidelobes under uniform power allocation or on the normalized global expected integrated sidelobe level (EISL), whereas it becomes an effective design degree of freedom for reshaping sidelobes under nonuniform power allocation. We further derive the globally optimal two-level pilot-data power allocation for global EISL minimization and characterize pilot placement for peak sidelobe suppression. For region-of-interest (ROI) sensing, we jointly optimize pilot placement and subcarrier power allocation to minimize the normalized ROI-EISL. A strictly uniform pilot placement is proved globally optimal under specified conditions, while a bisection-and-branch-and-bound algorithm is developed for general resource configurations. Numerical results validate the theoretical analysis and demonstrate that the proposed design reduces the ROI sidelobe level by up to 2.19 dB compared with optimized uniform-pilot OFDM, while improving ranging accuracy at an SNR of -10 dB by 84.3$\%$. These results reveal when and how pilot placement can be exploited to regulate the sensing behavior of random OFDM communication signals, providing practical resource-design principles for OFDM-based ISAC.

We use reflection and rotation signatures of dihedral groups to develop a Smith normal form method for computing the conjugation quandle colorings across all dihedral groups. Let $L = L_1 \cup \cdots \cup L_\ell$ be a link with link components $L_1, \ldots, L_\ell$, and let $\operatorname{Conj}(D_n)$ be the conjugation quandle of the dihedral group $D_n$. We use the semidirect-product structure $D_n \cong \mathbb{Z}_n \rtimes \mathbb{Z}_2$ to separate the coloring into a component signature. This component signature tells us whether each link component is colored by rotations or reflections, and the exponents are assigned separately under modulus $n$. Once the component signature is determined, every crossing relation becomes an equation that determines an integer matrix $M_\tau$. The abelian group $A_\tau(L)$, which we call the signature module, is represented by an integer matrix $M_\tau$. We prove that the free rank and nonunit Smith normal form entries, or equivalently the isomorphism class of $A_\tau(L)$, are preserved under the Reidemeister moves. The complete family of signature modules also determines the entire family of counting invariants for all $D_n$. The all-reflection system is the same as the Fox-$n$ colorings, while the mixed rotation-reflection signatures contain information that gives a stronger invariant. We also show that each signature module is the $t_k = (-1)^{\tau_k}$ specialization of the multivariable Alexander module, and extend the coloring interpretation of the module to generalized dihedral groups $\operatorname{Dih}(B)$ for every abelian group $B$.

Given a graph $F$, a family $\mathcal F$ of graphs on $[n]$ is \emph{$F$-intersecting} if $G\cap H$ contains a copy of $F$ for every $G,H\in\mathcal F$. We prove that there exists an absolute constant $\varepsilon>0$ such that every $P_4$-intersecting family $\mathcal F$ satisfies $|\mathcal F|\le\left(\frac12-\varepsilon\right)2^{\binom n2}$, which resolves a conjecture of Alon. Combined with Alon's reduction, this proves that a graph $F$ admits $F$-intersecting families of asymptotic density $1/2$ if and only if $F$ is a star forest. Multi-colored versions are also considered.

We solve Serre's problem on the specialisation of Brauer group elements in all cases where the number of variables is sufficiently large.

We introduce a new family of trickledown theorems, a.k.a., local to global technique to bound the spectral gap of the Glauber dynamics for multi-state spin systems. In this technique instead of upper-bounding the influence matrix of a link of co-dimension 2 by $\lambda I$ (where $\lambda$ is the second eigenvalue of the link), we upper-bound the influence matrix after a carefully chosen rank-1 shift. The rank-1 shift allows for a significantly smaller upper-bound but it comes at the cost of bounding the average loss due to rank-1 perturbations. As an application we use this method to show that the natural Glauber dynamics mixes in polynomial time to generate samples from the Sherrington-Kirkpatrick model for $\beta\leq \tfrac{1}{2}+\varepsilon$, for an absolute constant $\varepsilon>0$. At the heart of the proof we manage to bound the loss due to rank-1 perturbations by averaging over all links of co-dimension 2.

Let $2\leq k<n/2$ and let $(M^n,[g])$ be a smooth, closed, connected, and locally conformally flat Riemannian manifold with a $k$-admissible metric in the conformal class $[g]$. We prove a stability result of the $\sigma_k$-curvature inequality on $M$, in the sense that if equality is almost satisfied for some conformal metric, then its conformal factor is close to a minimizer of the inequality. Closeness is measured quantitatively in terms of Sobolev norms of the conformal factor, namely with respect to the $W^{1,2}$- and the $W^{1,2k}$-norm. In the non-degenerate case, these norms come with optimal exponents $2$ and $2k$, respectively, whereas in general the exponents are $2+\gamma$ and $\max\{2k,2+\gamma\}$ for some $\gamma\geq 0$ originating from a Łojasiewicz inequality. This extends a previous result by Frank and the author from $k=2$ and the sphere to $2\leq k<n/2$ and the full class of manifolds originally considered by Viaclovsky. It also extends a previous result by Engelstein–Neumayer–Spolaor from $k=1$ to the setting of fully non-linear scalar curvatures.

In 2021, Chern and Tang introduced two families of two-parameter modular functions associated with the Rogers-Ramanujan continued fraction. They established recurrence relations that express the members of these families in terms of eta quotients, and used these expressions to obtain dissection formulas. Motivated by their work, we construct a three-parameter extension and derive the corresponding recursive eta quotient representations. Our result also has applications to dissection formulas. In particular, it is used in separate work to obtain $5$-dissection formulas for overpartitions with restricted odd differences.

Let $d,K,N\in \mathbb{N}$ with $K\geq 3$ and $d\geq 4K+4$. Let $\Delta\subset \mathbb{Z}^d$ be the vertex set of a nondegenerate $(K-1)$-simplex, and let $A\subseteq[N]^d$ contain no nontrivial similar copy of $\Delta$. We prove that \[ |A|\ll_{\Delta,d} N^d\exp\!\left(-c_{\Delta,d}\sqrt{\log N}\right) \] improving upon a polylogarithmic bound due to Magyar. We perform a density increment argument using the circle method, and we introduce a “cut operator” method to decouple the weighted exponential sum over the system of quadratic forms describing the simplex. Our proof combines ideas from graph theory, functional analysis, and the geometry of numbers. In the process, we apply Finner's fractional form of Hölder's inequality, the analytic large sieve, and Kim's mean value formula for primitive lattice flags.

Let $g(n)$ denote the minimum number of peeling sequences among all $n$-point sets in general position in the plane. Dumitrescu and Tóth proved an exponential upper bound with base $12.29$, and Simon subsequently lowered the base to $9.78$. Using the same recursive construction, we prove \begin{equation*} g(n) \le (8+o(1))^n. \end{equation*}

We prove optimal trace-norm relaxation for the one-dimensional Lindbladian associated with the Witten differential $a=h\partial_x+V'$ which annihilates the classical Gibbs density: if $\lambda_1(h)$ is the first positive eigenvalue of $H=a^*a$, then the relaxation rate is $\gamma_h=\lambda_1(h)/(2h)$. It applies to initial operators whose Schwartz kernels, after a Gibbs conjugation, satisfy $ L^2 $ estimates for the restriction to the diagonal and for the normal derivative. An appendix by Chat GPT 6 presents a stronger result specialised to positive initial data.

We study archimedean smooth transfer for the Jacquet–Rallis relative trace formula comparison, in particular, identities between orbital integrals on GL(n) and its unitary forms. We work with Lie algebras and a (g,K)-module setting. Our main result states that for n=2, meaning GL(2) acting on gl(3), every polynomial type Schwartz function has a transfer to the unitary side and vice versa. Our proof relies on the Weil representation and the study of invariant differential operators. It also suggests certain structural properties of the (g,K)-modules in question which we formulate as conjectures.

We study the zero sets of the Zak transform $Z_\lambda B_n(x,\nu)$ of B-splines for $\lambda > 0$. Specifically, we provide a full characterization of the zero set for the hat spline for all positive values of the parameter $\lambda$ and for higher order B-splines when $\lambda > 1$. Finally, we apply these results to establish the frame property of integer-oversampled Gabor systems generated by B-splines. In particular, we characterize the frame property for integer-oversampled Gabor systems generated by the hat splines.

We prove three results as part of the program of diameter-free estimates initiated in [CDW26]. The first two are reverse square function estimates for the light cone in $\mathbb{R}^3$. We first establish an abstract $L^4$ inequality of independent interest, under an ordered superorthogonality hypothesis: for every four distinct indices, only the two nonalternating pairings are required to vanish. The loss is $C(1+\log N)^2$, where $N$ is the number of functions. An alternating-determinant argument verifies this hypothesis for separated cone sectors. This gives a diameter-free estimate for arbitrary disjoint angular intervals, at the thickness determined by their smallest width, with no upper restriction on the radial parameter. For the canonical equal-width partition on a fixed radial annulus, we obtain the loss $C(1+\log N)^{1/4}$, which is sharp up to constants. This refines the estimate by Guth-Wang-Zhang, via a different approach. The improvement uses additional orthogonality between diagonal and off-diagonal differences, together with bounded overlap of dyadic difference shells. Our third result is the diameter-free $\ell^2L^6$ decoupling for arbitrary partitions of the parabola, with an $N^\varepsilon$ loss independent of the interval widths. The argument is an adaptation of the method from \cite{Cushman-Demeter-Wu} and implies their three-fold additive-energy estimate for the parabola. The three proofs use variants of interlacing and special orthogonality in place of wave packet analysis, multilinearity and parabolic/Lorentz rescaling. Together, these results provide further evidence for the scope of the paradigm introduced in [CDW26].