TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/P1-W8-LOCAL-MAP-AND-TWOPOINT-TEST.txt
Original file SHA-256: 072b6588666504ec4f04024e279030bbdf83b724102bac181ee3696133c1be28
Inclusive original lines: 120-173
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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000120 | 5. Full source answer and an independent Ward check
000121 | ----------------------------------------------------
000122 | Let N denote the actual quotient relative radial generator and |d> either
000123 | of the original quarter charge states. Set q=w/z and X=q/(1-q)^2. The result is
000124 | 
000125 |  z^8 w^8 <d|Y(W8,z)Y(W8,w)|d> = sum_(r=0)^8 A_r X^r.    (4)
000126 | 
000127 | All nine rational coefficients are listed in W8-SOURCE-TWOPOINT.txt and
000128 | w8-source-twopoint-results.json. The full Wick/pole proof and finite exact
000129 | reconstruction are in that report; no physical free coefficient is fitted.
000130 | Two immediate independent constraints close:
000131 | 
000132 |  A0=(25123488075/357824462848)^2,
000133 |  A8=||W8||^2=65028096000/43.
000134 | 
000135 | There is also an independent STRESS-TENSOR Ward test. The W8 OPE coefficient
000136 | of T is (2h_W/c)||W8||^2=2||W8||^2, with h_W=8 and c=8. There is only one
000137 | weight-two E8 invariant, the invariant quadratic form T. Since
000138 | <T(z)>_d=h_d/z^2, h_d=85/256, this forces
000139 | 
000140 |  A7=2h_d||W8||^2=43182720000/43.                         (5)
000141 | 
000142 | The sparse result satisfies (5) exactly. It tests the source normalization
000143 | independently of its triangular pole reconstruction. All new checker
000144 | conditions use explicit exceptions and survive python -OO.
000145 | 
000146 | 6. A concrete failed candidate: replacing the shifted source by vacuum
000147 | ----------------------------------------------------------------------
000148 | For q=exp(-tau), subtract the disconnected term A0 and obtain
000149 | 
000150 |  C_conn(tau)=sum_(r=1)^8 A_r/[4sinh^2(tau/2)]^r.           (6)
000151 | 
000152 | Every A_r in this actual source is strictly positive. Its lowest spectral
000153 | coefficient is
000154 | 
000155 |  a1=A1=93237295693374937125/15629680934138150912 >0.
000156 | 
000157 | For the unshifted vacuum the exact answer is only ||W8||^2 X^8, whose
000158 | spectral coefficients at levels1 through7 vanish. Thus a map preserving
000159 | the state, the same radial generator and this W8 insertion cannot replace
000160 | the actual quarter source by that vacuum. This rejects a specific proposed
000161 | dictionary, not every original P1 seed.
000162 | 
000163 | The generator conversion is explicit:
000164 | 
000165 |  N=Lhat0-85/256=(Lcal0^alpha-1/32)/8=K_alpha,
000166 |  H0=Lcal0^alpha-1/32=8N
000167 | 
000168 | on the folded neutral sector. The lowest shifted connected response has
000169 | N-level1 and original H0-grade8. The homogeneous unshifted W8 state has
000170 | N-level8 and H0-grade64. Conformal weight, the grade of a vacuum state and
000171 | the first response on a shifted source must not be confused. Neither tau
000172 | nor the circle convention is an absolute calibration to physical seconds.
000173 |