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

Measuring Harness-Induced Belief Divergence in Multi-Step LLM Agents

Software-agent benchmarks usually report whether an agent solves a task, but the agent reaches that outcome through a harness that controls what it sees, which actions it can take, which failures are repaired, which states are verified, and which evidence is logged. We show that this harness can change the agent's multi-step beliefs even when the task, environment, and base LLM are fixed. We introduce a belief-rollout diagnostic that elicits structured K-step trajectories over progress, risk, recoverability, constraints, failure mode, uncertainty, future success, repair cost, and next action under alternative harnesses. We define a cross-harness belief divergence and decompose it into an arrival term for immediate interface shifts and a growth term for horizon-dependent belief changes. On controlled coding tasks and public-benchmark stress tests, blocked actions, compressed repairs, selective verification, and cost-aware evidence pruning often preserve terminal success while changing the beliefs that drive later decisions. We further introduce BIWM, a no-training protocol that canonicalizes observations, logs censored branches, expands repair traces, records verification masks, executes risky branches in shadow, and aligns belief trajectories across harness views. The results suggest that harness design is an experimental variable in agent evaluation, not an implementation detail. Our code is available at https://github.com/Hik289/Harness-induce-bias.git.