Flag Varieties: A Geometric Framework for Deep Network Alignment

Alignment, the tendency of adjacent weight matrices in deep networks to develop compatible subspace orientations, underlies gradient flow, Neural Collapse, and representation similarity across architectures. Despite extensive empirical documentation, these phenomena have resisted unified theoretical treatment: existing explanations are post-hoc, each fitted to a specific observation with whatever mathematics is at hand. We reverse this direction by deriving the mathematical structure that layerwise alignment inherently demands. Using geometric invariant theory, we prove that alignment geometry has a canonical closed, polystable stratum given by a flag variety, and that subspace intersection dimension is its unique reparameterization-invariant observable, establishing that subspace metrics are not empirical conventions but mathematical necessities. This unified framework yields two dynamical consequences: ridge regularization drives subspace alignment at an exponential rate set by weight decay, whereas nonlinear activations induce a commutator obstruction to exact basis alignment, generically present in nonlinear networks and absent in linear ones. Together these give a geometric explanation of the Level-2/3 hierarchy in Neural Collapse from first principles rather than post-hoc analysis. The commutator magnitude and head subspace overlap further serve as weight-space windows into internal alignment structure, requiring no forward passes. Experiments on multilayer perceptrons, residual networks, and pretrained language models support the proposed diagnostics and delineate their scope.

Geometric and Spectral Alignment for Deep Neural Network II

This paper develops the angular and static-channel component of Geometric and Spectral Alignment for residual Jacobian chains. Starting from Cartan-coordinate rigidity and fitted effective-rank windows, we study how dominant singular subspaces are transported across adjacent layers and how the resulting finite matrices can be displayed in physical channel coordinates. The main results are deterministic, margin-verified results. We bound the error between full interface transport and its dominant-window truncation, add fitted-tail errors so that empirical spectra can be certified against the Gibbs--Cartan tail model, and distinguish source-mode incidence from fully physical input-output channel incidence. Given row groups and active supports, the Physical Alignment Matrix decomposes orthogonally as core plus overlap plus noise. Active-column gaps, pairwise overlap margins, and noise bounds combine into a static certificate radius under which the full transport and the truncated transport induce the same active supports, pairwise incidence graph, SRS sets, hub columns, and core/overlap/noise masks. The finer SC/SA/ST labels of the Invariant Channel Mapping require additional row-energy and profile-correlation margins, stated as explicit perturbation tests. The empirical section reports the matrices and block-energy heatmaps that measure these certificate quantities across CNNs, language models, and vision/diffusion backbones. The figures are interpreted as finite-dimensional measurements; complete membership in the Physical GSA certificate domain requires checking the numerical margin protocol stated in Section 10.

What Did They Mean? How LLMs Resolve Ambiguous Social Situations across Perspectives and Roles

People increasingly turn to large language models (LLMs) to interpret ambiguous social situations: a delayed text reply, an unusually cold supervisor, a teacher's mixed signals, or a boundary-crossing friend. Yet in many such cases, no stable interpretation can be verified from the available evidence alone. We study how LLMs respond to these situations across four domains: early-stage romantic relationships, teacher--student dynamics, workplace hierarchies, and ambiguous friendships. Across 72 responses from GPT, Claude, and Gemini, only 9 (12.5\%) genuinely preserved uncertainty. The remaining 87.5% produced interpretive closure through recurring pathways including narrative alignment, narrative reversal, normative advice under uncertainty, and hedged language that still supported a single conclusion. We further find that narrator perspective shapes the path to closure: first-person accounts more often elicited alignment, while third-person accounts invited more detached interpretation, even when the underlying situation remained comparable. Together, these findings show that LLMs do not simply assist interpersonal sensemaking; they tend to resolve ambiguity into coherent and actionable narratives. These results suggest that the central risk is not only that LLMs may misinterpret social situations, but that they may make unresolved situations feel prematurely settled. We frame this tendency as a design challenge for uncertainty-preserving social AI.

From Pixels to Nucleotides: End-to-End Token-Based Video Compression for DNA Storage

DNA-based storage has emerged as a promising approach to the global data crisis, offering molecular-scale density and millennial-scale stability at low maintenance cost. Over the past decade, substantial progress has been made in storing text, images, and files in DNA -- yet video remains an open challenge. The difficulty is not merely technical: effective video DNA storage requires co-designing compression and molecular encoding from the ground up, a challenge that sits at the intersection of two fields that have largely evolved independently. In this work, we present HELIX, the first end-to-end neural network jointly optimizing video compression and DNA encoding -- prior approaches treat the two stages independently, leaving biochemical constraints and compression objectives fundamentally misaligned. Our key insight: token-based representations naturally align with DNA's quaternary alphabet -- discrete semantic units map directly to ATCG bases. We introduce TK-SCONE (Token-Kronecker Structured Constraint-Optimized Neural Encoding), which achieves 1.91 bits per nucleotide through Kronecker-structured mixing that breaks spatial correlations and FSM-based mapping that guarantees biochemical constraints. Unlike two-stage approaches, HELIX learns token distributions simultaneously optimized for visual quality, prediction under masking, and DNA synthesis efficiency. This work demonstrates for the first time that learned compression and molecular storage converge naturally at token representations -- suggesting a new paradigm where neural video codecs are designed for biological substrates from the ground up.

Analysis of Converged 3D Gaussian Splatting Solutions: Density Effects and Prediction Limit

We investigate what structure emerges in 3D Gaussian Splatting (3DGS) solutions from standard multi-view optimization. We term these Rendering-Optimal References (RORs) and analyze their statistical properties, revealing stable patterns: mixture-structured scales and bimodal radiance across diverse scenes. To understand what determines these parameters, we apply learnability probes by training predictors to reconstruct RORs from point clouds without rendering supervision. Our analysis uncovers fundamental density-stratification. Dense regions exhibit geometry-correlated parameters amenable to render-free prediction, while sparse regions show systematic failure across architectures. We formalize this through variance decomposition, demonstrating that visibility heterogeneity creates covariance-dominated coupling between geometric and appearance parameters in sparse regions. This reveals the dual character of RORs: geometric primitives where point clouds suffice, and view synthesis primitives where multi-view constraints are essential. We provide density-aware strategies that improve training robustness and discuss architectural implications for systems that adaptively balance feed-forward prediction and rendering-based refinement.

SCONE: A Practical, Constraint-Aware Plug-in for Latent Encoding in Learned DNA Storage

DNA storage has matured from concept to practical stage, yet its integration with neural compression pipelines remains inefficient. Early DNA encoders applied redundancy-heavy constraint layers atop raw binary data - workable but primitive. Recent neural codecs compress data into learned latent representations with rich statistical structure, yet still convert these latents to DNA via naive binary-to-quaternary transcoding, discarding the entropy model's optimization. This mismatch undermines compression efficiency and complicates the encoding stack. A plug-in module that collapses latent compression and DNA encoding into a single step. SCONE performs quaternary arithmetic coding directly on the latent space in DNA bases. Its Constraint-Aware Adaptive Coding module dynamically steers the entropy encoder's learned probability distribution to enforce biochemical constraints - Guanine-Cytosine (GC) balance and homopolymer suppression - deterministically during encoding, eliminating post-hoc correction. The design preserves full reversibility and exploits the hyperprior model's learned priors without modification. Experiments show SCONE achieves near-perfect constraint satisfaction with negligible computational overhead (<2% latency), establishing a latent-agnostic interface for end-to-end DNA-compatible learned codecs.

Thinking Like Van Gogh: Structure-Aware Style Transfer via Flow-Guided 3D Gaussian Splatting

In 1888, Vincent van Gogh wrote, "I am seeking exaggeration in the essential." This principle, amplifying structural form while suppressing photographic detail, lies at the core of Post-Impressionist art. However, most existing 3D style transfer methods invert this philosophy, treating geometry as a rigid substrate for surface-level texture projection. To authentically reproduce Post-Impressionist stylization, geometric abstraction must be embraced as the primary vehicle of expression. We propose a flow-guided geometric advection framework for 3D Gaussian Splatting (3DGS) that operationalizes this principle in a mesh-free setting. Our method extracts directional flow fields from 2D paintings and back-propagates them into 3D space, rectifying Gaussian primitives to form flow-aligned brushstrokes that conform to scene topology without relying on explicit mesh priors. This enables expressive structural deformation driven directly by painterly motion rather than photometric constraints. Our contributions are threefold: (1) a projection-based, mesh-free flow guidance mechanism that transfers 2D artistic motion into 3D Gaussian geometry; (2) a luminance-structure decoupling strategy that isolates geometric deformation from color optimization, mitigating artifacts during aggressive structural abstraction; and (3) a VLM-as-a-Judge evaluation framework that assesses artistic authenticity through aesthetic judgment instead of conventional pixel-level metrics, explicitly addressing the subjective nature of artistic stylization.

TVC: Tokenized Video Compression with Ultra-Low Bitrate

Tokenized visual representations have shown great promise in image compression, yet their extension to video remains underexplored due to the challenges posed by complex temporal dynamics and stringent bitrate constraints. In this paper, we propose Tokenized Video Compression (TVC), the first token-based dual-stream video compression framework designed to operate effectively at ultra-low bitrates. TVC leverages the powerful Cosmos video tokenizer to extract both discrete and continuous token streams. The discrete tokens (i.e., code maps generated by FSQ) are partially masked using a strategic masking scheme, then compressed losslessly with a discrete checkerboard context model to reduce transmission overhead. The masked tokens are reconstructed by a decoder-only transformer with spatiotemporal token prediction. Meanwhile, the continuous tokens, produced via an autoencoder (AE), are quantized and compressed using a continuous checkerboard context model, providing complementary continuous information at ultra-low bitrate. At the Decoder side, both streams are fused using ControlNet, with multi-scale hierarchical integration to ensure high perceptual quality alongside strong fidelity in reconstruction. This work mitigates the long-standing skepticism about the practicality of tokenized video compression and opens up new avenues for semantics-aware, token-native video compression.