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: 139-174
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.

000139 | 5. Link to the OLD native octic and complete root OPE
000140 | -----------------------------------------------------
000141 | The checker reuses the original O,M,U+ matrices, and verifies that all 240
000142 | old Construction-A roots map to exactly the standard root set used in (1).
000143 | No Cartan embedding is fitted to make the answer work.
000144 | 
000145 | In the actual original Gaussian polarization, the already known identity is
000146 | 
000147 |   p8|_(H+) = (15/2) I8.
000148 | 
000149 | Every Q11 term restricts to zero on this isotropic four, and higher oscillator
000150 | modes are orthogonal to F8hat. Consequently
000151 | 
000152 |   <F8hat,W8> = 720 sqrt140,
000153 |   |<F8hat,W8/||W8||>|^2 = 43/896.                        (3)
000154 | 
000155 | Thus the original octic is an actual nonzero component of the complete
000156 | operator. The fraction in (3) is a Hilbert-space overlap, not an empirically
000157 | selected probability or an identification of the two states.
000158 | 
000159 | Retain the OLD full root OPE coefficient, including all 22 partitions:
000160 | 
000161 |   S8=sum_beta [z^8]exp(sum_(n>=1) beta(-n)z^n/n)Omega,
000162 |   ||S8||^2=15600.
000163 | 
000164 | Pairing a polynomial state with that coherent exponential evaluates every
000165 | h_j(-n) at beta_j. All roots are Weyl conjugate, so (1) gives exactly
000166 | 
000167 |   <W8,S8> = 240 W8(beta,...,beta) = 48384000/1333,
000168 |   proj_(C W8) S8 = W8/41664,
000169 |   |<W8/||W8||,S8/||S8||>|^2 = 30/537199.                (4)
000170 | 
000171 | S8 is not replaced by W8. Equation (4) is an orthogonal decomposition in
000172 | the existing source space, not a new physical projection or preparation.
000173 | The old nonzero F8hat matrix elements and full S8 calculation are reused.
000174 |