MaddyMathematics by RIMO

Long-form proof · Lie theory

Why the Dynkin
list ends.

Positive-definite Cartan forms leave room for four infinite families, five exceptional diagrams, and nothing else.

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The exceptional boundary
1/(a+1)
+ 1/(b+1) + 1/(c+1)
> 1

Strict inequality gives finite type. Equality is affine.

The argument

Four movements

IntegralityRank-two geometry restricts every edge to multiplicity 0, 1, 2, or 3PositivityThe Cartan form turns finiteness into a spectral constraintBranchingOne reciprocal inequality produces D and the three E diagramsWeighted edgesA determinant recurrence leaves only B, C, F, and G

Current essay

Structure and classification

Article index
ProofAdvanced

Why the Dynkin List Ends

A proof-driven route from simple roots to the finite Dynkin diagrams, using positive-definite Cartan forms, spectral radius, Schur complements, and a determinant recurrence to explain both the infinite families and the exceptional cases.

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Central question

How much graph can a positive form support?

Follow the boundary ↗