TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/source-localized-spinor-20260919/PROOF.txt
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Inclusive original lines: 121-149
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000121 | specified twisted-section convention. It is not a periodic global gauge
000122 | transformation. On the selfdual doubled CAR space use G_a and its conjugate
000123 | on conjugate components; the resulting map respects the CAR real structure.
000124 | 
000125 | Cell-averaging G_a f_NS gives exactly Q_N^a(exp(i theta/2)f_NS).
000126 | This defines the source lift L_N of reference test functions. The preceding
000127 | norm and convergence arguments apply componentwise and to their adjoints.
000128 | 
000129 | Physical time is retained. On each sourced top-edge charge/oscillator core,
000130 | 
000131 |  H_a = L0 + (a-1/2)Q,
000132 | 
000133 | so at a=1/4 the source time generator is L0-Q/4. The local covariance
000134 | identification does not erase this chemical/holonomy term, choose an E8
000135 | vacuum, or identify its conformal rotations with physical time.
000136 | 
000137 | 4. The bounded spinor operation and its source lift
000138 | ---------------------------------------------------
000139 | Primary input: J. Boeckenhauer, Rev. Math. Phys. 8 (1996), Sections 5–7,
000140 | https://arxiv.org/html/hep-th/9507047 . Equations 23–28 give a real
000141 | Bogoliubov unitary V for an even number of Majorana components. Theorem
000142 | 5.4 identifies its restriction to the even vacuum net with a spinor
000143 | sector. This established sector theorem is cited, not re-proved here.
000144 | 
000145 | For reproducibility, the operation inside its localization interval can
000146 | be specified pairwise. Write o,e for an odd/even Majorana pair, f_n for
000147 | the interval integer Fourier basis and f_r for its half-integer basis:
000148 | 
000149 |  V f_0^o = i(f_(1/2)^o-f_(-1/2)^o)/sqrt(2),