TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/recursive-mirrored-source-20261103/GAUSSIAN-REFLECTION-PHASE-RADIAL.txt
Original file SHA-256: e51580087db4b7765f931f872916b64a2d2fa21d5417292048a7362455577d1e
Inclusive original lines: 191-229
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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000191 | The exp(-s mu) factor belongs to the positive original mixing weight.
000192 | Assume the original kernel exists, A=int A_s dw>0, B=int B_s dw and
000193 | C=int C_s dw are finite, with the needed DCT remainder control (finite
000194 | inverse moments through4 suffice for a fixed mass and finite charged
000195 | sum). Then the whole kernel has
000196 |  R_w(x)=A/x+B/x^2+C/x^3+o(x^-3).
000197 | For each individual heat/mass contribution,
000198 |  B_s^2-A_s C_s
000199 |  =[-384-1536mu s-512mu^2s^2]/s^6<0.
000200 | Because all weights are positive,
000201 |  B=int B_s dw < int sqrt(A_s C_s)dw
000202 |               <=sqrt[(int A_s dw)(int C_s dw)]=sqrt(A C),
000203 | where the second inequality is Cauchy--Schwarz. The strict first
000204 | inequality holds for a nonzero genuine positive mixture. Thus
000205 |  B^2-A C<0.                                            (10)
000206 | 
000207 | If the actual mixed radial kernel defines a stable standalone Gaussian,
000208 | its inverse after its complete polynomial contact x/A-B/A^2 has the
000209 | same negative1/x remainder and therefore the same RP obstruction.
000210 | If it is not stable, a real positive standalone Gaussian with that
000211 | quadratic weight already fails at the covariance-definition stage.
000212 | This argument includes positive mixtures over actual charged masses;
000213 | it does not replace the neutrino/Majorana block by such a mixture.
000214 | 
000215 | 7. Primary mathematical context and preserved exclusions
000216 | --------------------------------------------------------
000217 | Arthur Jaffe, Stochastic PDE, Reflection Positivity, and Quantum Fields,
000218 | https://www.arthurjaffe.net/Assets/pdf/SPDE-QFT-v3.pdf , sectionIII,
000219 | states the equivalence of Gaussian RP with covariance RP and constructs
000220 | countertests using positive-time distributions followed by legitimate
000221 | test-function limits. The explicit one-pulse negative kernels above are
000222 | derived directly, not attributed to that paper.
000223 | The exact numerator heat kernels used here were independently derived in
000224 | PHASE-FORM-FACTOR-REVIEW.txt. The original unsquared primitive f(D/chi),
000225 | its actual bound reference, eta/domain variations, physical selector,
000226 | full Majorana response, coupled bosonic sectors and complete interacting
000227 | weight are NOT replaced or disposed of by these Gaussian calculations.
000228 | No universal TFPT RP failure, primitive-action solution, common physical
000229 | spectrum or ledger promotion is claimed.