TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/DIRECT-DEFECT-CLOSURE.txt
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000020 | 1. The original pair, with its explicit orientation repair
000021 | -----------------------------------------------------------
000022 | fullv21, section001:230-251 supplies the twisted circle, i partial_theta,
000023 | u=exp(i theta), and displayed eigenvalues n+a. Those eigenvalues belong to
000024 | B=-i partial_theta, so fix B=-D_Gamma rather than mixing the printed signs.
000025 | For 0<a<1 set
000026 | 
000027 |   B_a psi_n=(n+a)psi_n,     u psi_n=psi_(n+1),
000028 |   C=1_(-infinity,0)(B_a),   P_+=I-C,
000029 |   Pi=|psi_0><psi_0|.
000030 | 
000031 | The complete Fourier basis, twisted H1 domain and strong covariance were
000032 | checked in the preceding older-paper contract. Directly on that basis,
000033 | 
000034 |   u C u*=C+Pi,               P_+-uP_+u*=Pi,
000035 |   [u,C]=|psi_0><psi_(-1)|,   ||[u,C]||_2=1.             (1)
000036 | 
000037 | Thus Pi is a consequence of the old operator pair. This is stronger than
000038 | inferring a positive rank-one defect from the index alone, which is invalid.
000039 | 
000040 | 2. Connection to the already retained CAR source
000041 | -------------------------------------------------
000042 | Under the existing full-CAR/same-positive-time interface, the ground state
000043 | has covariance C. This is the ground-state reduction already proved in
000044 | paper-source-reconstruction-20261025/GROUND-STATE-SOURCE-REDUCTION.txt;
000045 | no additional independent quasifree state is chosen.
000046 | 
000047 | The Hilbert-Schmidt condition in (1) permits a Fock implementer. With
000048 | U c_n U*=c_(n+1), its phase-independent action is
000049 | 
000050 |   U Omega ~ c_0*Omega=:Xi,    Q Xi=Xi,    H Xi=a Xi.    (2)
000051 | 
000052 | The inverse shift gives the hole c_(-1)Omega, charge-1 and energy1-a.
000053 | The displayed one-particle M_z is precisely the mode shift of the existing
000054 | SOURCE W1. The physical identification of the old geometric circle and the
000055 | original field/time image is still required; equality of Fourier formulas
000056 | does not by itself make that physical identification.
000057 | 
000058 | For the state of Xi, the exact neutral two-point response is
000059 | 
000060 |   omega_Xi(c(f)*c(g))-omega_C(c(f)*c(g))=<g,Pi f>.     (3)
000061 | 
000062 | Pi acts on the ONE-PARTICLE space. The Fock density Pi_F=|Xi><Xi| is a
000063 | different operator. There is an explicit isometry on the particle sector,
000064 | 
000065 |   V:H_+ -> Fock,       V f=c(f)*Omega,
000066 |   V*V=I_(H_+),         V Pi V*=Pi_F,                  (4)
000067 | 
000068 | with e^(itH)V=V e^(itB) on H_+. In particular V Dom(B|H_+) lies in Dom(H)
000069 | and HVf=VBf there. Thus the two rank-one traces can be transported when
000070 | this is the actual source map.
000071 | They are not literally identical operators on the same Hilbert space.
000072 | 
000073 | The charged U need not be an observable preserving the vacuum sector.
000074 | For a gauge-invariant A, the full word U*A U is gauge invariant and maps
000075 | charge-zero back to charge-zero. Therefore no demand that U itself be a
000076 | neutral preparation follows from using (3). Its physical permission and
000077 | localization still have to be checked on the full word in the original net.
000078 | 
000079 | 3. All equal-time CAR moments of this defect, without a new source model
000080 | ------------------------------------------------------------------------
000081 | Both C and C+Pi are projections, so the known Wick rule applies. For any
000082 | finite normal-ordered word with k creators followed by k annihilators put
000083 | 
000084 |   M_ij=omega_C(c(f_i)*c(g_j)),
000085 |   v_i w_j=omega_Xi(c(f_i)*c(g_j))-M_ij.
000086 | 
000087 | The matrix change v w^T has rank at most one. Hence
000088 | 
000089 |   omega_Xi(c(f_1)*...c(f_k)*c(g_k)...c(g_1))
000090 |     =det(M+v w^T)
000091 |     =det(M)+sum_(i,j)cof_ij(M) v_i w_j.               (5)
000092 | 
000093 | The cofactor formula also holds if M is singular; no inverse is needed.
000094 | Adjunction and CAR reduce other finite words to these. The entire finite
000095 | word response is fixed by the existing source and the line in (1). This
000096 | does not identify it with every field or interaction of S_adm.
000097 | 
000098 | 4. The exact relative determinant
000099 | ----------------------------------
000100 | C has infinite rank, so det_F(I-zC) alone is not defined. For z!=1 define
000101 | 
000102 |   A(z)=I-zC,        A_u(z)=I-z(C+Pi).
000103 | 
000104 | Because C Pi=0,
000105 | 
000106 |   A_u A^(-1)=I-zPi,
000107 |   A A_u^(-1)=I+[z/(1-z)]Pi.                           (6)
000108 | 
000109 | Both right sides are identity plus trace class. Their Fredholm determinants