TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/e-series-incidence-algebra-20261024/THEOREMS.txt
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000057 | 3. All pair relations and their full commutant
000058 | 
000059 | Fix an E6 cut E. The other 1119 cuts fall into exactly four classes:
000060 | 
000061 |     common roots    type         number
000062 |            40       D5               81
000063 |            24       D4              270
000064 |            20       A4              648
000065 |            12       A2+A2           120.
000066 | 
000067 | The actual simple-root Cartan matrices identify these types. The checker
000068 | first proves Weyl transitivity on all 1120 cuts. Reflections in the fixed
000069 | E6 and its A2 normal fix E; their explicitly computed action is transitive
000070 | on EACH of the four classes. Thus the five relations, including equality,
000071 | are exactly the orbitals of the Weyl action, not merely five intersection
000072 | sizes. Their adjacency matrices span the FULL permutation commutant.
000073 | 
000074 | Let A0=I, A1=A_D5, A2=A_D4, A3=A_A4, A4=A_(A2+A2). Define
000075 | 
000076 |     p[k][i][j] = #{Z : (E,Z) in Ri, (Z,F) in Rj}, (E,F) in Rk.
000077 | 
000078 | The orbital transitivity makes this independent of the representative pair.
000079 | All 125 integers are stored in results.json, along with their complete
000080 | 5 by 5 multiplication matrices. Their products commute. Since this is the
000081 | full commutant, the permutation representation is multiplicity-free.
000082 | Its full character table on the five relations is
000083 | 
000084 |     dimension       I      D5      D4      A4      A2+A2
000085 |             1       1      81     270     648        120
000086 |            35       1      33      30     -24        -40
000087 |            84       1      21     -30     -12         20
000088 |           300       1      -3      18     -24          8
000089 |           700       1      -3      -6      12         -4.