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: 241-286
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000241 | 9. Real EM response, phases, and the reference sector
000242 | -----------------------------------------------------
000243 | 
000244 | For the literal reflected covariance I-C, the same winding gives the inverse
000245 | factor (1-z)^(-1). The ordinary full relative pair, and the norm of THAT
000246 | inverse pair, are therefore one. Taking a square root of that norm cannot
000247 | produce 1-z. A positive pair with determinant (1-z)^2 would be a different
000248 | operator and needs an original source identity.
000249 | 
000250 | This does not say that a full Dirac field has zero real EM susceptibility.
000251 | Its actual D_A*D_A determinant is a different object. The even spectral
000252 | action, eta phase, and the chiral determinant line must remain typed.
000253 | 
000254 | The inverse-pair result is a scoped control, not an imposed physical source:
000255 | v42 starts with the labeled one-sided datum (A_+,H_+,D_+,B_Sigma), and its
000256 | minimal EM transfer contains no reflected inverse partner. Nevertheless
000257 | tr_adm lives on Ran(P_adm) inside H_OS; the source does not identify it with
000258 | the labeled graph marginal used in section4. Neither P_C nor Gamma supplies
000259 | that face selection. A new chiral measure is therefore not imposed as a
000260 | prerequisite for the real EM response. The required equality is the actual
000261 | state/time-preserving one-sided source map and its response functional.
000262 | 
000263 | Let b=B gamma=5/4 and q(alpha)=delta_top exp(-2 alpha), as in the originals.
000264 | The current operator theorem proves the candidate
000265 | 
000266 |   -b Log det_F R_C(q(alpha))=-b Log(1-q(alpha)).          (12)
000267 | 
000268 | The exponent b and the identification with the physical EM effective action
000269 | are not derived by (1). A sufficient actual Dirac-source comparison is
000270 | 
000271 |   Re[Gamma_F(alpha)-Gamma_F(alpha_*)]
000272 |    =-(1/2)log det_rel^+(D_alpha^*D_alpha,D_*^*D_*)
000273 |    =-b log[(1-q(alpha))/(1-q(alpha_*))].                 (13)
000274 | 
000275 | Here D_*=D_(alpha_*).
000276 | The symbol det_rel^+ denotes a properly defined positive RELATIVE
000277 | determinant with the original local counterterms/regularization retained;
000278 | no multiplicativity of arbitrary separate zeta determinants is assumed.
000279 | The reference alpha_* must be stated. In particular q(0)=delta_top is NOT
000280 | zero. A seam-free reference q_ref=0 is a different reference, and must be
000281 | named as such. The holomorphic branch is Log(1)=0 on |z|<1.
000282 | 
000283 | The old complete-sector criterion R(alpha)=R(alpha_*) remains necessary
000284 | when the rest of the source is integrated out. The Schur-complement identity
000285 | does not prove that residual ratio constant or remove a genuine source
000286 | normalization term.