TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/QUANTUM-OCTIC-COMPLETION.txt
Original file SHA-256: 2a075b2f0fc18c4133175eec20ef9b173fff7b23c3b66ef18c80c8505785f6c1
Inclusive original lines: 100-138
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

000100 | 4. Explicit answer
000101 | ------------------
000102 | The vacuum on the right is understood in this creation-mode formula:
000103 | 
000104 |  W8 = H8 +(1/1333) [
000105 |        28800 Q17 -100800 Q26 +201600 Q35 -126000 Q44
000106 |       -80640 Q11 Q15 +201600 Q11 Q24 -130368 Q11 Q33
000107 |           +0 Q12 Q14 -12096 Q12 Q23 +34944 Q13^2
000108 |       -46368 Q13 Q22 +21924 Q22^2
000109 |       +40656 Q11^2 Q13 -33516 Q11^2 Q22 +3024 Q11 Q12^2
000110 |        -2178 Q11^4 ] Omega.                              (1)
000111 | 
000112 | Here Q17 means Q_(1,7), not a new field or an adjustable coupling. Every
000113 | coefficient in (1) is fixed by the simultaneous constraints after specifying
000114 | the leading coefficient. Uniqueness is claimed in the declared 17-dimensional
000115 | ansatz. The proof does not classify every possible weight-eight source state.
000116 | 
000117 | The complete positive Fock norm, retaining every oscillator mode, is
000118 | 
000119 |   ||W8||^2 = 65028096000/43.                              (2)
000120 | 
000121 | There are 1539 nonzero real PBW monomials. Their norms use
000122 | product_(j,n) n^(m_jn) m_jn!, not just the mode-one factorial rule.
000123 | 
000124 | W8 is a finite-dimensional E8 Lie algebra ZERO-MODE singlet and a Virasoro primary of weight eight.
000125 | It is not asserted to be annihilated by all affine current modes.
000126 | The known W(E8) free-field construction gives the same type of spin-eight
000127 | generator at c=8; this is not a claim that W algebras were newly discovered.
000128 | The new Atlas connection is the explicit completion of its retained native
000129 | octic and its concrete old CAR source, with all coefficients and overlaps.
000130 | 
000131 | Primary literature: D. Ide, K. Ito and S. Miyazaki, Free field realization
000132 | of W-algebra associated with exceptional Lie algebras, arXiv:2609.14345v2,
000133 | sections 3 and 6, especially equations (21), (75)-(78):
000134 | https://arxiv.org/html/2609.14345v2#S6
000135 | The paper uses Q=b-b^-1. At b=1 its Q=0 and c(E8)=8. This Q must not be
000136 | confused with the contraction symbols Q_ab above. Our polynomial identities
000137 | are checked directly and do not depend on generic-parameter specialization.
000138 |