Topic
Energy in Statistics and Machine Learning
A terminology map separating energy-based models, variational free energy, energy distance, distance covariance, energy scores, Langevin potentials, OOD energy, and literal compute energy.
TheoremPath Know what to learn next.
A study map for machine learning foundations. Pick a goal, find gaps, save useful notes, and come back to a clear review path.
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Choose your first useful move.
TheoremPath gives you a path, a gap check, and a place to come back to. Start broad, then go deeper when the next topic needs it.
Featured lesson
The loop inside the model.
Recurrent depth lets a language model apply the same block again before it speaks. Dial the executed depth on a reachability task, watch a state-only loop drift from its prompt, and see what a learned halt rule does and does not decide.
Advanced45 minInteractive
Run the reasoning-loop labThree loops, not one
which loop got the compute?
Depth loop
01S(t+1) = R(S(t); M)
A continuous state; the shared core is reused
grows: executed depth
Token loop
02z(t) ~ p(z | x, z(<t))
One token plus the continuous KV cache
grows: context length
Agent loop
03s(t+1) = U(s(t), a(t), o(t+1))
Goal, persistent state, tool output
grows: logged history
Mechanism established, superiority open. Reusing a shared core buys executed depth without new parameters. Whether a continuous loop beats tokens, pauses, or search under a matched budget has not been measured.
Reading paths
See what blocks the next topic.
Reading paths keep the study loop concrete: what comes first, what to verify next, and which theorem trail follows.
662
graph topics
6980
prerequisite edges
0A to 5
layers
Reading pathsThe next useful stepSee where a topic sits, what blocks it, and what it opens next.ClaimsEvidenceInspect claim labels, source support, and verification boundaries when you want the proof layer.MethodMethodologySee how pages, claims, diagnostics, learner state, and proof records fit together.ProjectProjectsSee the topic library, evidence ledger, learner loop, and iOS companion as one system.ProofsLeanReview checked declarations, proof scopes, axiom profile, and manifest status.
Concentration inequalitiesMarkov, Chebyshev, Hoeffding, BernsteinUniform convergenceFinite-class bounds, ERM, and generalization gapsVC dimensionShattering, growth functions, and sample complexityRademacher complexitySymmetrization and data-dependent capacityTransformer architectureSelf-attention, FFN blocks, and residual streamsScaling lawsCompute allocation, data scaling, and fit limits
Recent work
Recent topics and labs.
Topic
Jacobian Lens and Global Workspace Interpretability
A 2026 Transformer Circuits study maps intermediate activations into final-layer coordinates, then uses interventions to test a small token-aligned broadcast component.
Topic
Ring vs. Tree Attention
An interactive comparison of ring and balanced-tree schedules for combining distributed online-softmax states, with exact-arithmetic equivalence and scoped communication costs.
Method
Go deeper when you need the proof layer.
The main product helps you study. The evidence pages, source roles, and Lean wrappers are there when you want to inspect why a topic or claim is trustworthy.
01
Claims are scoped
The site separates a theorem statement, its assumptions, and the page-level explanation so evidence attaches to the claim it actually supports.
02
Diagnostics route gaps
Missed items map to prerequisite concepts, not broad topic pages. The next step is a graph repair, not another generic lesson.
03
Lean evidence stays narrow
Formal wrappers appear only when the Lean theorem matches the governed claim scope and the manifest records the exact proof object.
Labs
Learn with diagrams, not only pages.
Labs make the mechanics visible: gradients moving, random vectors concentrating, and matrix maps changing geometry.
Browse all demosOptimizationOptimizer pinballCompare GD, Momentum, Nesterov, and Adam across convex and non-convex surfaces.
Probability geometryHigh-dimensional probabilityWatch concentration, spectra, and spike separation as dimension changes.
Linear algebraMatrix mechanicsManipulate maps, rank, eigenvectors, determinants, trace, and SVD directly.
Free core
The library and core study loop stay free.
The public library, reading paths, diagnostics, saved topics, notes, and review tools are free. TheoremPath Plus covers Advanced Review at USD 4.99/month, and while it is in development every account has it at no charge.
Start with one next step.
The fastest route is not more tabs or another syllabus. It is a visible path, saved context, and one useful move when you return.