math-foundations

A canonical mathematical foundations graph — 42 concepts spanning sets through optimal transport — used as the shared substrate for the study-paper Claude Code skill's per-study learning maps.

🌐 Interactive views

View	URL	Best for
🔵 D3 force graph — drag, zoom, filter	pleyva2004.github.io/math-foundations/html/	Exploring relationships visually
🗺️ Clickable SVG dependency graph	pleyva2004.github.io/math-foundations/html/graph.svg	Click any node → jump to that concept
📓 Aggregate Jupyter notebook	notebook/foundations.ipynb	Cell-by-cell linked index

Note: the live URLs above are served by GitHub Pages. If they don't work, enable Pages: Settings → Pages → Source: Deploy from a branch → main → /(root) → Save.

The dependency graph

Click the image above → opens the live SVG where each node is a working link to its concept folder.

📖 Notation glossary

Every distinct mathematical symbol used across the 42 concept lessons is indexed in a single auto-generated glossary — the canonical "what does this symbol mean and where does it first appear?" lookup:

• 📑 Markdown (browse on GitHub): NOTATION.md
• 📄 PDF cheatsheet (CI-compiled): notation.pdf — source: notation.tex
• 🧩 Single source of truth (machine-readable): notation.json

Regenerate after editing notation.json:

python generate_notation.py

Each concept ships four artifacts

Every concept lives in concepts/<NN>-<slug>/:

1. README.md — plain-English intro + LaTeX math + cross-links.
2. lesson.tex → lesson.pdf (CI-rendered) — standalone formal exposition with theorem + proof.
3. code.py — runnable demonstration (<30 s on CPU; finite/discrete witness for abstract concepts).
4. notebook.ipynb — interactive Jupyter form, runnable in Colab.

Pick your entry point

Order	Level	Audience
0	novice	Novice (high-school+)
1	intermediate	Intermediate (calc + linear algebra)
2	advanced	Advanced (analysis + probability)
3	graduate	Graduate (measure theory + stochastic)
4	frontier	Frontier (information geometry + OT)

Curated tours

Tour	Audience	File
Novice	"I don't know what a function is"	tours/novice.md
CS undergrad	"I know calc, want probability"	tours/cs-undergrad.md
Math grad	"I know measure theory"	tours/math-grad.md
Researcher	"Skip to a paper's foundations"	tours/researcher.md

Concept index (clickable)

The full 42-concept list, by level. Every entry below is a working markdown link.

Novice (high-school+)

• 00 Sets and Functions — The two foundational objects: collections and rules that map elements between them.
• 01 Logic and Proof — Propositional and predicate logic, truth tables, the basic proof strategies (direct, contrapositive, contradiction, induction).
• 02 Counting and Combinatorics — Counting principles: permutations, combinations, the pigeonhole principle, inclusion-exclusion.
• 03 Relations and Orderings — Equivalence relations, partial and total orders, equivalence classes.

Intermediate (calc + linear algebra)

• 04 Groups, Rings, Fields — Abstract algebraic structures with binary operations satisfying axioms.
• 05 Vector Spaces — Sets closed under linear combinations over a field; the universal object of linear algebra.
• 06 Linear Maps and Matrices — Structure-preserving functions between vector spaces; the matrix representation.
• 07 Eigenvalues and Eigenvectors — Directions preserved by a linear map up to scaling; the building blocks of spectral analysis.
• 08 Inner Product Spaces — Vector spaces equipped with a notion of angle and length via an inner product.
• 09 Norms and Metrics — Generalised notions of distance; the foundation of analysis on vector spaces.

Advanced (analysis + probability)

• 10 Sequences and Limits — Convergence of sequences; the epsilon-N definition; Cauchy sequences and completeness.
• 11 Continuity — The epsilon-delta definition; continuity as preservation of limits; uniform vs pointwise continuity.
• 12 Derivatives (Univariate) — The local linear approximation; the limit of a difference quotient; differentiation rules.
• 13 Multivariate Calculus — Partial derivatives; the chain rule in multiple variables; directional derivatives.
• 14 Gradient and Jacobian — The gradient as the vector of partial derivatives; the Jacobian as the matrix of partial derivatives of a vector-valued function.
• 15 Integration (Riemann) — The Riemann integral as a limit of step-function approximations; the fundamental theorem of calculus.
• 16 Series and Convergence — Convergence of infinite sums; absolute vs conditional; power series and Taylor expansions.
• 17 Sample Spaces — The set of all possible outcomes of an experiment; the starting point of probability.
• 20 Conditional Probability — P(A | B) = P(A and B) / P(B); the update rule for new information.
• 21 Independence — P(A and B) = P(A) P(B); events that do not influence each other.
• 22 Random Variables — Measurable functions from the sample space to the real line; the bridge from sets-of-outcomes to numerical values.
• 23 Probability Distributions — The full description of a random variable: its cumulative distribution function, its discrete probability mass function, or its continuous probability density function.
• 24 Probability Density Functions — A non-negative integrable function whose integral over a set gives the probability that a continuous random variable lands in that set.
• 25 Expectation — The weighted average of a random variable's values, weighted by their probability; the integral of the random variable against its distribution.
• 26 Variance and Covariance — Second-order moments measuring spread (variance) and linear association (covariance).
• 28 Change of Variables (Probability) — How densities transform under invertible smooth maps; the Jacobian determinant appears as the volume-correction factor.
• 29 Ordinary Differential Equations — Equations relating a function to its derivatives; the basic objects of dynamical systems.
• 35 Self-Information (Surprisal) — I(x) = -log p(x); the surprise associated with observing an outcome under a probability distribution.
• 36 Shannon Entropy — H(X) = E[-log p(X)]; the average information content of a random variable; measures uncertainty.
• 37 Cross-Entropy — H(p, q) = -E_p[log q]; the average code-length when using distribution q to encode samples drawn from p.
• 38 KL Divergence — D_KL(p || q) = E_p[log p - log q]; the relative entropy / information gain in moving from q to p. Non-symmetric, non-metric.
• 39 Mutual Information — I(X; Y) = D_KL(p(x,y) || p(x) p(y)); the reduction in uncertainty about X from observing Y.

Graduate (measure theory + stochastic)

• 18 Events and Sigma-Algebras — A sigma-algebra is a collection of subsets closed under complement and countable union; the rigorous basis for which events can be assigned probabilities.
• 19 Probability Measures — A function from a sigma-algebra to [0,1] satisfying countable additivity and P(Omega) = 1.
• 27 Conditional Expectation — The expected value of a random variable given the value of another (or given a sub-sigma-algebra); the foundation of regression and martingale theory.
• 30 Existence and Uniqueness (Picard-Lindelöf) — Sufficient conditions (Lipschitz) under which an ODE has a unique solution; the foundation of well-posed dynamical systems.
• 31 Stochastic Processes — Indexed families of random variables; the time-evolution of randomness.
• 32 Brownian Motion / Wiener Process — The canonical continuous-time stochastic process with independent Gaussian increments; the building block of stochastic calculus.
• 33 Stochastic Differential Equations — Differential equations driven by Brownian motion; the continuous-time analogue of random walks.

Frontier (information geometry + OT)

• 34 Itô Calculus — The calculus of stochastic integrals where the integrator is non-smooth (Brownian); Itô's lemma replaces the chain rule.
• 40 Information Geometry (Fisher Metric) — Treats probability distributions as points on a Riemannian manifold; the Fisher information matrix as the natural metric tensor.
• 41 Optimal Transport (Wasserstein) — Wasserstein distance W_p(mu, nu) = inf over couplings of E[|X - Y|^p]^(1/p); the cost of optimally moving one distribution to another. Provides a meaningful metric on distributions even when they have disjoint supports.

How the graph evolves

When the study-paper skill encounters a paper whose math deep dive defines a concept not yet in this graph, Stage 7 of the skill adds it here. This repo is the durable shared substrate.

Regenerating views from manifest.json

python generate.py            # refresh README, html/data.json, aggregate notebook
python generate_svg.py        # refresh html/graph.svg
python generate_notation.py   # refresh NOTATION.md and notation.tex from notation.json
python validate_bodies.py     # quality-gate the 42 concept folders

License

MIT.