TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/DET16-ORIGINAL-MAP.txt
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Inclusive original lines: 94-200
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000094 | gives exact CAR on the ENTIRE Fock space, and annihilates that vacuum.
000095 | For phase theta the same algebraic formula replaces F by exp(i theta)F;
000096 | this is the formula with the already specified phase, not phase selection.
000097 | Under the explicitly inherited neutral full-determinant premise, U_det
000098 | commutes with the actual gauge action, so a_i retain their charges.
000099 | Products of these even onsite unitaries preserve cross-site CAR.
000100 | This is a genuine answer to the old invalid Hubbard-field inference:
000101 |  |phi><e_i| do not satisfy CAR and their creators cannot form two particles.
000102 | 
000103 | The same original explicitly states
000104 |  N_a=U_det N U_det^*, Q<=N_a<=nQ,
000105 | and Delta N_a is NOT Delta Q, although the unique ground line and gap agree.
000106 | One may transport the whole old source by U_det, including every old
000107 | coupling, gauge map, state, time and adjoint. In that coordinate change
000108 | 2Q becomes 2(I-P_0), still a flat parent. Keeping other terms fixed while
000109 | substituting Delta N_a selects another Hamiltonian. The signed-wall result
000110 | in that source concerns a fixed c-number background, not operator-valued
000111 | quantum links; its scope is explicit. No new field or parent is needed to
000112 | obtain canonical coordinates for the given DET vacuum.
000113 | 
000114 | 3A. POSITIVE APPLICATION TO THE WHOLE ORIGINAL R104 SOURCE
000115 | 
000116 | This gives a concrete full field/vacuum identification inside the SAME
000117 | original Hamiltonian class, not merely a conditional single-factor name.
000118 | For a finite original lattice put
000119 |  T=I_R tensor product_x U_det,x^*, H'=THT^*, P_G'=T P_G T^*.
000120 | Then EXACTLY
000121 |  H'=H_R+2 sum_x(I-P_0,x).
000122 | For every original field/gauge insertion O put O'=T O T^*. For an ordered
000123 | Euclidean word, cyclic trace invariance proves, including its scalar,
000124 |  Tr[P_G e^(-s_0 H) O_1 e^(-s_1 H)...O_k e^(-s_k H)]
000125 |  =Tr[P_G' e^(-s_0 H') O_1' e^(-s_1 H')...O_k' e^(-s_k H')].
000126 | The same equality holds with the actual original reference Hamiltonian,
000127 | reference Gauss data and reference source transported by T. Thus the
000128 | unnormalized source, normalized source, partition function and actual
000129 | relative quotient all agree. No denominator or vacuum normalization is
000130 | freely selected. This equality holds for every beta and every insertion;
000131 | it does not assert that beta has a physically selected value.
000132 | 
000133 | No commutation of U_det with the original gauge representation is needed
000134 | for this theorem. If its actual action is G(g), use G'(g)=TG(g)T^*.
000135 | Then P_G' is the SAME Gauss projector in new coordinates, and any relation
000136 | G(g)psi=psi is equivalent to G'(g)Tpsi=Tpsi. Likewise
000137 |  alpha_t'(O')=T alpha_t(O) T^*, Theta'=T Theta T^*
000138 | for the actual original time and anti-linear reflection. Their original
000139 | relations, grades, positivity and source support are transported, not
000140 | chosen again. Every ground spectral projection is transported:
000141 |  1_B(H')=T1_B(H)T^*.
000142 | Thus the full physical ground-state SET and its word functional are
000143 | equivalent in the two coordinate systems. No phase or state is discarded.
000144 | 
000145 | The distinction of field legs is essential:
000146 |  original c_i -> U_det^* c_i U_det;
000147 |  dressed a_i=U_det c_i U_det^* -> canonical c_i.
000148 | Original c sources are high-degree local operators in the transported
000149 | chart; dressed a sources are canonical c sources. Both insertion catalogs
000150 | are explicit and exact. Under the original determinant-singlet premise
000151 | the onsite U_det is gauge invariant, so the dressed a carry the same
000152 | original charge representation. Otherwise the full gauge action still
000153 | transports exactly, but it must not be left fixed by declaration.
000154 | 
000155 | In the local pure mirror ground state phi, the dressed a are a Fock vacuum
000156 | and obey equal-time vacuum Wick identities. This does NOT make their
000157 | time-dependent full source a free additive-parent source. For i!=j,
000158 |  psi_ij=a_i^*a_j^*phi is normalized and orthogonal to phi, hence
000159 |  <psi_ij,e^(-t H_m)psi_ij>=exp(-2t).
000160 | The additive coefficient2 number parent would instead give exp(-4t).
000161 | The original flat-parent Heisenberg evolution is retained, including its
000162 | nonlinear action on multiparticle fields. This is the smallest decisive
000163 | fingerprint that the positive coordinate transport did not switch the
000164 | original parent. This local matrix element is not promoted to a physical
000165 | Gauss-invariant field unless its required charge dressing is included.
000166 | 
000167 | All these statements are exact finite-dimensional algebra on the actual
000168 | old source, so no numerical check is required. Products of the even
000169 | onsite maps define a quasilocal CAR automorphism. A thermodynamic state
000170 | is pulled back by that automorphism; its GNS spaces have the corresponding
000171 | state-preserving unitary. An infinite product need not be a unitary in
000172 | one preselected old Fock representation. At the original fixed phase
000173 | the local map is field independent. If that phase is instead varied as
000174 | an actual background field, the full transformed kinetic terms contain
000175 | the corresponding derivative connection; dropping it would not be this
000176 | source transport.
000177 | 
000178 | This closes the actual R104 field/vacuum JOIN in its original given source
000179 | class. It does not choose a particular vector from a degenerate physical
000180 | ground space and does not prove that a separate P1 source is this R104
000181 | source. The section5 comparison is preserved under T, so it remains the
000182 | same physical selection question rather than another vacuum choice.
000183 | 
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000193 | 
000194 | 4. WHAT THE ORIGINAL SEAM AND GAUGE DICTIONARIES ACTUALLY SAY
000195 | 
000196 | The original admissibility text defines
000197 |  P_str=P_Sigma,+ P_sing P_Theta,
000198 |  P_sing=(I+Z_c+Z_c^2)/3, P_Theta=(I+Q_Theta)/2.
000199 | Here P_sing is the COLOR centre projection, and Q_Theta is its declared
000200 | determinant involution on the original tensor factor. It is not written