# Index by Mathematical Area

Cross-references to where each theorem appears in the chronological chapters.

For a **cross-cutting index by discovery technique** (diagonalization, infinite descent, completing the square, fixed-point iteration, cross-field transfer, etc.), see [`09_discovery_techniques.md`](09_discovery_techniques.md) — 10 technique clusters with theorem examples, 8 inheritance chains, a decision-tree playbook, and a catalog of provable impossibilities.

For a **structured toolbox** (same content as Ch. 9 but as a tree / graph / function dictionary), see [`10_toolbox.md`](10_toolbox.md) — every technique as a callable "function" with inputs, process, outputs, preconditions, and examples, plus Mermaid-rendered cluster tree, inheritance graph, and decision flowchart.

For a **knowledge graph** that wires the ~100 theorems and 57+ techniques into one bipartite directed structure (state nodes → technique nodes → state nodes, with 12 compound-technique subgraphs and JSON at `knowledge_graph.json`), see [`11_knowledge_graph.md`](11_knowledge_graph.md).

## Number Theory
- Euclid's theorem (infinitude of primes) — Ch. 1
- Fundamental theorem of arithmetic (unique factorization) — Ch. 1
- Chinese remainder theorem — Ch. 1
- Fermat's little theorem — Ch. 2
- Fermat's theorem on sums of two squares — Ch. 2
- Fermat's Last Theorem (statement) — Ch. 2; (proof, Wiles) — Ch. 6
- Euler's totient theorem — Ch. 3
- Wilson's theorem — Ch. 3
- Lagrange's four-square theorem — Ch. 3
- Quadratic reciprocity (Gauss) — Ch. 3/4
- Prime number theorem — Ch. 4
- Mordell conjecture / Faltings's theorem — Ch. 6
- Green–Tao theorem — Ch. 6
- Zhang's bounded gaps — Ch. 6
- Helfgott's weak Goldbach — Ch. 6

## Geometry
- Pythagorean theorem — Ch. 1
- Thales's theorem — Ch. 1
- Archimedes' results (sphere, circle) — Ch. 1
- Heron's formula — Ch. 1
- Ptolemy's theorem — Ch. 1
- Menelaus's theorem — Ch. 1
- Brahmagupta's formula and theorem — Ch. 1
- Desargues's theorem — Ch. 2
- Pascal's theorem (mystic hexagram) — Ch. 2
- Gauss's Theorema Egregium — Ch. 4
- Gauss–Bonnet theorem — Ch. 4
- Poincaré conjecture / Perelman — Ch. 6
- Kepler conjecture — Ch. 6

## Algebra
- Quadratic formula (al-Khwarizmi) — Ch. 1
- Omar Khayyam's cubic solutions — Ch. 1
- Cardano's cubic formula — Ch. 2
- Ferrari's quartic solution — Ch. 2
- Viète's formulas — Ch. 2
- Descartes' rule of signs — Ch. 2
- Newton's generalized binomial theorem — Ch. 2
- Newton's identities — Ch. 2
- Fundamental theorem of algebra — Ch. 3
- Abel–Ruffini theorem — Ch. 4
- Fundamental theorem of Galois theory — Ch. 4
- Cayley–Hamilton theorem — Ch. 4
- Sylow theorems — Ch. 4
- Hilbert's basis theorem — Ch. 5
- Hilbert's Nullstellensatz — Ch. 5
- Noether's isomorphism theorems — Ch. 5
- Classification of finite simple groups — Ch. 6

## Calculus and Analysis
- Fundamental theorem of calculus — Ch. 2
- Rolle's theorem — Ch. 2
- Wallis product — Ch. 2
- Taylor's theorem — Ch. 3
- Euler's formula e^(iπ) + 1 = 0 — Ch. 3
- Basel problem (π²/6) — Ch. 3
- Lagrange's mean value theorem — Ch. 3
- Fourier's theorem — Ch. 4
- Cauchy's integral theorem and formula — Ch. 4
- Cauchy–Schwarz inequality — Ch. 4
- Bolzano's intermediate value theorem — Ch. 4
- Bolzano–Weierstrass theorem — Ch. 4
- Liouville's theorem — Ch. 4
- Riemann mapping theorem — Ch. 4
- Weierstrass approximation theorem — Ch. 4
- Stone–Weierstrass theorem — Ch. 5
- Hahn–Banach theorem — Ch. 5
- Banach fixed-point theorem — Ch. 5
- Dominated convergence (Lebesgue) — Ch. 5
- Ergodic theorem — Ch. 5

## Topology
- Euler's polyhedron formula V − E + F = 2 — Ch. 3
- Brouwer fixed-point theorem — Ch. 5
- Tychonoff's theorem — Ch. 5
- Stone representation theorem — Ch. 5

## Vector Calculus / Differential Geometry
- Green's theorem — Ch. 4
- Stokes' theorem — Ch. 4
- Divergence theorem — Ch. 4
- Riemann–Roch theorem — Ch. 4
- Atiyah–Singer index theorem — Ch. 6

## Probability and Statistics
- Bernoulli's theorem (law of large numbers) — Ch. 3
- De Moivre's theorem — Ch. 3
- De Moivre–Laplace theorem — Ch. 3
- Bayes's theorem — Ch. 3
- Laplace's central limit theorem — Ch. 3

## Logic and Foundations
- Zermelo's well-ordering theorem — Ch. 5
- Gödel's completeness theorem — Ch. 5
- Gödel's incompleteness theorems — Ch. 5
- Gödel's consistency of CH — Ch. 5
- Cohen's independence of CH — Ch. 6
- Turing's halting theorem — Ch. 5
- Banach–Tarski paradox — Ch. 5
- Cantor's theorem — Ch. 4
- Cantor–Bernstein–Schroeder theorem — Ch. 4

## Combinatorics and Discrete Math
- Euler's seven bridges / graph theory — Ch. 3
- Ramsey's theorem — Ch. 5
- Four color theorem — Ch. 6
- Szemerédi's theorem — Ch. 6
- Robertson–Seymour theorem — Ch. 6

## Applied Mathematics / Mathematical Physics
- Kepler's laws — Ch. 2
- Torricelli's theorem — Ch. 2
- Noether's theorem (symmetries & conservation) — Ch. 5

## Dynamical Systems
- Poincaré recurrence theorem — Ch. 4
- Birkhoff / von Neumann ergodic theorem — Ch. 5
