TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/KMS-AND-CARRIER-TRANSPORT.txt
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Inclusive original lines: 161-192
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000161 | 6. Why ordinary finite compression loses the factor
000162 | ----------------------------------------------------
000163 | 
000164 | First reuse the pure test independently proved in GRADED-DETERMINANT-SOURCE:
000165 | for any orthogonal F, v=F psi_0,
000166 | 
000167 |   C'_F-C_F=|v><v|,
000168 |   det_rel(C'_F,C_F;z)=1-z<v,(I-zC_F)^(-1)v>.             (8)
000169 | 
000170 | This equals 1-z on an open z range exactly when psi_0 lies in Ran F. Then
000171 | C_F has eigenvalue zero and C'_F has eigenvalue one. Mere rank one does not
000172 | imply a normalized projector: its nonzero eigenvalue is ||F psi_0||^2.
000173 | 
000174 | For the faithful thermal source there is a stronger finite-carrier test.
000175 | Let F be any finite-rank orthogonal projection on the SAME circle H and
000176 | C'_beta=uC_beta u*. Because D_beta is strictly positive on every nonzero
000177 | vector, Q=I-F is nonzero and
000178 | 
000179 |   0 <= Tr(F D_beta F)<Tr D_beta=1.
000180 | 
000181 | The derivative at z=0 of the naively compressed relative determinant is
000182 | 
000183 |   d/dz det_rel(F C'_beta F,F C_beta F;z)|_(z=0)
000184 |    =-Tr(F D_beta F)>-1.                                (9)
000185 | 
000186 | It cannot equal 1-z on an open range. Both finite compressions are faithful,
000187 | so the same conclusion follows from their nonzero relative value at z=1.
000188 | 
000189 | This is specifically a finite-rank compression of the circle source. It
000190 | does not apply to an infinite-rank projection onto a finite INTERNAL factor
000191 | with the spacetime modes retained. The active P_F requires an actual map
000192 | before either interpretation can be assigned to it.