A code world model accepted by a sampling gate can be exactly right on everything the gate can see and arbitrarily wrong beyond it. We characterize what a certified model can know, and what its errors can cost, when the omission is an annular freeze mode enclosing an unreachable interior. The gate quotient makes the question precise: acceptance-with-certainty determines the model exactly on the reachable query set; beyond reach is gauge. On a minimal ring instrument we prove the extreme case (a wrong-topology filled-disc artifact unfalsifiable by any sampling gate and bitwise harmless at play) and measure, with LLM synthesis across three model families, how one knob (a channel of width gamma) walks the same artifact through three regimes: unfalsifiable-and-harmless, falsifiable-and-costly, and instantly falsified. Three principles organize the empirics. First, danger is topology relative to reach: a channel the planner can use collapses the blind model's exploitation (play cost 1.09 to ~0 over a knee at gamma ~ 0.1), while a hidden channel with the same first Betti number keeps it at full strength (1.12). Second, repair is parameter-bound and sensor-bound: no family recovers the region from outside evidence; from inside, models pose the right topology but cannot pin its parameters, and the posed topology tracks the guiding persistent-homology summary's wrong beta_1 (a sensor with a measured geometric resolution limit), not the truth. Third, mitigation must match the error's dimension and direction: point fences fail against the one-dimensional boundary, a dimension-matched persisted fence collapses exploitation to a two-lesson transient (0.999 to 0.058), and the dual freedom certificate collapses the invented-mode failure symmetrically (1.769 to 0.029). In n dimensions the shell makes misidentification near-certain while the danger stays fully exploitable: the two axes are independent.
Machine learning and data analysis techniques have recently emerged as powerful tools for identifying patterns and formulating conjectures in mathematical research, most notably in the field of low-dimensional topology. In this paper, we initiate a systematic approach to handling mathematical data structured as (truncated) infinite $q$-series, or equivalently, infinite series of integers. To apply this data analysis pipeline, we construct a comprehensive dataset of $\widehat{Z}$-invariants (homological blocks) for plumbed 3-manifolds. We demonstrate that neural networks can reliably extract essential topological information, such as homology class and underlying graph structure, directly from the $q$-series coefficients. A central feature of our methodology is a focus on interpretability; by contrasting local gradient sensitivity with global feature relevance, we reveal that the networks learn to bypass complex topological rules in favor of specific spectral and geometric proxies. Finally, we apply this pipeline to probe homology cobordism, discovering a high-accuracy predictive relationship between the $\widehat{Z}$-invariant exponents and the Heegaard Floer $d$-invariant (correction term). These results suggest that $\widehat{Z}$-invariants capture subtle geometric information regarding cobordism equivalences, warranting a new direction for the study of quantum invariants.
Accurate foreground masks can still form an incorrect surgical-instrument instance set: duplicate, fragmented, merged, missed, or empty-frame predictions may preserve favorable pixel overlap while violating object identity and count. Final query selection is therefore a relational, variable-cardinality problem rather than a collection of independent candidate decisions. We evaluate topology-aware query selection, which represents the nonempty candidates of a fixed Mask2Former as a complete graph, learns relational candidate and pair representations, predicts set cardinality, and solves an exact structured subset problem. The formal comparison is the complete relational path versus a node-feature-matched path; it evaluates the combined effect of pairwise geometry, message passing, and the additional relational-path capacity, not an isolated component. On the sealed 22-case source test, all three discovery seeds supported instance-set performance improvement with segmentation fidelity and predefined technical-safety preservation: instance F1 increased by 0.0504--0.0612 and positive-frame set-failure rate decreased by 0.0848--0.1060. Direct ROBUST-MIPS transfer reproduced the complete result in all three seeds. Endoscapes supported only one of three seeds and therefore did not establish stable direct transfer. Taken together, the results support a bounded conclusion: the evaluated complete path improved coherent instance-set construction from fixed Mask2Former candidates in specified native-instance contracts, while stable cross-domain transfer and component-specific effects remain unestablished.
Extracting structured, parametric 3D representations from raw images remains a fundamental challenge in computer vision and graphics. While recent advancements in the 3D Gaussian Splatting (3DGS) pipeline integrate planar primitives to yield compact and editable geometry, these approaches typically treat planes as isolated, discrete sets. This lack of topological connectivity hinders robust geometric reasoning, leading to fragmented reconstructions and misaligned boundaries that fall short of the precision for rigorous spatial analysis and professional design workflows. To address this, we introduce TopoGS, the first 3DGS framework to explicitly integrate both planar and topological constraints for coherent 3D reconstruction. Specifically, we extract global 2D topological relationships from multi-view image segmentations and anchor Gaussian primitives to these structural elements. This formulation enables the joint optimization of plane parameters, rendering fidelity, and topological adjacency. By enforcing strict multi-view consistency alongside these topological constraints, our method significantly mitigates geometric misalignments and produces connected, structured 3D models. Extensive evaluations on the ScanNet++ dataset demonstrate that TopoGS achieves state-of-the-art performance, providing a highly robust solution for generating accurate, topologically sound, and visually faithful scene representations.
The Minkowski functionals of a field's excursion sets -- area, boundary measure, and Euler characteristic -- describe its level-set morphology; the Euler characteristic is the cheapest handle on topology. We derive smooth Monte-Carlo estimators for all three of a continuous neural field, evaluated at scattered points via the co-area formula and Gauss-Bonnet, using only autodiff: no grid, no complex, no persistence. The estimator is accurate to 1-3% against exact topology in 2D and 3D, and costs about 3 ms per iteration where a persistent-homology (PH) loss on a cubical grid costs 650-1000 ms -- a 250x gap. We establish four design rules without which these losses silently fail: a dense level ladder (invariants are flat in the parameters away from transitions), a $C^2$ backbone (ReLU nets hide curvature in kinks), the full Minkowski vector (Euler characteristic alone is an alternating sum, gamed by debris-hole cancellation; pricing perimeter closes the channel), and sampling-scale coverage. In 2D the vector-valued cap is the only method in a controlled comparison that both repairs topology (3/3 seeds) and preserves fidelity -- uniform smoothing repairs at 11-17x the fidelity cost, and the Euler term alone repairs nothing. In 3D neural-SDF fitting, however, a failure mode we believe general to any sampled soft topology objective appears: gradient descent adversarially hides topological noise below the sampling density, where the estimator is blind -- spurious-feature counts are invariant to 4x more samples, and closing the window needs cubically many points, erasing the cost advantage. A grid-based PH baseline, whose complex is the evaluation resolution, solves the same benchmark ($4/9$ exact; median $b_1$ error 1 vs. ours above $10^4$). The 250x cost of persistence is, at present, the price of having no null space. We release estimators, receipts, and benchmarks.
Jilles S. van Hulst, Jakub M. Tomczak, W. P. M. H. Heemels +1cs.LG cs.CV math.AT stat.ML
Variational autoencoders (VAEs) learn low-dimensional latent representations of high-dimensional data. When the data lies on a manifold with non-Euclidean topology, the standard Gaussian prior introduces a topological mismatch that degrades reconstruction quality and prevents faithful representation. We present a constructive mathematical framework that resolves this mismatch for all manifolds that admit a product covering space. These are manifolds expressible as products of elementary factors (circles, intervals, or lines) or as quotients of such products by a finite symmetry group. The class includes cylinders, tori, Möbius strips, Klein bottles, and real projective spaces. Factorized distributions over the elementary factors yield product topologies with closed-form, decoupled KL divergences, so that each latent factor can be shaped independently while keeping training tractable. We catalogue reparametrizable encoder-prior pairs for periodic, bounded, and unbounded supports, and provide coordinate transformations that allow standard neural networks to output non-Euclidean parameters with smooth gradients. For quotient manifolds, the decoder receives group-invariant features of the covering-space coordinates, so that identified points produce identical outputs. Anchor constraints fix the coordinate system relative to the data or create soft topological holes. Experiments on synthetic manifolds and real-image datasets (rotated and cyclically shifted MNIST) confirm that a topology-matched prior aligns KL regularization with the data manifold. The resulting topology-aware models outperform the Gaussian baseline at all practically relevant regularization strengths. The code is available at https://github.com/JvHulst/VAE-Topology.
We present a single-image head mesh reconstruction framework that addresses the longstanding challenge of simultaneously preserving facial identity and producing industry-grade topology. Our framework adopts a coarse-to-fine optimization pipeline that refines a rigged template across three stages -- rig, joint, and vertex -- achieving stable convergence and consistent topology. To mitigate the ill-posed nature of single-image 3D face reconstruction and ensure identity preservation, we employ a normal consistency objective jointly with landmark alignment. To further preserve local surface structure and enforce topological regularity, we introduce geometry-aware constraints based on Gaussian curvature and conformal consistency, along with auxiliary regularizations that correct fine artifacts such as lip seams and eyelid discontinuities. Our hierarchical optimization with geometry-aware regularization yields meshes with semantically meaningful edge flow and industry-grade topology. After geometry reconstruction, we extract UV-space texture and normal maps to preserve appearance details for visualization and downstream use. In a user study with 22 professional technical artists, our results were assessed as approaching industry-grade usability, and 95% of participants ranked our method as the top-performing approach, underscoring its effectiveness for real-world digital human production.