TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/W8-SOURCE-TWOPOINT.txt
Original file SHA-256: f8f03e184f8813ecaf7439f47903d3835896475e272bd0ea03f4e8d85c12f688
Inclusive original lines: 146-224
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.

000146 | 
000147 | The field is bosonic and mutually local, so its separated analytic
000148 | continuation obeys F_d(q_rad)=F_d(1/q_rad).  After the displayed powers of z
000149 | and w are removed, Wick contractions have only the collision pole q_rad=1.  Its order is at
000150 | most 16 because W8 has weight eight.  The source terms d/z add no new separated
000151 | pole.
000152 | 
000153 | Every q_rad<->1/q_rad invariant rational function is rational in
000154 | q_rad+q_rad^-1.  Since
000155 | 
000156 |   q_rad+q_rad^-1-2 = (1-q_rad)^2/q_rad,
000157 | 
000158 | finiteness at q_rad=0 and the absence of any pole except q_rad=1 imply that F_d is a
000159 | polynomial in
000160 | 
000161 |   X = q_rad/(1-q_rad)^2.
000162 | 
000163 | The collision bound fixes its degree to at most eight:
000164 | 
000165 |   F_d(q_rad)=sum_(r=0)^8 A_r X^r.                          (9)
000166 | 
000167 | For 1<=ell<=8, comparison of (8) and (9) is triangular:
000168 | 
000169 |   a_ell=sum_(r=1)^ell binom(ell+r-1,2r-1) A_r,
000170 |   a_0=A_0.                                                 (10)
000171 | 
000172 | Thus the nine exact Fock norms a_0,...,a_8 determine the entire separated
000173 | function.  Equation (9) is a consequence of locality, exchange symmetry, and
000174 | the pole bound, rather than a fit to nine samples.  All higher creation modes
000175 | are thereby retained in the resulting rational function.
000176 | 
000177 | Exact answer
000178 | ------------
000179 | 
000180 | Both original quarter sources give the same coefficients:
000181 | 
000182 |   A_0 = 631189653054667205625/128038346212459732271104,
000183 |   A_1 = 93237295693374937125/15629680934138150912,
000184 |   A_2 = 32613852269143537875/30526720574488576,
000185 |   A_3 = 18040045268721320775/238490004488192,
000186 |   A_4 = 3872007396271830375/1863203160064,
000187 |   A_5 = 5291949560625/170624,
000188 |   A_6 = 337111976250/1333,
000189 |   A_7 = 43182720000/43,
000190 |   A_8 = 65028096000/43.                                   (11)
000191 | 
000192 | The decisive collision check is A_8=||W8||^2.  The disconnected term is
000193 | 
000194 |   A_0=w^2.
000195 | 
000196 | Therefore the connected function is obtained from (9) by deleting A_0.
000197 | 
000198 | Why the two quarters agree
000199 | --------------------------
000200 | 
000201 | Let R reverse the eight original source coordinates.  The exact matrix
000202 | 
000203 |   g=-M R M^-1
000204 | 
000205 | acts in standard coordinates as
000206 | 
000207 |   g(x0,x1,x2,x3,x4,x5,x6,x7)
000208 |     =(x3,-x4,-x5,x0,-x1,-x2,x7,x6).
000209 | 
000210 | The checker verifies directly that g permutes the actual 240 standard E8
000211 | roots, sends d_1/4 to d_3/4, and sends the first level vector in the first
000212 | quarter to the first level vector in the second.  Since W8 is invariant under
000213 | this root action and the Fock metric is orthogonal, the equality of the entire
000214 | two-point functions is structural.  It does not identify the complete odd CAR
000215 | modules or their physical interpretations.
000216 | 
000217 | Scope
000218 | -----
000219 | 
000220 | The result is the full separated neutral W8 two-point function in the already
000221 | given shifted current modules.  It establishes a nonzero lowest connected
000222 | source response and an exact common quarter polynomial.  It supplies no new
000223 | physical preparation, measurement, coupling, P1 derivation, or identification
000224 | with the unshifted vacuum.