Technical Labs
Interactive systems for understanding the mathematical structure behind machine learning.
Knowledge State Demo
No edge without proof.
A learning-state engine that separates canonical topic structure from what a learner has actually demonstrated. Edges are earned through evidence: a topic can be visible without being trustworthy, and recommendations follow the blocked dependency rather than the topic label.
Powered by Performance Factor Analysis (Pavlik et al. 2009) over a per-edge dependency graph. Provider-seam refactor pending merge.
Spectral PDE Lab
Make PDE structure executable.
Browser-side Fourier solvers for the classical PDE archetypes that show up in diffusion models, flow matching, PINNs, and neural operators. Exact under stated assumptions: periodic boundary conditions, smooth initial data, evaluated at FFT-resolved Fourier modes.
Lab page foregrounds the verification scope (what's exact vs approximate) and links back to the PDE fundamentals topic. Interactive solver lands when codex/knowledge-state-preview merges.
Proof Tasks
Earn dependency edges through evidence.
Graded MCQ checkpoints anchored to real definitions and theorems on canonical TheoremPath pages. A correct answer writes evidence to your assessment-attempt history; that evidence flows through the PFA model and the knowledge-state engine to update the corresponding edge state. Initial set covers softmax-axis, the heat-equation Fourier multiplier, chain-rule → backprop, attention shape trace, and Fokker-Planck vs SDE.
5 tasks shipped in Run 4. Each tagged with real topic slugs so evidence flows through the existing pipeline without remapping.
Attention Mechanics Lab
Shapes, softmax, and Q/K/V attention.
Trace the exact tensor operations that turn dot products into attention weights. Identify which axis softmax normalizes over in attention logits with shape [batch, heads, query_tokens, key_tokens], and verify what breaks if you normalize over heads instead.
Initial coverage is provided by the softmax-axis and attention-shape-trace proof tasks. A dedicated interactive lab with shape diagrams and per-axis simulation is queued for a follow-up.
What these labs are for
Most ML learning tools focus on isolated lessons or coding tasks. TheoremPath focuses on the structure underneath: prerequisites, proof obligations, interactive labs, and diagnostics that show what to study next.
Labs are how that structure becomes inspectable. Spectra, gradients, tensor shapes, solver assumptions, and failure modes are visible objects in a lab, not paragraphs in an article. Each lab states the assumptions under which its computation is exact and links back to the topic page where the underlying theory is developed.
Open references. Earned progress.