TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/DET16-ORIGINAL-MAP.txt Original file SHA-256: f36de20b0c3996161122372291c9869d7bb103b3464a3c42f53ee677868b5f3f Inclusive original lines: 1-200 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000001 | J: ORIGINAL DET16 FIELD, GAUSS AND VACUUM MAP 000002 | 3 November 2026. origin_topology independent source review. 000003 | Entry: plan90; Atlas457bcebe946f1d5ac3b6f403. 000004 | Read only canonical sources. No new Hamiltonian, gauge representation, 000005 | temperature selection, numerical run or source harness. G/H/I are retained. 000006 | 000007 | RESULT 000008 | 000009 | The original DET16 model has a complete finite Fock field definition and 000010 | R104 already gives its whole non-Gaussian ordered source. Later v1027 000011 | also supplies a genuine canonical field dictionary for its chosen local 000012 | vacuum, including the actual charge action. These results must be counted 000013 | as delivered. The original coupled source has its own state/time/reflection 000014 | data; the full G transport carries those data and does not need a new 000015 | independent vacuum choice after that source is supplied. 000016 | 000017 | The literal v1013 DET row is a separate mirror Fock factor. Neither its 000018 | rank-one projector nor its coefficient2 is identified by the reviewed 000019 | originals with the original admissibility projector or P1 collar generator. 000020 | The earliest valid positive arrow here is therefore 000021 | given original R104 whole CAR source + specified H_m 000022 | -> exact WHOLE unitary source transport -> canonical DET vacuum fields. 000023 | It is not an inverse derivation of those choices from P1. The later 000024 | canonicalization repairs the field coordinates and explicitly changes 000025 | the parent if its additive number operator is used. 000026 | 000027 | For the SAME actual R104 Hamiltonian and Gauss projector, the physical 000028 | vacuum problem can now be reduced exactly, without another model, to 000029 | charge-resolved rest energies. This reduction is given in section5. 000030 | It explains precisely when the local DET vacuum is also physically forced 000031 | and preserves the entire rest process rather than renaming it "mirror". 000032 | 000033 | 1. ACTUAL ORIGINAL OPERATORS AND THE COEFFICIENT 000034 | 000035 | v1002 defines sixteen CAR modes on a 2^16 Fock space. With 000036 | |F>=c_0^*...c_15^*|0>, phi=(|0>+exp(i theta)|F>)/sqrt2, 000037 | P=P_phi, Q=I-P, Omega=|0>)/Z_m. 000062 | It retains all CAR words, not just covariance. For instance 000063 | =delta_ij/2, =0 for every beta, 000064 | =1/4+b/(4Z_m) for i!=j, 000065 | =b exp(i theta)/(2Z_m). 000066 | The ordered heat/letter source is reduced exactly by a renewal expansion. 000067 | The actual reference scalar and grading are not replaced by normalized 000068 | two-point data. A covariance match alone would lose the displayed 000069 | 16-field response and the connected density response. 000070 | 000071 | In the literal v1013 table, 000072 | H_Lambda=H_R,Lambda+sum_x 2Q_x. 000073 | The physical edge, two outgoing gauge links and four-state seam clock 000074 | belong to R. The seam charge term couples the clock to the PHYSICAL Q 000075 | density; it does not mix this DET factor into those fields. Per-cell R 000076 | dimension256 is not the dimension of a whole lattice. 000077 | 000078 | Sources: research-source:52f35ed085f7e4e10c1bc207:1-200 and 000079 | research-source:e1f6b2dddcade38e6bd3aad1:201-217, 000080 | DET16-WHOLE-SOURCE-REDUCTION.txt; 000081 | SHAfc43fcfef453bb278b8f5c28123078331a36d1e4350672b45c47a5d99c992897. 000082 | Independent R104-REVIEW.txt:1-57, 000083 | research-source:13876ae098790631ccdd50be; 000084 | SHAb2706681f27fda41d93fe760499f45ed561cc6d4da980a320b366b30a64110e8. 000085 | Existing R104 execution evidence is reused; it was not rerun for J. 000086 | 000087 | 3. THE LATER FIELD AND CHARGE DICTIONARY REALLY EXISTS 000088 | 000089 | [OMITTED: non-scientific conversation or execution metadata] 000090 | [OMITTED: non-scientific conversation or execution metadata] 000091 | [OMITTED: non-scientific conversation or execution metadata] 000092 | [OMITTED: non-scientific conversation or execution metadata] 000093 | [OMITTED: non-scientific conversation or execution metadata] 000094 | [OMITTED: non-scientific conversation or execution metadata] 000095 | [OMITTED: non-scientific conversation or execution metadata] 000096 | [OMITTED: non-scientific conversation or execution metadata] 000097 | [OMITTED: non-scientific conversation or execution metadata] 000098 | [OMITTED: non-scientific conversation or execution metadata] 000099 | [OMITTED: non-scientific conversation or execution metadata] 000100 | [OMITTED: non-scientific conversation or execution metadata] 000101 | [OMITTED: non-scientific conversation or execution metadata] 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 | =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 | 000184 | [OMITTED: non-scientific conversation or execution metadata] 000185 | [OMITTED: non-scientific conversation or execution metadata] 000186 | [OMITTED: non-scientific conversation or execution metadata] 000187 | [OMITTED: non-scientific conversation or execution metadata] 000188 | [OMITTED: non-scientific conversation or execution metadata] 000189 | [OMITTED: non-scientific conversation or execution metadata] 000190 | [OMITTED: non-scientific conversation or execution metadata] 000191 | [OMITTED: non-scientific conversation or execution metadata] 000192 | [OMITTED: non-scientific conversation or execution metadata] 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