Lie Algebra Census

rank(E8) = 8. All root counts are 7-smooth.

In 1894, Killing and Cartan classified all simple Lie algebras. Four infinite families (A, B, C, D) and five exceptions: G2, F4, E6, E7, E8. Every exceptional root count factors into {2, 3, 5, 7} only -- no prime above 7 appears. Their Coxeter numbers h are all divisible by 6, and h+1 is always prime. These patterns connect them to the ring Z/214,414,200.

The Root Count Theorem

Every exceptional Lie algebra has a 7-smooth root count: factors entirely into {2, 3, 5, 7}. The prime 11 is absent from every root count.

AlgebraRankRootsFactorizationh+1
G2212 = 2^2 * 3h+1 = 7 (prime)
F4448 = 2^4 * 3h+1 = 13 (prime)
E6672 = 2^3 * 3^2h+1 = 13 (prime)
E77126 = 2 * 3^2 * 7h+1 = 19 (prime)
E88240 = 2^4 * 3 * 5h+1 = 31 (prime)

The factor 7 appears only in E7 (rank 7). The factor 5 appears only in E8 (rank 8). Coxeter h+1 is prime for all five, and all five have h divisible by 6.

Coxeter Divisibility

6-Divisibility
For all five exceptional Lie algebras, h is divisible by 6: h/6 = {1, 2, 2, 3, 5}. The half-Coxeter numbers h/2 = {3, 6, 6, 9, 15} are all multiples of 3. And h+1 is always prime: {7, 13, 13, 19, 31}. These boundary primes are Cunningham images: 7=2*3+1, 13=2*6+1, 19=2*9+1, 31=2*15+1. At the non-6-divisible Coxeter numbers (h = 2, 4, 10, 16), no exceptional algebra appears.

The Eight Tiers

Group the simple Lie algebras by Coxeter number h where h+1 is prime. Eight tiers emerge:

hh+16 | h?Algebras
23NoA1
45NoA3, B2
67YesA5, B3, C3, D4, G2
1011NoA9, B5, C5, D6
1213YesB6, C6, D7, E6, F4
1617NoB8, C8, D9
1819YesA17, B9, C9, E7
3031YesE8 (alone)

Why E8 Is Terminal

E8 has rank 8 and Coxeter number h = 30. Here is why it is the last exceptional algebra.

E8 rank
8
phi(30) = 8: the rank equals Euler's totient of the Coxeter number.
E8 Coxeter
h = 30 = 2 * 3 * 5
Exponents = totatives of 30: {1, 7, 11, 13, 17, 19, 23, 29}.
E8 roots
240 = 2^4 * 3 * 5
Only primes 2, 3, 5 appear. Primes 7, 11, 13 absent.
E8 dim
248 = 8 * 31
rank * (h+1).

Coxeter exponents of E8: {1, 7, 11, 13, 17, 19, 23, 29} = totatives of 30. Sum = 120 = half the roots. phi(30) = 8 = rank. The number of exponents IS the rank.

The Dimension Ladder

dim = rank * (h+1). Every dimension factors through the boundary prime:

Algebrarank * (h+1)dimFactorization
G22 * 7142 * 7
F44 * 13524 * 13
E66 * 13786 * 13
E77 * 191337 * 19
E88 * 312488 * 31

Dimension differences: F4-G2 = 38 = 2*19. E6-F4 = 26 = 2*13. E7-E6 = 55 = 5*11. E8-E7 = 115 = 5*23.

Hypothetical E9

If the E-series continued, E9 would have h = 42 = 2*3*7. Then h+1 = 43, a Heegner number and Cunningham image c(21) = 2*21+1. But E9 is infinite-dimensional (Kac-Moody). The boundary h+1 = 43 marks where finite exceptional Lie algebras end.

b1-Lie Correspondence

Betti-Lie Correspondence
First Betti number b1(Z/p x Z/q) = (p-1)(q-1). For products of ring moduli: b1(8, 9) = 7*8 = 56 = dim(fund E7). b1(25, 11) = 24*10 = 240 = |roots(E8)|. PSL ladder: b1(3,7) = 12. b1(7,11) = 60 = icosahedron. b1(8,25) = 168 = |PSL(2,7)|. These are exactly the products of adjacent CRT moduli Z/8, Z/9, Z/25, Z/49, Z/11, Z/13.

Explore: Coxeter Tier Lookup

Enter a Coxeter number h (even). See h+1, 6-divisibility, and which algebras live on that tier. The eight tiers where h+1 is prime: h = {2, 4, 6, 10, 12, 16, 18, 30}.

Coxeter number h:

Try: 6 (G2 + 4 classical), 12 (F4 + E6 + 3 classical), 30 (E8 alone).

Contrast Table

Five exceptionsG2, F4, E6, E7, E8 are sporadic anomaliesAll five have h divisible by 6 and h+1 prime. Not sporadic -- structurally linked.Why rank 8?E8 rank is 8. No structural reason usually givenphi(30) = 8. The rank equals Euler's totient of the Coxeter number.Root countsComputed case by case from Dynkin diagramsAll five exceptionals are 7-smooth. Prime 11 absent from every root count.Betti numbersTopological invariants of CW complexesb1(Z/p x Z/q) = (p-1)(q-1). Products of ring moduli give Lie dimensions.

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