TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/native-octic-seam-origin-20261025/NATIVE-FOLDED-OCTIC-ORIGIN.txt
Original file SHA-256: c694cb1cee45e65145fdbf144925e45178a7d0ee29adb7142400a00a970e259b
Inclusive original lines: 145-189
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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000145 | collar embedding. Profiles and strengths specify interventions; they do
000146 | not need to be selected as fundamental constants by P1.
000147 | 
000148 | 4. Bounded root OPE result retained
000149 | -----------------------------------
000150 | 
000151 | For completeness, the checker preserves the already reached smallest test
000152 | of the phase-correct dual-root bilocal
000153 | 
000154 |   B(z)=sum_(beta in Phi) epsilon(beta,-beta)^(-1)
000155 |        Y(e^beta,z)e^(-beta)
000156 |       =z^-2 sum_beta exp(sum_(n>=1) beta(-n)z^n/n) Omega.
000157 | 
000158 | Its z^6 coefficient is the complete weight-eight state
000159 | 
000160 |   S8=sum_beta sum_(sum n m_n=8)
000161 |        product_n beta(-n)^m_n/(n^m_n m_n!) Omega.          (4)
000162 | 
000163 | Equation (4) retains all 22 oscillator partitions.  In the same original
000164 | U_+ polarization the physical roots alpha=r/sqrt(2) obey
000165 | 
000166 |   sum_beta (beta_+.x)^2=sum_beta (beta_+.x)^4
000167 |    =sum_beta (beta_+.x)^6=0,
000168 |   sum_beta (beta_+.x)^8=(15/2) I8(x).
000169 | 
000170 | Therefore
000171 | 
000172 |   <F8hat,S8>=sqrt(140)/56 != 0.                            (5)
000173 | 
000174 | The complete state norm, including every mixed oscillator mode, follows
000175 | from
000176 | 
000177 |   <E_beta,E_gamma>_q=(1-q)^(-beta.gamma)
000178 | 
000179 | and the exact E8 root-pair census.  It is
000180 | 
000181 |   ||S8||^2=15600.
000182 | 
000183 | Equation (5) proves a canonical full-field root OPE component.  It does not
000184 | prove that Virasoro projection leaves only F8hat, and no physical projection
000185 | is performed here.  The folded-source result above already gives the needed
000186 | positive original matrix element without making that exclusivity claim.
000187 | 
000188 | Original paths used
000189 | -------------------