Papers

In this paper, we introduce and study the (involutive) representation homology of involutive algebras, following the development of representation homology of associative algebra in our earlier papers. Our main motivation is to understand the hyperoctahedral homology of involutive algebras. We show that over a field of characteristic $0$, the hyperoctahedral homology of an involutive algebra is isomorphic (as a graded commutative algebra) to its one dimensional involutive representation homology. This result enables the explicit computation of the hyperoctahedral homology of the universal enveloping algebras of involutive (DG) Lie algebras. For group algebras $k[\Gamma]$ equipped with the involution $g \mapsto g^{-1}$, we relate the involutive representation homology of $k[\Gamma]$ to the representation homology of the group $\Gamma$ with coefficients in $\mu_2$, and we show that the latter is the homology of the loop space of the mod $2$ Dold-Thom space of $B\Gamma$; more generally, we compute the representation homology of $\Gamma$ with coefficients in an arbitrary diagonalizable group scheme. As a consequence, the hyperoctahedral homology of $k[\Gamma]$ is concentrated in degree $0$ in characteristic $0$, for every $\Gamma$.

Let $E$ be a vector bundle on a normal complex projective variety. We prove that slope stability of $\mathop{Sym}^2E$ and Zariski density of the reductions of $E$ modulo $p$ that are not strongly geometrically stable imply that the bundle $E$ is projectively unitary flat. This generalizes Shepherd–Barron's result from rank $2$ to arbitrary rank. We also show that if $E$ has rank $\le 3$ we can replace stability of $\mathop{Sym}^2E$ by stability of $E$ but we cannot do so if $E$ has rank $4$. We also generalize these results to reflexive sheaves on some mildly singular normal complex projective varieties.

Let $\Phi$ be an irreducible root system normalized so that long roots have length $\sqrt{2}$, and let $\mu\leq\lambda$ be dominant weights. We associate to $(\Phi,\mu,\lambda)$ the convex polytope $$P^{\Phi}_{\mu,\lambda} = (\mu+C)\cap(\lambda-C)\cap(\overline{D}-\rho),$$ where $C=\mathbb{R}_{\geq 0}\Phi^+$ and $\overline{D}$ is the closed dominant Weyl chamber. We prove that if $P^{\Phi}_{\mu,\lambda}$ and $P^{\Phi'}_{\mu',\lambda'}$ are isometric, then $$K^{\Phi}_{\lambda\mu}(q)=K^{\Phi'}_{\lambda'\mu'}(q),$$ where $K^{\Phi}_{\lambda\mu}(q)$ is Lusztig's $q$-analogue of the $\mu$-weight multiplicity in the irreducible highest-weight representation $V(\lambda)$, often called the generalized Kostka-Foulkes polynomial. Moreover, we construct the polynomial directly from the metric shape of the polytope, and we classify when two such polytopes are isometric. Such coincidences are abundant and can occur in different Dynkin types, and in different Dynkin families. By the Lusztig-Kato theorem, the main theorem gives a metric analogue, in the spherical setting, of the combinatorial invariance conjecture for Kazhdan-Lusztig polynomials.

We construct an explicit realization of finite-dimensional irreducible representations $V_{\lambda}$ of $\mathfrak{gl}_{n}$ using the representation theory of $\mathrm{Y}(\mathfrak{gl}_{2})$. We consider $V_{\lambda}$ as the evaluation representation of $\mathrm{Y}(\mathfrak{gl}_{n})$ and restrict it to the subalgebras $\mathrm{Y}(\mathfrak{gl}_{2})^{(i)}$. Then $V_{\lambda}$ is completely reducible as a representation of $\mathrm{Y}(\mathfrak{gl}_{2})^{(i)}$. It allows us to use the explicit formulas for irreducible representations of $\mathrm{Y}(\mathfrak{gl}_{2})$.

We extend the study of Hamilton-Jacobi equations on networks from a single equation to a weakly coupled system of two equations on a star graph, and we develop a numerical scheme for it. Instead of prescribing the boundary values and imposing a junction condition, we compute the solution on each branch from its outer vertex toward the junction and determine the boundary values for which these branchwise solutions agree at the junction. On each branch, we replace the right-hand sides by their cell averages and solve the resulting problems exactly on every cell. This gives an iteration whose limits we compute in closed form. From these limits, the values of the discrete solution at the junction are affine functions of the boundary values on the branch, with a $2 \times 2$ matrix whose rows sum to one and whose determinant factors into explicit contributions of the cells. We use these formulas to characterize the boundary values that give solutions continuous at the junction and monotone on every branch: when the matrices are nonsingular, the admissible boundary values of one branch form a convex polygon, bounded by conditions on the difference of the two components and by positivity conditions, and the boundary values on the other branches follow from them. We also treat the singular case, and we illustrate the results with numerical experiments.

We establish characterizations of planar quasiconformal mappings in weighted Bergman spaces $A_\alpha^p(\mathbb D)$. For $p>0$ and $\alpha>-1$, our characterization involves the convergence of certain integrals involving superlevel set areas, the Hausdorff measure of boundary superlevel sets, and the averaged derivative. These include quasiconformal analogues of characterizations established by Karafyllia and Karamanlis for conformal mappings. Moreover, for $-1<\alpha\leq 0$, we obtain an additional equivalent characterization in terms of the Hausdorff measure of level sets. The power $\alpha$ is sharp: for every $\alpha>0$, we construct a bounded $K_\alpha$-quasiconformal mapping for which the weighted level-set length integral with exponent $\alpha+2$ diverges.

Motivated by the role of piecewise linear functions in ReLU neural networks, we study systems of dilated functions in the Lebesgue spaces $L_q([0,1]^d)$, $q\in(1,\infty)$. For a sequence $\tau\in\ell_1$ we consider the operator $T_\tau g(x)=\sum_{k=1}^\infty \tau_k g(kx)$ and the function system $\mathcal Z^\tau$ obtained by applying it to the trigonometric system. We show that $\tau\mapsto T_\tau$ is an isomorphism of algebras from $\ell_1$ with the Dirichlet convolution as multiplication onto an algebra of operators and connect the inverse of $T_\tau$ to the inverse of $\tau$ with respect to the Dirichlet convolution. We give explicit conditions on $\tau$, under which the resulting system $\mathcal Z^\tau$ is a Schauder basis of $L_q([0,1]^d)$ and, for $q=2$, a Riesz basis for which we estimate the Riesz constants in terms of $\tau$. We characterise the Riesz bases of this type in terms of a sequence space of Dirichlet convolution multipliers, and we treat mixed systems of cosine and sine functions with different sequences. Finally, we apply these results to the system generated by two piecewise linear functions inspired by ReLU networks and improve the known lower Riesz constant to $2/3$, independently of the dimension.

Under the conditional framework breaking the longstanding O(n log n) integer multiplication barrier (originating from OpenAI Preprint #109), the asymptotic time complexity exponent T(n) = O(n (log n)^{1-\kappa}) is governed by bounding the second moment E[X^2] < 1 of recursive branching processes over trilinear tensor carrier graphs. In this work, we present a constructive combinatorial optimization over the alternating carrier bipartite matching graphs of the Schönhage trilinear tensor decomposition at dimensions h=23 and h=25. Through an alternating augmenting path search, we uncover a non-trivial cluster-unlocking cascade that eliminates 185 carrier roles (95 roles in h=23 and 90 roles in h=25), reducing the global physical wire width by 377,890 wires from the reviewed baseline W = 177,284,805 down to W = 176,906,915. Evaluated over the exact 10^{-14} rational grid, this wire reduction certifies an improved second-moment coefficient AB = 5,164,059 / 125,000,000,000 and establishes the sharpened conditional exponent \kappa = 826,215,307 / 20,000,000,000,000 \approx 4.131076535 \times 10^{-5}, strictly surpassing the preceding record on this architecture (\kappa \approx 4.123864 \times 10^{-5}). All 47 polynomial constraints and 7 rational security margins are audited in exact Q-arithmetic with zero floating-point error, and the fundamental wire reduction identity is formally verified in the Lean 4 interactive theorem prover without unproven axioms.

Let $\kappa_{\mathbb{K}}$ be the residue at $1$ of the Dedekind zeta function of a number field $\mathbb{K}$. We prove explicit upper and lower bounds for $\kappa_{\mathbb{K}}$ under the Generalized Riemann Hypothesis (GRH). Previous explicit bounds of comparable generality are weaker and require a weak form of Artin's holomorphy conjecture: our bounds improve the constants while assuming only GRH for the relevant Dedekind zeta functions. The new lower bound enables all imaginary quadratic fields with class number $\leq 2000$ to be detected (assuming GRH). Furthermore, we demonstrate that the upper bound can be significantly reduced when considering extensions of any fixed, nontrivial field.

Self-assembled monolayers (SAMs), formed by the adsorption of a monomolecular layer of adsorbate onto a surface, have emerged as leading candidates for plat- forms for electrochemical biosensors. In order to characterize the working age of a SAM-based biosensor, it is necessary to understand the long-term degradation of the monolayer, with particular emphasis on the intermolecular forces present in the self-assembly process. In this work, discrete- and continuous-time proba- bilistic cellular automaton (PCA) models for the degradation of SAMs involving adsorbate-adsorbate interactions are proposed in which adsorbate molecules desorb at a rate dependent only on the number of adsorbed neighbors. Impor- tantly, we make no assumptions on the relationship between degradation rate and adsorbed neighbors. Using techniques from generating function theory and complex analysis, the expected degradation curve is solved for explicitly in one dimension and shown to be a sigmoid curve related to the Gompertz function.

The second Jones problem (2011) consists in determining whether there exists a convergent positive series for which the ratio of consecutive terms has a limit and the set of subsums is a Cantorval; in the survey by Gł{\k a}b and Prus-Wiśniowski (2026) this problem is included in the list of open problems. In this paper, a family of such series is constructed explicitly as combinations of blocks of the form $\frac{2^k-1}{2^t} + \frac{2^{k-1}}{2^t} + \frac{2^{k-2}}{2^t} + \cdots + \frac{2}{2^t}$, where $t$ is the index of the last term of the block in the series and $k\ge2$ is the length of the block. For each series of the family it is proved that its set of subsums contains an interval and is not a finite union of closed intervals, and hence is a Cantorval. The proof relies on mixed-radix numerical systems with a redundant alphabet and uses an idea of Guthrie and Nymann (1988). If the lengths of the blocks tend to infinity, then the limit of the ratio of consecutive terms of such series is $\tfrac{1}{2}$, which, by a result of Jones, is the only possible value of the limit for which the set of subsums of a series can be a Cantorval. Thus, an affirmative answer to the second Jones problem is obtained.

In his paper, ”Sur la distribution des fractions irréductibles de dénominateur $n$ ou de dénominateur au plus égal à $x$”, Hubert Delange considers the family integrals defined by \begin{equation*} \int_0^1\left(\frac{N(n,t)}{\phi(n)}-t\right)^{2q}dt, \end{equation*} where $n>1$ and $q>0$ are positive integers, $\phi$ is the Euler totient function, and for any real number $t>0$, $N(n,t)$ counts the number reduced fractions whith denominator $n$ that belong to the interval $]0,t]$. He was able to evaluate the integral for $q=1$ but stated that ”Malheureusement, nous n'avons pas su jusqu'à présent faire cette évaluation pour $q>1$”. In addition to its intrinsic interest, this calculation would help understand the asymptotic distribution of the $\phi(n)$ totatives of $n$ relative to $\phi(n)$ equidistributed numbers in $[0,n]$. In this paper, we first provide some general properties of the integral for any $q>1$ and reduce the problem to its calculation for odd square-free integers with at least two distinct prime factors. We then analyse into more detail the case $q=2$ for which we provide a formula that reduces the problem to the evaluation of the fourfold Möbius sum, of integer valued modular sums related to generalized Dedekind sums.

We determine the dynamical degrees of bimeromorphic automorphisms of hyperkähler manifolds of all known deformation types with irreducible characteristic polynomial.

By the Strong Perfect Graph Theorem, every graph whose chromatic number exceeds its clique number contains an odd hole or an odd antihole. We investigate the structure of odd antiholes in critical graphs with fixed maximum degree. We prove that if $G$ is a $k$-critical graph with maximum degree $\Delta$ then every odd antihole in $G$ has length at most $2(\Delta-k)+7$, and conjecture that, in the odd-hole-free setting, equality forces the graph itself to be the corresponding odd antihole. We verify this extremal characterization when $\Delta\geq 2k-5$, extending recent results for $\Delta=k$, and construct $k$-critical odd-hole-free graphs showing that the next possible odd antihole length $2(\Delta-k)+5$ below the extremal bound can occur. Our proofs use a list-coloring theorem for graphs on $2\ell-1$ vertices whose complements are hypomatchable. This motivates the conjecture that every odd antihole $\overline{C_{2\ell-1}}$ is strongly $\ell$-chromatic-choosable; we show that this conjecture would imply the full extremal characterization. We also obtain degree and attachment restrictions for odd antiholes in critical graphs.

This article presents an introductory study of $p$-differential forms in spinor variables defined on the Minkowskian (or Lorentzian) manifold. Starting from the algebraic and geometric foundations of spinor spaces and Clifford algebras, we establish the correspondence between Minkowski spacetime and Hermitian matrices in spinor coordinates. Within this framework, we construct differential 1-forms and 2-forms expressed in spinor notation, and formulate the Dirac operator in terms of exterior calculus. Furthermore, we extend this approach to the case of non-monogenic functions, following the method proposed in~\cite{Jaramillo2024}. The results illustrate the consistency of the spinorial exterior calculus and its potential applications in the study of relativistic field equations and geometric formulations of quantum theory.

A classical uniformization theorem of Ahlfors from 1963 asserts that every bounded turning planar arc is the image of the unit segment under a quasiconformal self-map of the plane. We prove an analogous uniformization theorem for planar trees by showing that every planar quasiconformal tree with uniform branch separation is quasiconformally equivalent to a quasiconvex planar tree with uniform branch separation and Hausdorff dimension arbitrarily close to 1. Along the way, we show that every quasisymmetric (resp. bi-Lipschitz) homeomorphism between planar quasiconformal trees that respects the counterclockwise orientation of the branches at every branch point can be extended quasisymmetrically (resp. bi-Lipschitz) to the whole plane. As a corollary of the extension theorem we construct a large class of quadrivalent quasiconformal trees, which includes the Vicsek fractal, so that each tree in this class is the Julia set of a uniformly quasiregular map of $\mathbb{R}^2$.

For $1\leq r\leq n-1$ we study shrinking self-similar ends of the $r$-mean curvature flow, $S_r=c\langle X,N\rangle$ with $c<0$, that are asymptotic to a cone. For every rotational cone of positive slope we construct a smooth embedded rotational end and prove that its profile is the only positive solution of the profile equation with the prescribed slope at infinity. Its angular defect $xu'-u$ is strictly negative, which yields $r$-admissibility, and the profile has a complete asymptotic expansion for $u(x)/x$ in powers of $x^{-(r+1)}$, valid after arbitrary differentiation. The profiles depend real analytically on the slope and, on sufficiently distant exterior regions, foliate the image of the corresponding family. Our main result removes the symmetry assumption: two ends that converge smoothly, with multiplicity one, to the same circular cone coincide outside a compact set. The proof transforms the equation for their difference into an almost-eigenfunction inequality for a drift Laplacian on an auxiliary conical metric and applies Bernstein's asymptotic analysis on weakly conical ends. The argument extends to strictly $r$-admissible cones whose links have constant principal curvatures.

We construct a symmetric trace-free tensor for constant-mean-curvature hypersurfaces in $\mathbb{S}^n\times\mathbb{R}$ and $\mathbb{H}^n\times\mathbb{R}$ that extends the Abresch-Rosenberg tensor. Within its natural class, it is characterized by the linearity of its Codazzi defect, and its complete connected zero set consists exactly of the rotational CMC spheres. An exact integrated Simons identity yields explicit gap theorems, valid for every $h>1$ in $\mathbb{H}^n\times\mathbb{R}$ and every $h>(n\int_0^{\pi/2}\sin^{n-1}r\,dr)^{-1}$ in $\mathbb{S}^n\times\mathbb{R}$. We compare the pinching thresholds with the pointwise algebraic level that implies convexity. In $\mathbb{H}^n\times\mathbb{R}$ the explicit pinching conditions imply strict convexity, so there the method gives an independent proof with explicit constants. In $\mathbb{S}^n\times\mathbb{R}$ the pinching region exceeds the convexity level for $n\in\{3,4\}$ and every admissible $h$, and for $n\geq5$ whenever $h^2\geq n/(6(n+2))$; there the gap theorem does not require convexity to be known in advance. We also determine second-order large-$h$ asymptotics of the thresholds.

We give a complete classification of tight contact structures on manifolds obtained from any surgery along any torus knot, which provides the first such classification on an infinite family of knots. The main tools are the classification of non-loose torus knots, bordered contact invariants via their immersed curve interpretations, and non-thickenable tori in knot complements. As a byproduct of this classification, we also give the first classification of contact structures on some surface bundles over the circle with fibers of genus larger than $1$. The symplectic fillability of these contact structures is also studied and leads to, among other things, the disproof of a conjecture of Ghiggini and Van Horn-Morris on Stein fillability of some contact structures on a Brieskorn sphere.

A digraph $D = (V,E)$ is 3-transitive (respectively, 3-quasi-transitive) if whenever $(u,v,w,z)$ is a directed path of length 3, $(u,z) \in E$ (respectively, $(u,z) \in E$ or $(z,u) \in E$). These digraphs extend the concepts of transitive and quasi-transitive digraphs, respectively. For 3-transitive and 3-quasi-transitive digraphs, only the strong case has been characterized. In this paper, we introduce the notion of semi-acyclic digraphs and use it to describe completely the structure of 3-transitive and 3-quasi-transitive digraphs. As an application of this type of structure description, we prove that the path partition conjecture holds for the class of extended semi-acyclic digraphs, which generalizes two previously known results on the conjecture.

Transmitters in adaptive communication systems may switch their modulation scheme at an unknown time, and a receiver must detect this switch quickly without raising frequent false alarms. Existing modulation classification methods label a finite block of received data and do not provide stopping rules with controlled false-alarm and delay performance. We study the sequential detection of a switch from one phase shift keying (PSK) scheme to another over an additive white Gaussian noise channel. We build cumulative-sum (CUSUM) and moving-sum (MOSUM) detectors from the log-likelihood ratio and its low-SNR approximation. For a switch from quadrature PSK to binary PSK, we derive the moment-generating function and probability density function of the approximate statistic. We use them to relate the CUSUM threshold to the average run length to false alarm and the worst-case detection delay. For a switch between any two PSK orders, we approximate the vector of overlapping MOSUM statistics as multivariate Gaussian and obtain bounds on both measures. Monte Carlo simulations show that our derived expressions for ARL2FA and WADD closely match the empirical results across thresholds, validating the accuracy of the proposed analytical approximations and theoretical bounds.

We say that a group $G$ has the $R_{\infty}$ property if the number $R(\varphi)$ of twisted conjugacy classes is infinite for any automorphism $\varphi$ of $G$. The $R_{\infty}$-nilpotency degree is the least integer $c$ such that $G/\gamma_{c+1}(G)$ has property $R_{\infty}$; in case this integer does not exist, we say that $G$ has infinite $R_\infty$-nilpotency degree. In this paper, we determine the $R_{\infty}$-nilpotency degree of some lollipop GBS groups by using different approaches, such as Sylow subgroups, normal forms and presentation of nilpotent quotients, and diagrams of exact sequences. We also compute the $R_\infty$-nilpotency degree of all 2-generated lollipop GBS groups and of all GBS groups whose associated graph is a tree.

For the bilinear spherical averages in dimension $d\geq2$, we show that the lacunary maximal operator is not bounded from $L^1(\mathbb R^d)\times L^q(\mathbb R^d)$ to $L^r(\mathbb R^d)$ whenever $1\leq q\leq\infty$ and $1/r=1+1/q$. The obstruction is realized by fixed nonnegative inputs; a second construction proves quantitative failure on smooth compactly supported functions and finitely many dyadic scales. The same conclusions hold for the full maximal operator. We give a sharp Lorentz-index criterion in the range $q>d/(d-1)$ and extend the negative result to all multilinear spherical maximal operators with an $L^1$ input. We indicate the precise strong-type boundary statements to which these conclusions apply, without drawing conclusions about separate weak-type or interior estimates.

Toric stacks enrich the study of toric varieties by accounting for quotients of actions of diagonalizable groups. In practice, toric stacks are constructed and understood via "stacky fans" generalizing the fans appearing in toric geometry. Previous attempts to make this construction into an equivalence of categories have either run into crucial errors or have been restricted to special cases (e.g. only considering separated Deligne-Mumford stacks). In this paper, we solve this problem by introducing a new 2-category $\mathsf{StkFan}$ of stacky fans and constructing an equivalence of 2-categories between $\mathsf{StkFan}$ and a 2-category $\mathsf{TStk}$ of toric stacks. Along the way, we develop a theory of "toric categories" which categorifies the classical theory of toric monoids. We also explain how working with toric categories provides a simple axiomatic framework for Gale duality.

The modified Anderson-Bjorck (modAB) method was first proposed in 2023 by Ganchovski and Traykov for solving nonlinear equations in structural mechanics. It was improved further in 2026 by introducing better switching criteria with account to function symmetry, and bisection fallback based on measuring the bracket reduction rate. Although the algorithm demonstrated superior performance across a great variety of functions, it lacked formal mathematical analysis and proof of convergence. In addition, some discrepancies were noted by L. Tomov in the latest version that need to be addressed: premature application of the Anderson-Bjorck (AB) correction on the first AB step, caused by retaining the endpoint-update record from the preceding bisection step, and using the corrected endpoint ordinates in symmetry factor calculation instead of the true residuals. The purpose of the present paper is to provide rigorous mathematical proof of global convergence in exact arithmetic, remove the above discrepancies, and provide floating-point safeguards for interpolation, switching and stopping, with explicit treatment of infinite and NaN values, underflow, overflow and round-off errors. These safeguards do not certify the accuracy of the supplied function values. Additional performance improvement was introduced by allowing for early bisection fallback when AB fails to make progress on reducing either the bracket width or the residual, while retaining a finite bound on consecutive exceptions to the prescribed width condition.

The codeword stabilized (CWS) framework provides a systematic method for constructing quantum error correcting codes from a simple undirected graph and a classical code. When the underlying classical code is linear, the resulting quantum code is a stabilizer code. In this work, we investigate the conditions under which an arbitrary $[n,k,d]$ classical binary linear code can be lifted to an $[[n,k,d]]$ quantum stabilizer code using the CWS framework. We analyze the random CWS code constructed from a given $[n,k,d]$ binary linear code $\mathcal{C}$ and an Erdős-Rényi random graph $\mathrm{G}(n, \frac12)$. We establish a lower bound on the probability that the resulting quantum code preserves the minimum distance $d$ of the classical code. This probabilistic analysis yields a Gilbert-Varshamov-type condition on the $n,k,d$ parameters of the classical code that guarantees the existence of a graph resulting in a quantum code with the same $n,k,d$ parameters.

We construct an algebraic framework for the extended holomorphic anomaly equations following ideas of Alim, Movasati, Scheidegger and Yau. We choose the quintic threefold as our main model and use this framework to solve the equations and fix the ambiguity up to genus 5. We obtain predictions for real Gromov-Witten invariants, or, equivalently, real Gopakumar-Vafa invariants that are integers and agree with their complex counterparts modulo 2. In the end, we discuss how the framework could be extended and compute the same predictions for the bicubic up to genus 9.

We show how to perform exact diagonalization of SU($N$)-symmetric Bose-Hubbard models separately in each irreducible representation of SU($N$), by exploiting the Howe duality between U($L$) and U($N$). Three structural consequences follow for any lattice and any Hamiltonian built from the U($L$) generators: (i) the spectrum in a given orbital sector is independent of $N$ and of the particle statistics, (ii) the canonical partition functions for bosons and fermions obey the exact identity $Z_{\mathrm{b}}(-N)=(-1)^{N_\mathrm{b}}Z_{\mathrm{f}}(N)$, (iii) with unfrustrated hopping the groundstate is always a color ferromagnet, so that the color-resolved method we introduce is useful to study excited sectors, frustrated lattices and finite temperatures. As applications, we first derive the finite-temperature phase diagram of attractive SU($N$) bosons on few-site rings at large $N$ for arbitrary filling. We then show the emergence of singlet-cluster states from frustrated hopping, the counterpart of fermionic ferromagnetism, with the example of the sawtooth and kagome lattice.

We develop a homotopical framework for understanding the space of quantum spin systems based on topological $n$-nets, which assign associative unital algebras to local regions in an $n$-dimensional space in a manner that captures the locality structure of the underlying lattice theories. The net-theoretic perspective admits a natural categorification, in which the category of topological $n$-nets carries the structure of a higher Morita category. Moreover, the nets of greatest physical interest form a Karoubi-complete subcategory that shares many of the characteristic structural features of the category of finite TQFTs. Our main result identifies the group of locality-preserving symmetries of (3+1)-dimensional quantum spin systems (i.e. quantum cellular automata modulo stabilization, finite-depth quantum circuits, and translations) with the full Witt group of nondegenerate braided fusion categories, thereby answering a question posed by Haah.

Deterministic flows can contract Wasserstein distances between probability laws without being contractive in point-state space. We develop a differential contraction theory for pushforward flows on forward-invariant subsets of the 2-Wasserstein space that quotients out the gauge freedom of the continuity equation. Weighted Helmholtz-Hodge decomposition and the Wasserstein contingent cone define the weighted horizontal tangent cone (whtc). Requiring the contraction quadratic form to be nonpositive only on the whtc yields the Intrinsic Wasserstein Contraction Condition (IWCC), which generalizes the classical contraction LMI and recovers it on Dirac measures. An almost-everywhere Euler approximation based on the superposition principle and Banach-valued Lebesgue differentiation theorem proves that IWCC is sufficient for intrinsic Wasserstein contraction. Law contraction without point-state contraction is illustrated by a rotationally symmetric shear example. Finally, a new comparison lemma yields an incremental Wasserstein ISS theorem; for Brockett's integrator, we prove exponential stabilization of a class of axisymmetric laws to the uniform law on a circle via a smooth feedback.