Papers

We define a generalized monoid of words, an asymptotic quotient of moderate nets of finite words, to regularize divergent products of arithmetic origin. Divergent sequences are encoded as moderate nets over a finite alphabet, while prefix-dependent observables are evaluated along admissible asymptotic scales. We characterize the evaluation and renormalization schemes axiomatically. Once the asymptotic scale and class of admissible regular expansions are fixed, the constant-term prescription is uniquely determined. Thus the regularized value is canonical relative to the chosen scale and evaluation functional, without absolute uniqueness. The regularizing functionals arise as expectations of logarithmic-prefix functionals under empirical distributions of sign oscillations. The alternating product of integers is regularized to $\sqrt{2/\pi}$, matching zeta regularization, while the alternating product of factorials yields $(2/\pi)^{1/4}$ and satisfies $\mathfrak{F}_{\mathrm{ren}}=\sqrt{P_{\mathrm{ren}}}$. Products governed by dyadic block signs acquire finite values through logarithmic-scale averaging in the corresponding dyadic quotient, invariant under dyadic equivalence. For the Thue–Morse sequence, zeta regularization is well defined via meromorphic continuation of its Dirichlet series. A separate convergent functional associated with its logarithmic partial sums yields $\exp(\mathcal F)\approx0.9732$. We also study numerically a substitution-generated sign sequence with exact square-root growth on dyadic blocks and fractal log-periodic structure. Its Dirichlet series shows a candidate log-periodic pole structure, separating local scale constants from a candidate global regularized value. The choice of asymptotic scale is determined by the arithmetic structure of the sequence: logarithmic scales for ordinary alternating products and dyadic scales for block-automatic sequences.

Fault diagnosis and prognosis in discrete event systems are studied in the scenario where the observations are possibly received with delay. To address this scenario, two conditions for diagnosis and prognosis with delayed observations are proposed, where we show that the state-of-the-art notion of prognosability must be revised to avoid conservativeness. Diagnosability and prognosability conditions are then verified by introducing a delay observer and a new verification function. Theoretical analysis indicates the effectiveness of the verification method for fault diagnosis and prognosis in the system.

Translating solitons arise as models for type~II singularities of mean-convex mean curvature flow. We construct a smooth bounded uniformly convex domain \(\Om\Subset\R^2\) such that the zero-Dirichlet solution of the planar translating mean curvature equation has a nonconvex sublevel set. The construction is based on a corrected near-critical grim-reaper profile and explicit barriers on a long convex channel.

We study the average analytic rank in the family of $L$-functions $L(s, E_d)$ associated with the elliptic curves $E_d : y^2=x^3-dx$, as $d$ varies over fourth-power-free odd integers. Since this is a family of curves with complex multiplication, we have $L(s, E_d)=L(s - \frac12, \xi_d)$, where $\xi_d$ is a Hecke character over $\mathbb{Z}[i]$. Assuming the Generalized Riemann Hypothesis, we compute the one-level density of the low-lying zeros of this family for test functions whose Fourier transform is supported in $(-\frac35, \frac35)$. As a consequence, we obtain the upper bound $\frac{13}{6}$ for the average analytic rank $r(E_d)$ over the family. Under the additional assumption of a conjecture on the distribution of quartic Gauss sums at prime elements (a quartic analogue of Patterson's conjecture for cubic Gauss sums), we extend the admissible support to $(-1, 1)$ and improve the upper bound for the average analytic rank to $\frac32$. Both results imply that a positive proportion of twists satisfy $r(E_d) =1$, while the second also yields a positive proportion of twists with $r(E_d)=0$.

Following a suggestion in a paper by Erné, Picado and Pultr, we establish the category of basic zero-dimensional spaces and their basic continuous maps. The objects are closure spaces with a distributive closure system containing a specified meet-base of complemented members that make the category a common generalization of those of closure spaces, zero-dimensional topological spaces, Heyting semilattices, and locales (frames). We treat images, preimages, continuity, openness and closedness of maps between basic zero-dimensional spaces. Among our main results, we have a useful description of preimages for basic continuous maps (analogous to the one for localic preimage maps as coframe homomorphisms), and a Joyal-Tierney type theorem for basic open maps. A novel aspect of the category of basic zero-dimensional spaces is a duality principle that, among other things, enables results for basic closed maps to be obtained for free from those for basic open maps.

We show that the graded Betti numbers are not invariants of the tropicalisation of a homogeneous ideal, or the tropicalisation of a projective algebraic scheme over a valued field. We do this by producing a family of saturated ideals that all realise the same tropical ideal with one member having a distinct Betti table from the rest.

We study the problem of sampling from target distributions whose potentials are simultaneously non-smooth, subject to superlinear gradient growth, and non-convex. We introduce the Subgradient Tamed Unadjusted Langevin Algorithm (SG-TULA), a discretisation of the Langevin diffusion that operates directly on subgradients, without relying on computationally demanding smoothing procedures. To handle the superlinear regime, taming techniques are employed to produce a stable, explicit scheme. We derive non-asymptotic convergence bounds in Wasserstein-2 distance, with all constants tracked explicitly in terms of dimension and inverse temperature, improving upon the currently known rates for subgradient-based Langevin algorithms. We further provide excess risk estimates for the associated optimisation problem. We verify the assumptions, with explicit constants, for the regularized pretraining potential of a LLM in the GPT-2 lineage and the boosted coordinate-wise variant of SG-TULA pretrains the former competitively against finetuned AdamW and Muon, for which no comparable non-asymptotic guarantees are presently available.

The low-degree method predicts computational thresholds by asking whether low-degree polynomials can distinguish a planted distribution from a null one. Recent counterexamples by Buhai, Hsieh, Jain, and Kothari (FOCS 2025) and Mao (2026) show that low-degree indistinguishability does not, in general, rule out efficient algorithms, even on noisy inputs. Hsieh, Kane, Kothari, Li, Mohanty, and Tiegel (STOC 2026) asked what it does imply, and showed that for permutation-invariant distributions on $\{0,1\}^n$, low-degree indistinguishability from the uniform distribution implies statistical indistinguishability after noise. In this work, we study the same question for planted subgraphs in sparse random graphs. Let $Q_n=G(n,c/n)$, and let $P_n$ be obtained by planting a uniformly random copy of a deterministic graph $\Gamma_n$ into an independent sample from $Q_n$. In the supercritical regime $c>1$, we show that if $P_n$ is degree-$D_n$ indistinguishable from $Q_n$ and the treewidth of $\Gamma_n$ is $o(D_n/\log n)$, then a noisy version of $P_n$ is asymptotically indistinguishable from $Q_n$. In the critical and subcritical regimes $0<c\le 1$, we obtain the same conclusion without any treewidth assumption, provided $D_n=\omega(\log n)$. Our proof has two main ingredients. First, we identify the subgraph-count and automorphism factors in the Fourier expansion with counts of isomorphism triples. Second, we cut a tree decomposition of $\Gamma_n$ into subtrees, which splits each large Fourier support into low-degree pieces that meet at only a few interface vertices, and we use noise to absorb the cost of reassembling them. At and below criticality, we use the low-degree assumption to rule out short cycles and noise to destroy the remaining long ones.

Persistence diagrams (PDs) provide stable and interpretable summaries of multiscale topological structure. While substantial progress has been made in the statistical analysis of PDs, existing literature often treats diagrams as static objects and provide limited frameworks for probabilistic modeling and stochastic evolution on PD space. We introduce a reinforcement learning framework for stochastic dynamics on PD space, where diagrams evolve through topology aware local edit operations. The dynamics define controlled Markov processes on spaces of finite PDs with variable cardinality. We establish conditions under which the induced Markov chains are irreducible, aperiodic, and geometrically ergodic, implying the existence of unique stationary probability laws on PD space. To guide the dynamics toward scientifically relevant topological targets, we formulate objectives that encompass distribution matching, task specific topological statistics, and structure-preserving compression. The resulting rewards balance task specific distributional targets, diagram fidelity, and complexity reduction, and yield a framework for adaptive topological simplification and probabilistic modeling. Experiments on synthetic and neuroimaging PDs demonstrate that the proposed framework can preserve dominant topological structure while reducing diagram complexity.

Let $\cX\subseteq\Pj^n_{\Z}$ be a fixed integral quasiprojective subscheme, smooth over $\Z$ of relative dimension $r$. For each fixed $m\ge1$, we bound the probability that the $m$th principal-parts jet of the restriction of a uniform degree-$d$ form to $\cX_p$ has a positive-dimensional zero scheme. The bound is $C(d+1)^{N_m}p^{-\lambda_m(d)}$, where $N_m=\binom{r+m}{m}$ and $\lambda_m(d)=\floor{m(d+1)/(m+1)}$. For $m=1$, this gives the Bertini singular-locus estimate $C(d+1)^{r+1}p^{-\ceil{d/2}}$. It settles Poonen's arithmetic Bertini Conjecture~5.2 and, after increasing the degree threshold, yields $p^{-A}$ for every fixed $A>0$. For $c\le r$ independent hypersurfaces, the probability of a positive-dimensional Jacobian rank-degeneracy locus is bounded both by $C\sum_i(d_i+1)^{r+1}p^{-\ceil{d_i/2}}$ and by $C'(d_{\min}+1)^{r+1}p^{-\ceil{d_{\min}/2}}$. The proof uses filtered $Q$-adic decompositions, triangular normal Taylor blocks, and Jacobian-pivot charts of uniformly controlled complexity.

We give a negative solution to the inverse generator problem on Hilbert spaces. More precisely, we construct a bounded operator $A$ with dense range on a Hilbert space $H$ that generates a bounded, strongly stable $C_0$-semigroup, while $A^{-1}$ does not generate a $C_0$-semigroup. We also construct an exponentially stable generator $A$ with $0 \in \rho(A)$ such that the inverse semigroup is unbounded and grows at least double logarithmically. For the latter generator, every Cayley transform satisfies the ordinary Kreiss resolvent condition but is neither strongly Kreiss bounded nor power bounded. Moreover, its powers satisfy a doubly logarithmic lower bound. Therefore, the Crank–Nicolson scheme is unstable in operator norm both for every fixed step size over long times and under mesh refinement at any fixed final time. Our counterexamples are deduced from a common finite-dimensional construction. For $\alpha\in(0,1)$, we use explicit bases of $\mathbb C^{2n}$ whose partial-sum projections are uniformly bounded and whose unconditionality constants are comparable to $n^\alpha$. The matrices underlying the counterexamples are then obtained as Schauder multipliers with respect to these bases, using a sequence of eigenvalues whose moduli decay doubly exponentially.

Starting from the Standard Model Lie algebra $\mathfrak{g}_{\mathrm{SM}} = \mathfrak{sl}_3 \oplus \mathfrak{sl}_2 \oplus \mathbb{C}$ sitting inside the complex Lie algebra $\mathfrak{e}_7$, we show how to decompose $\mathfrak{e}_7$ into the direct sum of a Lie subalgebra containing $\mathfrak{g}_{\mathrm{SM}}$ and three 32-dimensional subspaces, each of which forms the same representation of $\mathfrak{g}_{\mathrm{SM}}$ as one generation of fermions and their antiparticles. The setting is due to Nasmith, and much of the mathematics is that underlying the $\mathrm{E}_7$ generation unification of Kugo and Yanagida. New features include the derivation, in which as many results as possible rely only on the embedding $\mathfrak{g}_{\mathrm{SM}} \subset \mathfrak{e}_7$, and also the description of each 32-dimensional subspace as a copy of the exterior algebra $\Lambda\mathbb{C}^5$.

We consider the problem of minimising an energy over an unknown manifold that is given implicitly by a point cloud. For a known manifold one can define a gradient descent scheme analogously to the classical construction in Euclidean spaces. However, when the manifold is not known one has to simultaneously estimate the manifold whilst minimising the energy. We define a gradient descent scheme which remains in a neighbourhood of the manifold and, under suitable stability and sampling assumptions, converges to a neighbourhood of a local minimiser whose size vanishes as the time step and sampling errors vanish. As an example we show the application of the methodology to learning operator corrections in inverse problems.

The evolution of the front of branching Brownian motion is determined by the limit of the derivative martingale. In this work, we characterise which particles contribute to this limit. Precisely, we establish a sharp almost sure path localisation result which shows that the limit is determined by those particles whose trajectory stays within a thin tube at distance $s^{1/2}$ from the extremal particle.

Model evaluations may fix all tests before observing any responses or select later tests using earlier responses. We study this choice in a conditional-query model on a finite outcome space $\mathcal{X}$ with $|\mathcal{X}|=N$. We first ask which pairs of distribution classes can be reliably distinguished. We then ask how many additional queries are required to match an adaptive tester when all queried events must be fixed in advance. We show that learnability holds if and only if the two classes have positive separation in their pairwise conditional probabilities. When this separation is zero, the optimal worst-case error is exactly $1/2$ at every finite query budget. For any $T$-query adaptive policy and any $\rho \in (0,1)$, we construct a randomized non-adaptive procedure using $O(N^2(T + \log(1/\rho)))$ pair queries chosen before any response is observed. Its simulated transcript is within $\rho$ in total variation of the adaptive transcript, uniformly over all distributions in the model. We also construct a matching family with constant adaptive query complexity and $\Omega_\varepsilon(N^2)$ non-adaptive query complexity. Consequently, the worst-case fixed-error adaptivity gap is $\Theta_\varepsilon(N^2)$. Thus interaction can reduce the required number of tests by a quadratic factor, but the apparent exponential branching of an interactive evaluation does not yield an exponential query advantage.

Resolving Babai's minimal Cayley graph problem, we construct finite minimal Cayley graphs with arbitrarily large chromatic number.

We study stabilization of multidimensional continu- ity equations with source terms on bounded domains by means of localized interior flux feedback. The feedback is prescribed through the divergence of the flux and is chosen so that the error with respect to a reference profile satisfies a transport equation with localized damping. The main geometric condition is a finite-time characteristic damping inequality, requiring relevant characteristics to accumulate a uniform amount of damping over a time horizon. This condition is shown to yield exponential stability in L2 of the error, under a gain condition relating localized damping to compressive amplification of the transport field. Lyapunov-type entrance conditions ensure characteris- tic damping on support-restricted families of trajectories. A weighted Lyapunov functional provides a differential Lyapunov criterion and an input-to-state (ISS) estimate with respect to additive perturbations. We also discuss elliptic right-inverse realizations of the feedback flux and extend the characteristic damping argument to velocity fields depending nonlinearly on the state. A two-dimensional example finally illustrates the geometric, gain, and realization conditions.

Suppose that $n=2m$, $k=2t$ and $n > 10 k^7$. We show that if a family $\mathcal F$ of $k$-subsets of an $n$-set has only even pairwise intersections then $|\mathcal F| \leq \binom{m}{t}$. Moreover, every extremal family has an atomic structure. This result was previously proved by Frankl and Tokushige for $n > n_{FT}(k)$, where $n_{FT}(k)$ is at least exponential. The main technique is Delsarte linear programming.

Determining the combinatorial structures that maximize the number of prescribed substructures is a central theme in extremal combinatorics. Grzesik, Král', Lovász and Volec showed that when $\ell$ is not divisible by $4$, the random tournament contains asymptotically the most directed cycles of length $\ell$ among all $n$-vertex tournaments. In the paper, we resolve the remaining cases where $\ell$ is divisible by $4$. We show that, in this regime, the so-called carousel tournament asymptotically maximizes the number of directed $\ell$-cycles among all $n$-vertex tournaments, and in particular contains strictly more such cycles than the random tournament. This confirms the conjecture of Bartley and Day.

We consider the initial-boundary value problem for the stochastic time-nonlocal telegraph equation with $(\mathcal{PC}_\varepsilon)$-type kernel $a$: \begin{align*} \gamma \partial_t \left( a \ast \partial_t (a \ast v)\right) =\Delta v-\partial_t (a \ast v)+ \Psi(v)+ \Phi(v) \frac{\mathrm{d}W(t)}{\mathrm{d}t}, \end{align*} where $W$ is a space-time Gaussian white noise, $\Psi$ satisfies a linear growth condition, and $\Phi$ is Hölder continuous and uniformly nondegenerate. This model characterizes high-frequency signal propagation in small-scale systems under stochastic fluctuations. We develop a new hereditary phase-space lifting framework for time-nonlocal telegraph equations. In addition, we propose a novel generalized coupling framework, which features a new construction of the damping control term for the velocity. Based on these analytic tools, we prove the first results on weak existence and uniqueness in law for mild solutions, valued in $L_{loc}^2(\mathbb R_+; H^{\delta})$, to the stochastic nonlocal telegraph equation. The regularity index $\delta$ can be arbitrarily close to $\min\{\frac{1}{2},\frac{\varepsilon}{2+\varepsilon} \}$ from below, and the admissible lower bound of the Hölder exponent $\kappa$ is quantitatively determined by the integrability exponent $\varepsilon$. For the corresponding IBVP of the stochastic damped wave equation, obtained by replacing $a$ with the Dirac measure $\delta_0$, $\kappa$ can be improved to any value in $(\frac{3}{5}, 1]$. More significantly, the generalized coupling framework also handles low-regularity nonlinearities depending on both displacement and velocity.

To probe a bipartite quantum system, one may use arbitrary global operators or restrict to product operators acting separately on the two subsystems. We determine the sharp universal comparison between the resulting norms. For every $z\in M_n\otimes M_m$, we prove $\|z\|_1\leq\sqrt{2}\min\{n,m\}\|z\|_\varepsilon$, where $\|\cdot\|_1$ is the trace norm and $\|\cdot\|_\varepsilon$ is the injective tensor norm associated with the trace norms on $M_n$ and $M_m$. To prove the upper bound, we establish an $L_1$ noncommutative Khintchine inequality whose random coefficients are the entries of a Haar unitary. We also show that the coefficient $\sqrt{2}$ is sharp. As applications, we show that the same sharp constant governs the gap between bipartite correlation measured in trace norm and that measured by a correlation function, and obtain an improved universal upper bound for quantum data hiding. The upper bound has also been formalized and machine-checked in Lean.

Let $(\mathscr U,w)$ be a smooth separated Deligne–Mumford stack of finite type over $\mathbb C$, with a regular function $w$. Following Yu, one can use a compactification to define a filtration on its twisted de Rham cohomology. The same compactification gives Kontsevich lattices that compute this filtration. The extension of $w$ on the compactification may be only rational. Only a local nondegeneracy condition near the polar divisor is imposed. We prove that the resulting filtration does not depend on the compactification. The proof uses good resolutions and stacky weak factorization. This reduces the comparison to blowups and roots along boundary divisors. Filtered comparison theorems for the Yu and Kontsevich complexes are established for both operations. Harder and Lee use orbifold irregular Hodge numbers in their study of stacky Clarke mirror pairs. Compactification independence applies sector by sector to their setting. Thus the resulting orbifold filtration and irregular Hodge numbers depend only on the stack Landau–Ginzburg model.

For each of the four classical discrete laws — binomial, Poisson, negative binomial and hypergeometric — the mean absolute deviation about the mean collapses to a single point mass. We give a common telescoping proof of these identities and interpret the resulting closed forms by size biasing. We then derive complete asymptotic expansions for the Poisson ($\lambda\to\infty$), negative binomial ($r\to\infty$, $p$ fixed) and hypergeometric ($N\to\infty$, margins in fixed proportion) cases, extending the binomial expansion from the companion papers. The coefficients are given in closed Bernoulli-polynomial form and carry the lattice displacement of the mean exactly. At integer means the expansions reduce to sign-alternating odd series, and a single Binet-kernel argument shows that these series envelop the logarithm of the normalised mean absolute deviation: successive partial sums bracket it.

Despite advances in artificial intelligence (AI) across multiple sectors, today's AI tools, including deep learning and generative AI, still fail when embedded into physical systems, such as robots and vehicles operating under real-world physical laws. This stems from their inability to maintain reliable world models for long-horizon planning under uncertainty and generalize to unseen scenarios. In this context, wireless networks, through pervasive sensing and communication, can orchestrate physical intelligence. However, current architectures optimize throughput, latency, and reliability and cannot support real-time physical AI coordination, requiring agents to maintain shared spatiotemporal context. To address these challenges, a network of holonic digital twins (HDT-Nets) framework is proposed to deliver real-time physical AI inference through holonic agents that actively reason about their environment rather than passively mirror physical assets. Each HDT is realized as a hierarchical structure spanning the physical agent and network edge, reasoning autonomously at the local level while cooperating with neighboring HDTs to form collectively intelligent units. In HDT-Net, causal Markov blankets spanning sensing, communication, and control determine which agents must coordinate and enable counterfactual reasoning over multi-domain interventions. Active inference within these boundaries unifies perception, action, and learning by minimizing expected free energy while deciding which beliefs to transmit based on their cognitive value to the receiver. Category theory ensures that transmitted beliefs preserve semantic structure across heterogeneous agents with incompatible representations. Finally, integrated information theory quantifies when collective intelligence exceeds independent operation and how network intelligence evolves through coordinated learning and information exchange.

We study the energy $\mathcal{E}(s) := E_{s}[\Phi_s]$ of the one-dimensional fractional Allen–Cahn layer solution $\Phi_s$, defined as the unique odd, increasing solution of $(-\Delta)^{s} \Phi_s = \Phi_s - \Phi_s^{3}$ with $\Phi_s(\pm\infty)=\pm 1$ and $\Phi_s(0)=0$, for $s\in(1/2,1]$. Our main results are a sharp qualitative and quantitative description of the energy $\mathcal{E}$ on this interval. We show that the energy is continuous and strictly decreasing, and we obtain explicit asymptotic expansions at both endpoints. At the upper endpoint we prove $\mathcal{E}(s) = \frac{2\sqrt{2}}{3} + \kappa_1 (1-s) + o(1-s)$ with an explicit formula for $\kappa_1$. At the lower endpoint we prove that the energy goes to infinity as $\mathcal{E}(s)=\frac{1}{\pi(s-1/2)}+O(1)$. The strict decrease is proved with computer assistance. On an interior subinterval it is reduced to finitely many inequalities verified by interval-arithmetic computations. The proofs combine the Cabré–Sire construction of the layer, the minimality theorem of Palatucci–Savin–Valdinoci, an identity for the $s$ derivative of the energy, and a computer-assisted coercivity estimate at explicit approximate layers.

Much work in knot theory has consisted of categorifying, and thereby strengthening, knot invariants. We take the opposite approach: decategorification, more commonly called combinatorics. We introduce a technique that relates the dual braid group generators, the Hecke images of pure braids, and factorization problems in reflection groups to knot invariants. We prove that the $(a,z{=}0)$-HOMFLYPT polynomial can be computed as a solution to such a problem. This technique was motivated by Coxeter–Catalan combinatorics. For example, we give a new proof of EL-shellability of the noncrossing partition lattice using the image of the full twist in the Hecke algebra; in a surprising sort of combinatorial reciprocity, its inverse computes the homotopy type. Similarly, noncrossing partitions naturally arise from our construction applied to positive powers of the full twist, while cluster complexes come from the same construction applied to negative powers. In crystallographic type, we exploit the conjugacy of all Coxeter elements to give the first reflection subword models for rational noncrossing Catalan objects. Our reciprocity gives two models: one generalizing noncrossing partitions, and one generalizing clusters. Applying the same method produces two (rational) noncrossing parking models.

This work builds on the breakthrough of OpenAI in finding the first nonsofic group. We analyze the underlying proof mechanism and find further applications. Let $\Gamma<G$ be such that $\{g\in G:g\Gamma g^{-1}\leq\Gamma\}$ generates $G$ as a group, and suppose that both $\Gamma$ and $G$ have property $(T)$. If $\Gamma$ is not normal, then the generalized wreath product $\bigl(\bigoplus_{G/\Gamma}\mathbb Z/2\mathbb Z\bigr)\rtimes G$ and the group double $G \ast_{\Gamma} G$ are nonsofic. These hypotheses hold for explicit pairs of elementary groups over polynomial and Laurent polynomial rings, in which both groups are residually finite and Kazhdan.

We study Muon, a recently proposed matrix-aware optimization method, in the context of the Stiefel manifold. This manifold consists of matrices with orthonormal columns and is ubiquitous in machine learning and scientific computing. Existing extensions of Muon to this manifold rely on heuristic, approximate, or iterative updates with varying computational efficiency. We show that the corresponding Stiefel Muon update admits an exact closed-form solution and use this result to develop Skewon, a practical algorithm for orthogonality-constrained optimization with an efficient implementation. We further establish first-order convergence guarantees for Skewon in the smooth non-convex setting.

In this paper, we build on the analysis initiated by Averkov and Schymura (2022) and prove that the shifted Heller constant for matrices with $m$ rows is at most $m^2 + m$ for all sufficiently large $m$. Consequently, the number of distinct columns of a $\Delta$-modular matrix $A \in Z^{m \times n}$ of rank $m$ is at most $O(m^2 \Delta)$. This improves upon the earlier bounds of $O(m^4 \Delta)$ and $O(m^2 \Delta^2)$.

We prove that both continuous K-theory and analytic K-theory of rigid analytic spaces (à la Kerz–Saito–Tamme) satisfiy descent with respect to the Nisnevich topology. Together with the fact that it is $\mathbb{A}^1$-invariant assuming resolutions of singularities, we deduce that it is representable in the $\mathbb{A}^{1}$-homotopy category of rigid spaces (à la Dahlhausen–Yaylali). We identifiy the representing object with both $\mathbb{Z}\times\mathrm{BGL}$ and the analytification of algebraic K-theory. As a consequence, we get a representability statement for coefficients in light condensed spectra. Moreover, we show Weibel vanishing and that continuous K-theory is $\mathbb{A}^1$-invariant on local Tate pairs (without any regularity assumption).