Asymptotic-Preserving A Posteriori Analysis of Diffusion and Flow-Matching Samplers

Diffusion and flow-matching samplers integrate a learned probability-flow ODE from a large noise scale down to a small terminal floor $σ_{\min}$, at which the score is stiff and the flow develops a boundary layer. We treat $σ_{\min}$ as a singular-perturbation parameter and determine which fixed-step samplers are asymptotic-preserving (AP), that is, stable and uniformly accurate as $σ_{\min}\to0$, casting the criteria as an a posteriori audit: residual functionals with $σ_{\min}$-uniform coefficients, computable on a pretrained checkpoint without ground-truth scores or exact trajectories. On the terminal layer, Euler in the $σ$-clock, the deterministic DDIM update, is the unique layer-exact discretization up to affine reparameterization, with rectified flow its flow-matching counterpart; the $λ$-clock is stable only for steps $h\le h_\star=1+W(1/e)$, and the uniform-$σ^2$ heat clock stalls a $σ_{\min}$-independent distance from the data. On two solvable models (rank-deficient Gaussian, symmetric two-point mixture), deterministic samplers remain first-order uniformly accurate with no $\log(1/σ_{\min})$ factor, even across a symmetric posterior-switching interface whose distributional budget is a universal constant; the logarithm is charged entirely to the Itô term of stochastic samplers, whose path-KL scales as $Λ^2/N$ against the ODE's $O(Λ^2/N^2)$ budget, with $Λ=\log(σ_{\max}/σ_{\min})$. On the EDM CIFAR-10 checkpoint, spectra measured once predict held-out residual budgets across step count, schedule, and noise level against pre-specified gates with no per-configuration refitting, and calibrate the Itô coefficient at $M_1=1.00\pm0.01$. The clock decides stability; the noise, not the geometry, charges the logarithm.

Covert Trait Propagation Is Representation Alignment: Mechanistic Evidence from Hidden-Channel Distillation

A student model trained on pure uniform noise can still inherit its teacher's digit-classification ability, provided the two share initialization. Previous work proves this transfer is guaranteed when the teacher's learning rate is small enough, but does not explain where in the network the channel lives or what sets its capacity. Working in an MLP distillation setting on MNIST, we show these channels are not purely informational: geometric alignment gates access to the information the channel carries. Shared initialization makes the output projection W_2 a common coordinate key, and KL gradients reshape the student's input projection W_0 until its hidden representations align with the teacher's. We call this covert trait propagation (CTP). Five experiments support this mechanism: channel closure tracks weight drift, not teacher accuracy; freezing W_0 destroys transfer while freezing W_2 leaves it intact; multi-teacher ensembles cancel out despite each teacher carrying comparable label information; and linear centered kernel alignment (CKA) tracks student accuracy at r=0.98 across a continuous initialization sweep. Applying the same geometric lens to cross-token behavioral entanglement (CTBE) in instruction-tuned LLMs, we find the effect appears to be activated by alignment training, acting on an inherited substrate, and that the standard log-ratio metric produces an apparent frequency bias that is largely a circularity artifact.

ManifoldFlow: SPD-Relaxed Stiefel Layers with Learnable Singular Spectrum

Orthogonal and Stiefel layers give neural weights exact spectral control, but they also impose a strong modeling constraint: all represented singular values are fixed at one. Many settings that benefit from an orthonormal basis still need direction-dependent attenuation or amplification. We introduce ManifoldFlow, a minimal relaxation of a fixed-spectrum Stiefel layer that keeps the basis on the Stiefel manifold while learning a bounded positive spectrum through W = Q S^{1/2}, with Q^T Q = I and S positive definite. Since W^T W = S, the eigenvalues of S are exactly the squared singular values of the realized weight, making eigenvalue clipping a direct singular-value control mechanism. Across paired sequence, tabular, and image experiments, the learnable SPD spectrum improves the fixed-spectrum Stiefel counterpart in the reported settings where the Stiefel prior is useful, with the largest gains in recurrent language-model projections. Boundary cases in convolutional classifier heads clarify the intended scope: ManifoldFlow is not a universal dense-layer replacement, but a spectrum-learnable Stiefel relaxation for settings where an orthonormal basis is a useful prior. When the basis should be orthonormal, its spectrum need not be frozen. Code available at https://github.com/Hik289/manifold_flow

LRX-PINN: A Layer-Resolving XNet Physics-Informed Neural Network with Integrated Cauchy Activations for Convection-Dominated Problems

Convection-dominated convection-diffusion problems often develop thin layers, where the solution has sharp transition profiles and its derivatives are highly localized. This creates a structural mismatch for standard physics-informed neural networks (PINNs), whose trial spaces are not designed to match the value--derivative structure of such layers. We propose a Layer-Resolving XNet Physics-Informed Neural Network (LRX-PINN) based on integrated Cauchy activations. The proposed basis is transition-type at the solution level, while its derivative recovers a localized Cauchy kernel. We show that this structure matches the scaling of convection-dominated layers, inherits the Cauchy approximation mechanism at the derivative-profile level, and identifies \(d/\|w\|\) as the effective physical width of a ridge neuron. For analytic layer profiles, this yields derivative-stable exponential approximation in the stretched coordinate and a layer-scaled estimate for the strong residual of the singularly perturbed operator. Numerical experiments on several convection-dominated benchmarks show that LRX-PINN achieves higher accuracy than PIKAN and Fourier-feature PINNs while using less than \(30\%\) of their trainable parameters. On more challenging benchmarks, embedding the proposed representation into hp-VPINN-based frameworks further improves the best results obtained by existing hp-VPINN-based baselines without changing their original loss functionals or stabilization strategies. These results show that neural representations aligned with layer structure provide a compact and effective approach for convection-dominated problems.

Finite-Lag Operator Geometry of Recurrent Representations

Recurrent representations are trajectories, but representation geometry is often measured from static snapshots. We develop finite-lag operator geometry for recurrent hidden states from observed source-successor pairs $(X_t,X_{t+Δ})$. The primitive is the conditional transport law $Q_Δ(dy\mid x)$, estimated by a dense Gaussian source-smoothing operator. From this directed finite-lag law we derive a source-centered transport tensor $G_Δ$, which decomposes exactly into conditional spread and coherent displacement, and an antisymmetric coordinate circulation $W_Δ^ρ$, which summarizes directed lagged flow. We prove affine covariance with explicit metric dependence of scalar summaries, dense estimator stability on bounded trajectory clouds, and a finite-lag separation result showing that source-centered transport detects deterministic recurrent motion not recorded by infinitesimal carre-du-champ geometry. A linear-Gaussian closed form calibrates the quantities in terms of the update $A_Δ$, source covariance, and innovation covariance. Controlled experiments validate the decomposition, circulation, covariance, and stability predictions. In performance matched repeat-copy networks, the framework reveals architecture dependent differences in total transport scale and coherent displacement trace, while coherent displacement fraction is metric and resolution dependent.

Expander Sparse Autoencoders: Parameter-Efficient Dictionaries for Mechanistic Interpretability

Sparse autoencoders (SAEs) decompose internal activations of neural networks into sparse linear combinations of learned features by fitting an overcomplete dictionary $\mathbf{W}\in\mathbb{R}^{m\times n}$ with $m<n$, and inferring a sparse code $\mathbf{x}\in\mathbb{R}^n$ from $\mathbf{h}\approx\mathbf{W}\mathbf{x}$. This inference problem closely resembles the canonical setup of compressed sensing, but dense decoders requires $O(mn)$ learned values, which becomes costly at large feature counts. We introduce Expander SAEs: TopK SAEs whose decoder and tied encoder are supported on a left-$d$-regular expander mask with $d\ll m$, learning only $dn$ decoder values while keeping the sparse-coding problem $(m,n,k)$ fixed. The same structure reduces storage and turns the matching-pursuit correlation step $\mathbf{W}^\top \mathbf{r}$ in OMP into an $O(dn)$ gather-and-reduce operation. Our experiments show that across Pythia-70M/160M, Qwen2.5-3B, and Llama-3.2-1B residual-stream activations, varying $d$ traces a consistent storage--fidelity frontier, and that at the most compressed modern-LM setting, Qwen2.5-3B with $d=7$ uses $293\times$ fewer learned decoder values than the full dense decoder while retaining $84$% of dense CE-loss recovered. Control experiments show that the improved storage--fidelity tradeoff is driven by sparse, diverse decoder support structure rather than by fewer learned decoder values, and that when sparse and dense decoders are compared at matched parameter count, part of the remaining gap comes from encoder amortisation. On the theoretical side, we show that expansion and column flatness are sufficient for identifiability of noiseless $k$-sparse codes, and we derive complementary sufficient conditions under which OMP recovers the support exactly.

Boundary-Aware Quantization: Finite-Scale Decision Geometry of Neural Classifiers

We measured quantization-induced decision-boundary changes using local logit-margin radii, first-order boundary displacement, normal variation, slice-boundary Jaccard distance, grid prediction changes, multiclass junction counts, and low-margin boundary-band flips. On the digits benchmark, 8-bit weight quantization preserved all test labels while producing boundary-mask Jaccard \(0.428\) on the PCA slice; at 4 bits, accuracy remained \(0.9733\), while boundary Jaccard rose to \(0.970\) and median local boundary shift reached \(0.0290\). Interpolation between adjacent quantization levels localized the visible reconfigurations at multiclass junctions, with 12, 34, and 17 triple-junction cells in the selected transitions. Calibration-to-test stopping reduced the digits held-out flip rate from \(0.0094\) to \(0.0022\) and boundary Jaccard from \(0.825\) to \(0.524\); the same stopping rule also reduced flips on MNIST and Fashion-MNIST. On official CIFAR-10 subsets, PTQ-W selected by accuracy gave 6-bit flip \(0.0367\) and boundary Jaccard \(0.184\), whereas boundary-aware stopping selected 8-bit flip \(0.0083\) and boundary Jaccard \(0.048\). On full CIFAR-10 with three seeds, 6-bit PTQ-W lost \(0.0029\) accuracy relative to float, changed \(5.3\%\) of held-out decisions, and changed \(24.5\%\) of low-margin boundary-band decisions. A fixed-bit boundary-gap rounding term changed the trade-off at 4 bits by reducing boundary Jaccard from \(0.457\) to \(0.435\) and boundary-band pair-order flip from \(0.3600\) to \(0.3558\), with an accuracy trade-off; the 3-bit stress test exposed the tuning limit of this surrogate. Calibration boundary Jaccard predicted held-out boundary Jaccard across PTQ-W and optimized rounding variants with \(r=0.947\)--\(0.994\).

When Context Compensates for Sparse Event History: AlphaEarth for Spatio-Temporal Point-Process Forecasting

Spatio-temporal point-process models must often generalise across space when local event histories are sparse. We study whether exogenous spatial context can compensate in such regimes. Using a fixed log-Gaussian Cox process backbone, we compare an event-only model with the same model augmented by AlphaEarth embeddings as linear spatial context. We evaluate spatial transfer on emergency medical services (EMS) forecasting across eight held-out regions, fixed forecast anchors, and a sweep over history length $w$, using only AlphaEarth (AE) embeddings available strictly before each anchor. AE improves out-of-region predictive performance across all history regimes, with the largest gains under scarce histories: approximately $2$--$6\times$ multiplicative improvements at $1-2$ weeks, tapering to roughly $10$--$20\%$ at $w=20$--$104$ weeks. These results show that contextual information can substantially stabilise spatially transferred point-process forecasts when event history is limited.

Review Residuals: Update-Conditioned Residual Gating for Transformers

Residual connections add every sublayer's proposed update with a fixed coefficient of one; the network never evaluates whether an update is reliable before committing it. Drawing on the human-factors principle of independent verification, we introduce Review Residuals, which scale each update by a learned, input-dependent gate conditioned on both the current state and the proposed update: h_l = h_{l-1} + r_l * u_l with r_l = sigmoid(W[RMSNorm(h_{l-1}), RMSNorm(u_l)]). Conditioning the gate on the update is the property that distinguishes it from prior gated and scaled residuals. We report two findings. First, a depth-stability result: a convex (Highway-style) form of the gate reintroduces vanishing gradients and fails to train beyond ~20 layers, whereas the additive, identity-preserving form trains stably at all depths we tested. Second, an emergence-with-scale result: trained from scratch across five sizes (60M-1B parameters, multi-seed), Review Residuals show no advantage at small scale but at 590M significantly outperform both a parameter-matched Highway gate and a parameter-matched standard residual (p<0.05), with a larger advantage at 1B. The benefit grows with model size rather than shrinking.

CORA: Per-Slice Coherent Orthogonal Rotation for SVD-based Low-Rank Adaptation

Parameter-Efficient Fine-Tuning (PEFT) commonly adapts pretrained weights through low-rank updates, and recent methods further exploit the singular value decomposition (SVD) of the base weight for initialization or subspace selection. However, these methods do not explicitly preserve the coupled geometry between the pretrained left and right singular bases. Motivated by recent minimum-perturbation theory, which shows that stable finetuning follows a coherent SVD rotation in which a single orthogonal $Q$ acts on both the left singular basis $U_0$ and the right singular basis $V_0$, we prove a per-slice analogue: each row slice of $W_0$ can be adapted by a shared orthogonal rotation $Q_i$ on its left basis $U_i$ and right basis $V_i$ together with a diagonal spectrum shift. We implement this form as CORA (Coherent Orthogonal Rotation Adaptation), which applies per-slice orthogonal rotations and a per-layer diagonal scale to the rank-$r$ SVD truncation of $W_0$. CORA uses $\tfrac{1}{2}m(r{-}1)$ trainable parameters per linear layer, about $4{\times}$ fewer than LoRA at the same rank. CORA outperforms LoRA, DoRA, PiSSA, and MiLoRA on commonsense reasoning and code generation while using about $8{\times}$ fewer parameters.