TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/QUANTUM-OCTIC-COMPLETION.txt
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000065 | 3. Solve the completion in the existing Heisenberg algebra
000066 | ---------------------------------------------------------
000067 | Use the original positive c=8 current normalization
000068 | 
000069 |   [h_j(m),h_k(n)] = m delta_jk delta_(m+n,0),
000070 |   L_n = (1/2) sum_(j,m) :h_j(n-m)h_j(m):.
000071 | 
000072 | Define, in the real orthonormal Cartan chart,
000073 | 
000074 |   Q_ab = sum_j h_j(-a)h_j(-b),
000075 |   p8(x) = sum_(alpha in Phi(E8)) (alpha.x)^8,
000076 |   H8(x) = p8(x)-30(x.x)^4.
000077 | 
000078 | H8 is harmonic. Indeed p6=30(x.x)^3, Delta p8=3360(x.x)^3
000079 | and Delta(x.x)^4=112(x.x)^3 in eight dimensions.
000080 | 
000081 | Consider precisely H8(h(-1)) plus all sixteen monomials in Q_ab of total
000082 | conformal weight eight. For a given polynomial P the root zero mode is
000083 | computed by the displayed vertex shift, and
000084 | 
000085 |   L_k P = sum_(n>=1,j) (n+k)h_j(-n) partial_(h_j(-n-k)) P
000086 |           +(1/2)sum_(a=1..k-1,j) a(k-a)
000087 |                          partial_(h_j(-a))partial_(h_j(-(k-a))) P.
000088 | 
000089 | Impose e_(e0-e1)(0)P=0 and L1 P=L2 P=0, keeping the H8 coefficient one.
000090 | The exact 1600 by 17 rational coefficient system has one-dimensional
000091 | kernel. The unique normalized-leading solution is the state W8 below.
000092 | All polynomials in this ansatz are Weyl invariant: H8 is a root sum and
000093 | the Q's are orthogonal contractions. Weyl transitivity therefore carries
000094 | the one root-zero-mode equation to all 240 roots. Cartan zero modes vanish
000095 | on the neutral Heisenberg space. Hence all finite E8 Lie algebra zero
000096 | modes annihilate W8. These 1600 coefficient equations have rank16; they
000097 | are not 1600 linearly independent constraints. L1 and L2 generate the positive Virasoro algebra;
000098 | the checker also verifies L1 through L8 directly.
000099 |