TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/KMS-AND-CARRIER-TRANSPORT.txt
Original file SHA-256: 9efcfb0d68e16db842aab97b7171b2d667e7b065cb4c6ab8138102acc4c8fb68
Inclusive original lines: 194-220
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000194 | 7. An exact sufficient repair using the same source
000195 | ---------------------------------------------------
000196 | 
000197 | Do not discard the complement. For ANY actually supplied finite-rank F,
000198 | let Q=I-F, A=I-zC, A'=I-z uCu*. In the decomposition F H direct_sum Q H,
000199 | the QQ blocks are invertible for |z|<1. Define
000200 | 
000201 |   S_F=A_FF-A_FQ A_QQ^(-1) A_QF,
000202 |   S'_F=A'_FF-A'_FQ A'_QQ^(-1) A'_QF.
000203 | 
000204 | These finite Schur complements are invertible because A,A' and their QQ
000205 | blocks are invertible. Block Gaussian factorization uses upper and lower
000206 | unipotent finite-rank factors with Fredholm determinant one. It gives
000207 | 
000208 |   1-z=det(S'_F S_F^(-1))
000209 |       det_F[A'_QQ A_QQ^(-1)].                           (10)
000210 | 
000211 | The second relative determinant exists because A'-A is trace class. No
000212 | commutation [F,C]=0 is assumed. In terms of the covariance,
000213 | 
000214 |   S_F=I_F-z C_F
000215 |       -z^2 C_FQ(I_Q-z C_QQ)^(-1) C_QF.                 (11)
000216 | 
000217 | The last term and the QQ determinant are both determined by the SAME
000218 | original C, once F is actually supplied. They add no fitted kernel or new
000219 | state. Equations (10)-(11) are the exact sufficient transport of the old
000220 | factor through a finite carrier. Keeping only C_F is a different operation.