Sets saved for each word
The number of saved sets changes.
Interactive notebooks for the math inside modern AI
A chatbot has 32 readers. Each keeps notes for every word. If every four readers share one set, what gets smaller?
Make a guess before the answer appears. Then change one input and keep the rule that survives.
One question
Choose what changes when 32 readers share notes in groups of four.
The setupThe conversation, model, and number size stay the same. Only the sharing rule changes.
Your job
Choose before the totals appear. A wrong guess is useful.Words remembered
The conversation becomes shorter.
Size of every number
Each saved number uses less space.
Number of model layers
Fewer layers keep a memory.
Calculated here
How much space the displayed memory rule requires.
Not measured
Answer quality, speed, and real-server behavior.
Saved where
Only in this browser when you choose to keep your guess.
Local route memory
Checking this browser for a saved route…
No route is being claimed as present or absent until the local read finishes.
Editorial Method
Every page follows the same teaching contract
The site is not an archive of notes. It is a repeatable notebook format for turning abstract ML ideas into something you can reason through from first contact to implementation.
Start with motion, shape, and analogy.
Each concept opens with a mental model you can carry before the notation starts.
Geometry, routing, density, and flow before formalism.
Study Dot ProductWrite the objects down precisely.
Definitions, symbols, and derivations stay close to the intuition instead of replacing it.
Derivatives, KL, vector spaces, and chain rule done step by step.
Study Chain RuleMatch notation to runnable Python.
The symbols on the page become short NumPy or PyTorch fragments you can actually run.
Code mirrors the math instead of hiding it behind frameworks.
Study Adam OptimizerManipulate the system and watch it respond.
Interactive diagrams turn abstract machinery into something you can stress-test, poke, and break.
Attention, diffusion, routing, and serving concepts become explorable.
Study Scaled Dot-Product Attention & Transformer LayersDomain Atlas
Navigate by mathematical territory
linear-algebra
Linear Algebra
Vectors, matrices, and linear maps: the language of representations, optimization, and modern deep learning.
6 concepts2 demos
Dot ProductVector Spaces
calculus
Calculus
Rates of change and accumulation. Calculus is the language behind gradients, optimization, continuous-time dynamics, and why backprop works as efficiently as it does.
6 concepts4 demos
BackpropagationComputation GraphsDerivatives
optimization
Optimization
How we train models: gradients, learning rates, curvature, and the practical tricks that make deep nets converge.
10 concepts4 demos
Adam OptimizerGradient DescentLearning Rate Schedules: Warmup, Decay & Cycling
probability
Probability
Uncertainty made precise: events, random variables, expectations, and the distributions that models learn.
6 concepts6 demos
Cross-EntropyDistributionsMaximum Likelihood
information-theory
Information Theory
How we measure information and mismatch between distributions: entropy, cross-entropy, KL divergence, mutual information, and why they appear everywhere in ML.
1 concepts1 demos
KL Divergence (Relative Entropy)
attention-transformers
Attention & Transformers
The sequence model backbone: tokenization, self-attention, positional encodings, and the transformer block that powers modern LLMs.
12 concepts10 demos
Scaled Dot-Product Attention & Transformer LayersEfficient Attention at Scale: KV Cache, GQA & FlashAttentionFlashAttention: IO-Aware Attention
representation-learning
Representation Learning
Embeddings and the geometry of meaning: similarity, normalization, contrastive objectives, and why vector spaces become usable interfaces for models.
2 concepts2 demos
Representation Learning & Embedding GeometrySparse Autoencoders: Feature Dictionaries for Mechanistic Interpretability
generative-models
Generative Models
How models generate: likelihood, latent variables, diffusion/score models, flows, and the training tricks that make sampling work.
5 concepts5 demos
Diffusion, Score-Based Models & Flow MatchingFlow Matching & Rectified FlowsNormalizing Flows: Tractable Density via Invertible Transforms
scaling
Scaling
How loss and capability change with parameters, data, and compute; how to allocate a training budget; and why some abilities appear suddenly at scale.
6 concepts6 demos
Overparameterization & Generalization (Double Descent)Scaling Laws & Emergent AbilitiesTest-Time Compute: Spending Inference Budget on Search
alignment
Alignment
How we shape model behavior: preference learning, reward modeling, KL-regularized fine-tuning, and the failure modes that appear when you optimize the wrong thing.
5 concepts5 demos
Direct Preference OptimizationKahneman-Tversky OptimizationProcess Reward Models: Step-Level Verifiers for Reasoning
efficiency
Efficiency
How we make models cheaper to train and serve: quantization, distillation, low-rank adapters, sparsity, and the memory/latency tradeoffs that dominate real deployments.
5 concepts5 demos
Efficiency: Quantization, Distillation, LoRA & Sparse MoEKnowledge Distillation: Learning from TeachersPruning: Removing Unnecessary Weights
llm-systems
LLM Systems
How models run in production: prefill vs decode, KV cache memory, batching and scheduling, and the techniques that make latency and throughput practical.
6 concepts6 demos
Decoding & Sampling: Temperature, Top-p & Inference-Time ControlLLM Serving at Scale: Prefill, Decode & Continuous BatchingMoE Serving & Scheduling: Token Dispatch, All-to-All, Disaggregated Parallelism
Curated Entries
Start with a thread, not a random page
You can browse the full atlas, but the fastest way in is to follow one editorial thread from prerequisites to modern applications.
track
Foundations First
Build the minimum mathematical language needed to understand optimization and modern model training.
Open track domaintrack
Transformer Systems
Move from attention mechanics to the engineering decisions that make long-context inference work.
Dot ProductScaled Dot-Product Attention & Transformer LayersEfficient Attention at Scale: KV Cache, GQA & FlashAttentionLong Context Engineering: RoPE Scaling, KV Compression & Memory Optimization
Open track domaintrack
Frontier Notebooks
Use the foundations to step into generation, alignment, and inspectable model representations.
Scaled Dot-Product Attention & Transformer LayersSparse Autoencoders: Feature Dictionaries for Mechanistic InterpretabilityDirect Preference OptimizationDiffusion, Score-Based Models & Flow Matching
Open track domainReader Lenses
One atlas, different depths of use.
A guided route through prerequisites, notation, code, and demos.
Start from a path, predict the demo, then ask the companion to repair the exact gap.
Start foundationsA quick way to inspect the mechanism, assumptions, and executable witness.
Use concept pages as small, testable models before jumping back to papers or experiments.
Inspect a bridgeA teachable sequence with visual hooks, derivation checkpoints, and failure regimes.
Use the same page as a lecture spine: intuition first, then math, code, and manipulation.
Open a lessonPaper Mapper
Turn a paper clue into a route, one equation, and one experiment.
Clickable equation
Mem_KV = B * N_layers * T * H_kv * d_head * 2 * bytesThe mapper turns the bottleneck into a calculator, then links the symbols back to attention and serving.
AI synthesis
What changed from old work?
The paper trades exact key-value retention for a bounded decode-time memory budget. Learn attention first, test the memory curve, then inspect where retrieval quality can break.
Extract equationsFind prerequisitesOpen a labAsk one question
Research Operating Loop
Turn claims into objects a serious researcher can falsify.
Continuous Function should help a researcher move from a frontier paper, model behavior, or systems result to the prerequisite repair, central equation, code witness, toy lab, source boundary, and next experiment without losing the thread.
01Paper
Paste the source
arXiv links, PDFs, blog posts, and model reports become a structured reading object instead of another tab.
02Map
Reveal prerequisites
The graph places the paper inside concepts, equations, prior work, and the ideas the reader should repair first.
03Lab
Test the mechanism
Small controlled experiments turn the central claim into sliders, curves, tensors, and failure cases.
04Discuss
Ask one sharper question
Discussion prompts attach to the exact concept, equation, lab, or paper claim so the next conversation has a clear anchor.
Flagship Proof Slice
Transformer systems, from attention to serving.
The first complete slice should help strong learners and researchers inspect transformer inference from the attention equation to KV memory, long-context pressure, serving latency, evaluation, and bounded claims.
Study moduleAsk Beside The Notebook
Specific help attached to the route.
Concept Coach
Answers from the current notebook, prerequisites, equations, demos, and learner context.
Paper Mapper
Extracts contribution, prerequisites, equations, novelty, limitations, and graph placement.
Lab Builder
Produces safe visualization specs and small experiments rather than arbitrary broken demo code.
Claim Checker
Keeps factual paper and model claims source-linked, confidence-aware, and last-verified.
Status
What is live, previewed, and not live yet.
The product stays trustworthy by separating what a learner can use now from what still needs accounts, storage, discussion infrastructure, or billing.
StatusAvailabilityForIncludes
Live nowPublicReaders todayPublic notebooks, paper-map preview, concept routes, graph routes, attention-serving module, local carried equations.
Local previewBrowserCurrent study loopSource check preview, equation extraction, generated lab specs, saved browser route, copyable study prompts.
Not live yetFutureSerious learnersAccounts, saved paths, private rooms, full discussion, team annotations, billing, and weekly briefings.
Ask beside the notebook
Get help without losing the page you are on.
The helper can carry the exact question, math, source, result, and next step from this notebook. You do not have to paste everything into a blank chat.
Object Companion
Plan beside the atlas
Choose a path, study a notebook, manipulate the demo, then use the selected object to explain, quiz, connect, or debug the idea.
Context prompt
You are my AI learning companion for Continuous Function. Current context: homepage learning atlas. Learning surface: Continuous Function atlas. What this page says: Choose a path, study a notebook, manipulate the demo, then use the selected object to explain, quiz, connect, or debug the idea. Suggested next step: Pick a track and ask the companion to turn it into a short plan.. Learner goal: Understand the idea. Learner comfort level: New to this. Preferred explanation style: Visual first. Task: Ask me 5 quick questions, then recommend the best Continuous Function learning path for my current level. Answer in a way that helps me learn: ask one clarifying question only if needed, use intuition before notation, and end with one thing I should try on the page.