TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/RELATIVE-PAIRING-REVIEW.txt
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Inclusive original lines: 192-263
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000192 | 4. The actual Quillen norm and the different positive-pair candidate
000193 | ---------------------------------------------------------------------
000194 | 
000195 | For a chiral Fredholm family D_z^+, a relative determinant-line section s_z
000196 | has a Hermitian Quillen norm. With the convention that its squared norm is
000197 | given by the zeta determinant of the positive Laplacian, the elementary
000198 | identity is
000199 | 
000200 |   Delta log ||s_z/s_0||_Q
000201 |     =(1/2) Delta log det_zeta((D_z^+)^*D_z^+).             (5)
000202 | 
000203 | For the actual cancelling pair, multiplicativity gives
000204 | 
000205 |   Det(R)=Det(R_+) tensor Det(R_-)
000206 |         =Det(1-z) tensor Det((1-z)^(-1)) congruent C,       (6)
000207 | 
000208 | and its Quillen norm is
000209 | 
000210 |   ||s_R||_Q=|1-z| |1-z|^(-1)=1.                           (7)
000211 | 
000212 | Equivalently, the positive operators associated with the two inverse relative
000213 | sections have reciprocal determinant ratios. Formula (5) therefore gives zero,
000214 | not Log(1-z). The same conclusion holds for complex z after the inverse phase
000215 | is included: the full determinant-line section is canonically the product of a
000216 | line and its dual.
000217 | 
000218 | A different positively adding pair could be written abstractly as
000219 | 
000220 |   R_pos(z)=diag(1-z,1-z),
000221 |   det R_pos(z)=(1-z)^2.                                  (8)
000222 | 
000223 | For real 0<=z<1 its Quillen norm would be 1-z, since the norm is the
000224 | square root of the squared norm. But R_pos is not R. It replaces the reflected
000225 | inverse update by a second copy of the same oriented update. The factor one
000226 | half is then only the standard conversion from norm squared to norm of this
000227 | different object. It is not a normalization that can be applied to the
000228 | cancelling pair.
000229 | 
000230 | The required original identification is absent at three exact points:
000231 | 
000232 |   1. The v42 master functional writes
000233 | 
000234 |        Re log det_zeta,Padm(D_xi^2+mu^2)
000235 |        +(1/2) eta_Sigma,Padm(D_xi)
000236 | 
000237 |      with coefficient one on the positive determinant, not one half:
000238 | 
000239 |        documents/archive--paper-latex--tfpt-42/sections/
000240 |        027-subsection-primitive-completion-from-the-boundary-kernel.tex:
000241 |        229-248.
000242 | 
000243 |      That source does not define D_xi as the covariance-pair deformation.
000244 | 
000245 |   2. The active Quillen contract defines the continuum line from a chiral
000246 |      family D_b^+ and requires a future comparison of its metrics and
000247 |      connections with the occupied Berry line. It writes the proposed
000248 |      functional with log det_zeta Delta_U(1), while explicitly retaining the
000249 |      continuum comparison and first-principles response identity as open:
000250 | 
000251 |        documents/tfpt_research_contracts/sections/020-source.tex:180-240,
000252 |        296-318.
000253 | 
000254 |   3. The already reviewed direct-defect contract records as open the equality
000255 |      between the original electromagnetic logarithmic response and the
000256 |      directed relative covariance determinant. Replacing that determinant by
000257 |      a Quillen norm changes the candidate but does not prove the missing source
000258 |      map.
000259 | 
000260 | Thus neither the Quillen metric nor Delta eta turns the actual inverse pair
000261 | into the single factor. To use (8), the source must first construct a positive
000262 | pair family and then identify its electromagnetic observable with that norm.
000263 |