Generating crystals has recently attracted significant interest due to their broad applications in materials science. However, existing generative models struggle to produce complete crystallographic specifications, limiting their ability to capture global symmetry and structural dependencies. In particular, current state-of-the-art approaches generate crystals only up to site symmetries and rely on sampling space groups from empirical distributions during generation. Inspired by \emph{spontaneous symmetry breaking} in physics, where crystals break symmetries under external conditions, we propose a novel diffusion-based framework that generates full structure specifications by reversing from the lowest-symmetry priors. Our method leverages a Markovian jump-diffusion process to model these symmetry-breaking dynamics, enabling it to traverse different space groups in a physically motivated manner. Our model, dubbed \emph{Symmetry-breaking Crystal Diffusion} (SbCD), introduces a principled approach to explicitly incorporate inter-space-group transitions into the generative process. In de novo generation experiments on MP20 and MPTS-52, SbCD outperforms its symmetry-preserving counterpart by a substantial margin, offering a promising perspective for generative modeling of crystalline materials.
Constraint programming is a core technology for solving complex combinatorial problems in scheduling, planning, configuration, and verification. Trusting its results therefore demands guarantees at two levels: that reformulations applied beforehand are semantics-preserving, and that solvers produce correct answers. In this work, we introduce a framework that addresses both verification levels in the Lean theorem prover: it can be used to prove formulation-level properties, such as equivalence, equisatisfiability, and the correctness of symmetry-breaking constraints, parametrically for entire problem families; and to check solver-produced certificates for individual instances via translation backends to external formats such as MiniZinc, SMT-LIB, and OPB. Combining both levels yields an end-to-end workflow that establishes the satisfiability or unsatisfiability of a constraint problem without trusting the external solver. Experimental results show that our framework's verified symmetry breaking also pays off in practice: a single parametric proof per problem family, reused across all instance sizes, reduces solver search effort by a factor of up to 2x10^7, while the entire in-Lean certification stays affordable, taking at most a few minutes for our largest instances.
We prove the identifiability of deep generative models (DGMs) with piecewise-affine (PWA) decoders and Gaussian mixture model (GMM) priors, in a purely unsupervised setting. We introduce three algebraic contrast principles for symmetry breaking: domain contrast, which trivializes the mixture symmetry group; mechanism contrast, which ensures every decoder branch is witnessed by a unique boundary; and interaction contrast, which forbids parameter conspiracies between latent components and decoder branches. Together they exploit the interplay between the discrete combinatorics of the PWA map and the continuous symmetry structure of the latent GMM. Continuity is replaced by algebraic symmetry conditions; injectivity is decoupled from structural identification and required only for pointwise inversion. Our results form a hierarchy: from law identifiability (LID; latent distribution up to a global affine map) through map identifiability (MID; decoder up to the same map) to posterior and pointwise identifiability. The ICA-form ambiguity emerges under conditions on diagonal component covariances. Assumptions are only on the data-generating process, not on learning methods, except for the interaction contrast. To our knowledge this is the first to make algebraic symmetry-breaking the engine of nonlinear identifiability, the first to admit discontinuous decoders, and the first to handle fully non-injective decoders, where every observation admits multiple latent codes.
Weight matrices in deep networks exhibit geometric continuity -- principal singular vectors of adjacent layers point in similar directions. While this property has been widely observed, its origin remains unexplained. Through experiments on toy MLPs and small transformers, we identify two mechanisms: residual connections create cross-layer gradient coherence that aligns weight updates across layers, and symmetry-breaking nonlinearities constrain all layers to a shared coordinate frame, preventing the rotation drift that would otherwise destabilize weight structure. Crucially, a nonlinear but rotation-preserving activation fails to retain continuity, isolating symmetry breaking -- not nonlinearity itself -- as the active ingredient. Activation and normalization play distinct roles: activation concentrates continuity in the leading singular direction, while normalization distributes it across multiple directions. In transformers, continuity is projection-specific: Q, K, Gate, and Up (which read from the residual stream) develop input-space ($\mathbf{v}_1$) continuity; O and Down (which write to it) develop output-space ($\mathbf{u}_1$) continuity; V alone, lacking an adjacent nonlinearity, develops only low continuity.