TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/RELATIVE-PAIRING-REVIEW.txt
Original file SHA-256: ef280950f70a108ecff5dde9f57c67d097494e3824fdd65aea2ca176b5b83a34
Inclusive original lines: 66-149
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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000066 | 1. Exact two-line calculation
000067 | -----------------------------
000068 | 
000069 | Assume the established one-face defect line Pi satisfies
000070 | 
000071 |   P_- Pi=0,
000072 |   P_+ Pi=Pi.
000073 | 
000074 | On the paired polarization put
000075 | 
000076 |   C_1=C_0+diag(Pi,-Pi).
000077 | 
000078 | For |z|<1 define the directed relative Fredholm operator
000079 | 
000080 |   R(z)=(I-z C_1)(I-z C_0)^(-1).
000081 | 
000082 | It is the identity away from the two defect lines. On the first line C_0=0 and
000083 | C_1=1, hence R_+(z)=1-z. On the reflected line C_0=1 and C_1=0, hence
000084 | 
000085 |   R_-(z)=(1-z)^(-1).
000086 | 
000087 | Consequently
000088 | 
000089 |   det_F R(z)=(1-z)(1-z)^(-1)=1.                 (1)
000090 | 
000091 | This cancellation is correct for the ordinary vector determinant of the full
000092 | occupation-covariance pair. It does not say that a real electromagnetic
000093 | susceptibility cancels. Particle and hole have opposite signed charge, while a
000094 | quadratic gauge response depends on charge squared and adds their
000095 | contributions. Equation (1) is therefore an occupation-response statement,
000096 | not an F^2 response theorem.
000097 | 
000098 | If one defines an additional face grading
000099 | 
000100 |   Gamma_sheet=diag(1,-1),
000101 | 
000102 | then its superdeterminant is
000103 | 
000104 |   sdet_Gamma_sheet R(z)
000105 |     =det R_+(z)/det R_-(z)
000106 |     =(1-z)^2,                                      (2)
000107 | 
000108 | and on the holomorphic branch fixed by Log(1)=0,
000109 | 
000110 |   (1/2) Log sdet_Gamma_sheet R(z)=Log(1-z).        (3)
000111 | 
000112 | Equations (2) and (3) are exact algebra. They do not establish that
000113 | Gamma_sheet is an original grading or that this superdeterminant is the
000114 | physical response.
000115 | 
000116 | 2. The decisive original type check
000117 | -----------------------------------
000118 | 
000119 | The original reduced datum requires the involution to commute with Gamma:
000120 | 
000121 |   documents/archive--paper-latex--tfpt-31/sections/
000122 |   002-subsection-primitive-relative-spectral-dynamics.tex:7-29.
000123 | 
000124 | The current source extract retains the same typed separation:
000125 | 
000126 |   documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/
000127 |   sections/001-section-primitive-core-and-carrier-interface-handoff.tex:94-179.
000128 | 
000129 | It distinguishes the geometric deck involution tau_dbl, the Calderon
000130 | polarization iC, and the restricted carrier involution eps_car. It explicitly
000131 | forbids identifying these three operators without a displayed restriction or
000132 | transport map.
000133 | 
000134 | For a two-face basis in which the deck operation exchanges faces,
000135 | 
000136 |                 [ 0  I ]
000137 |   tau_sheet  =  [      ],
000138 |                 [ I  0 ]
000139 | 
000140 | one has
000141 | 
000142 |   tau_sheet Gamma_sheet tau_sheet^*=-Gamma_sheet. (4)
000143 | 
000144 | This contradicts the original commutation type if Gamma_sheet is called the
000145 | original Gamma. The normal reflection can make the relative operator odd
000146 | under deck exchange, but it does not turn the spectral-triple grading into a
000147 | sheet sign. The proposed ratio orientation therefore needs its own oriented
000148 | relative line or face-label datum.
000149 |