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: 37-153
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000037 | 
000038 | 2. Fixed orientation and domains
000039 | --------------------------------
000040 | 
000041 | On H=L^2(S^1) with the original twisted domain, use
000042 | 
000043 |   B_a psi_n=(n+a)psi_n,    0<a<1,
000044 |   u psi_n=psi_(n+1),
000045 |   P=1_(B_a<0),
000046 |   u P u*=P+Pi,            Pi=|psi_0><psi_0|.
000047 | 
000048 | This includes a=1/4,3/4 and the full A4-induced NS cover a=1/2. The sign
000049 | is the previously repaired B_a=-i partial_theta convention. Reversing the
000050 | winding reverses the power of 1-z. The twist a is not alpha_EM.
000051 | 
000052 | Let C be any positive contraction on this SAME H such that
000053 | 
000054 |   C-P is trace class.
000055 | 
000056 | For |z|<1 define A_C=I-zC, A_P=I-zP and
000057 | 
000058 |   R_C(z)=u A_C u* A_C^(-1)
000059 |         =(I-z uCu*)(I-zC)^(-1).
000060 | 
000061 | The inverses exist by the Neumann bound. Moreover
000062 | 
000063 |   uCu*-C=u(C-P)u*-(C-P)+Pi
000064 | 
000065 | is trace class. Thus R_C-I is trace class, and its ordinary Fredholm
000066 | determinant is well defined. Neither separate infinite determinant det A_C
000067 | nor det A_P is introduced.
000068 | 
000069 | 3. Exact stability theorem
000070 | --------------------------
000071 | 
000072 | The following is an exact multiplication identity:
000073 | 
000074 |   R_C=(u A_C A_P^(-1)u*)
000075 |       (u A_P u* A_P^(-1))
000076 |       (A_P A_C^(-1)).
000077 | 
000078 | Every factor is I plus trace class. By unitary invariance and
000079 | multiplicativity of the Fredholm determinant, the first and last determinants
000080 | cancel. The middle factor is I-zPi because P psi_0=0. Therefore
000081 | 
000082 |   det_F R_C(z)=1-z.                                      (1)
000083 | 
000084 | This is a relative-determinant invariant of the SAME trace-class polarization
000085 | class. No empty or filled eigenvector of C itself is required. For winding
000086 | u^m, m in Z, the same proof gives
000087 | 
000088 |   det_F[u^m A_C u^(-m) A_C^(-1)]=(1-z)^m.                (2)
000089 | 
000090 | For m>0 the reference adds m orthogonal modes; for m<0 it removes them.
000091 | Unitary transport of the entire tuple C,P,u preserves (1). This permits the
000092 | already established change to the actual D8 CAR frame; it does not turn the
000093 | interacting E8 extension into an unproved quasifree state.
000094 | 
000095 | 4. The faithful original KMS representative
000096 | -------------------------------------------
000097 | 
000098 | Use the EXISTING circle functional calculus,
000099 | 
000100 |   C_beta=(I+exp(beta B_a))^(-1),    beta>0.
000101 | 
000102 | For the labeled marginal of the supplied product collar beta=2 ell. No
000103 | identification of beta or ell with alpha_EM is made. The occupations are
000104 | 
000105 |   n_n=(1+exp(beta(n+a)))^(-1).
000106 | 
000107 | Both C_beta and I-C_beta have trivial kernel. Their spectra accumulate at
000108 | 0 and 1, but neither endpoint is an eigenvalue. Functional calculus gives
000109 | 
000110 |   log(I-C_beta)-log C_beta=beta B_a on Dom(B_a).           (3)
000111 | 
000112 | This is a densely defined selfadjoint log odds operator, rather than the
000113 | endpoint singularity of a pure projector. The exponential Fourier tails
000114 | give the explicit trace norm bound
000115 | 
000116 |   ||C_beta-P||_1
000117 |    <= [exp(-beta a)+exp(-beta(1-a))]/[1-exp(-beta)].       (4)
000118 | 
000119 | Consequently (1) holds for EVERY finite beta>0. It is not a large-beta
000120 | approximation and needs no clipping at eigenvalues zero or one.
000121 | 
000122 | Directly, D_beta=uC_beta u*-C_beta is diagonal with
000123 | 
000124 |   d_n=n_(n-1)-n_n>0,       sum_n d_n=1.                  (5)
000125 | 
000126 | It is positive trace class and has infinite rank. Its relative determinant
000127 | telescopes on the actual open window [-L,L]:
000128 | 
000129 |   product_(n=-L)^L (1-z n_(n-1))/(1-z n_n)
000130 |    =(1-z n_(-L-1))/(1-z n_L) ->1-z.                     (6)
000131 | 
000132 | The difference R_Cbeta-I is trace class, so trace norm convergence of its
000133 | diagonal restrictions justifies the Fredholm limit. As beta tends to
000134 | infinity, (4) also implies ||D_beta-Pi||_1->0.
000135 | 
000136 | This closes the mathematical gap between the old pure defect and a faithful
000137 | thermal source ON THE SAME CIRCLE. It does not identify that source with the
000138 | active finite collar Hamiltonian or with the full physical EM measure.
000139 | 
000140 | 5. The full NS cover does not require a single quarter selector
000141 | ----------------------------------------------------------------
000142 | 
000143 | The already proved NS-cover unitary has
000144 | 
000145 |   B_cov=2(B_(1/4) direct_sum B_(3/4)),
000146 |   u_cov=[[0,u_base],[I,0]],
000147 |   u_cov^2=diag(u_base,u_base).
000148 | 
000149 | The faithful covariance must be the functional calculus of this full B_cov.
000150 | It is not two arbitrarily adjusted independent sources. Applying (1)-(2)
000151 | before the unitary decomposition gives
000152 | 
000153 |   det_rel(u_cov C_beta u_cov*,C_beta;z)=1-z,