TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/source_unification_20261027/review/PROJECTOR-CAR-DEFECT-LINE-REVIEW.txt Original file SHA-256: a74aad8f91c90bcbf26e6afbbe7567bd28a540a29cb71008422e44cdccd7f457 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 | PROJECTOR CAR DEFECT LINE REVIEW 000002 | The full CAR state fixes all words but its ket line is dual to the Hardy Fredholm line 000003 | 2026-10-27 000004 | 000005 | VERDICT 000006 | 000007 | The full projector covariance theorem applies and gives a genuine positive 000008 | result: once the original three-sheet boundary covariance is the shifted 000009 | projection, positivity fixes the complete charged CAR state and every CAR 000010 | word. No further Gaussian or higher-moment choice remains. 000011 | 000012 | The proposed determinant-line identification has the opposite orientation 000013 | for the already fixed source winding. In the old source convention the 000014 | occupied covariance is the negative sea C, while the positive Hardy 000015 | projection is its complement P_+=I-C. For the same ambient unitary U=M_z, 000016 | 000017 | U C U^*=C+Pi_V, 000018 | P_+-U P_+ U^*=Pi_V. (1) 000019 | 000020 | Thus the source shift fills the three-dimensional defect V. Its shifted 000021 | vacuum ket lies in det(V), with beta character -i. The Hardy Fredholm line 000022 | is det(V)^(-1), with beta character +i. It is the dual bra or anomaly line, 000023 | not the shifted ket line itself. 000024 | 000025 | The state functional is phase blind. Its complete CAR correlations therefore 000026 | do not choose between these two dual lines. A direct connection to the 000027 | inverse q norm is valid only if the original sewing or response functional 000028 | uses the dual bra, determinant or Pfaffian orientation. The choice between 000029 | covariant ket and contravariant dual is not supplied merely by projector 000030 | covariance and positivity. 000031 | 000032 | 000033 | 1. THE TWO PROJECTION FORMULAS HAVE OPPOSITE ROLES 000034 | 000035 | On the old oriented circle, with 0. 000141 | 000142 | Under Xi -> exp(i phi)Xi, the bra and ket phases cancel. The same cancellation 000143 | occurs in every covariance and every higher CAR word. Projector-state 000144 | rigidity therefore determines the ray and all state correlations, but it 000145 | cannot decide whether a separate charged boundary amplitude is represented 000146 | covariantly by the ket line or contravariantly by its dual. 000147 | 000148 | This is also visible in the old rank-one calculation. The source winding 000149 | creates c_0^* Omega, while the directed relative Fredholm determinant can be 000150 | taken in either order and gives 1-z or (1-z)^(-1). The state fixes the 000151 | rank-one covariance defect; the response contract fixes which directed 000152 | determinant is read. 000153 | 000154 | Consequently the full projector theorem removes every independent 000155 | higher-correlation or Gaussian assumption. It does not remove the typed 000156 | boundary-amplitude orientation needed to identify the CAR defect with 000157 | N_q^(-1). 000158 | 000159 | 000160 | 5. SMALLEST REMAINING SOURCE TEST 000161 | 000162 | No new covariance or model is required. The exact remaining question is: 000163 | 000164 | Does the original boundary, sewing, eta or Pfaffian functional pair with 000165 | the shifted CAR vacuum as a bra and therefore take values in 000166 | det(V)^(-1), or does it use the ket line det(V)? 000167 | 000168 | If the original amplitude is a covector on the shifted charged sector, then 000169 | (9) gives the desired inverse norm line, its beta phase +i and its odd 000170 | Fredholm parity directly from the actual CAR defect. If the source only 000171 | supplies the positive state omega_Xi, its phases cancel and no norm-line 000172 | character follows. 000173 | 000174 | An explicit reversal that declares the occupied projection to be P_H would 000175 | also produce a hole ket in det(V)^(-1), but that reversal is not licensed by 000176 | the inherited source winding. It would have to be proved by an actual 000177 | orientation-changing source map rather than introduced to match the desired 000178 | line. 000179 | 000180 | Thus the new CAR connection is real and useful: it identifies the Hardy 000181 | Fredholm line as the dual of the unique charged three-mode defect and fixes 000182 | all its correlation functions. It does not yet prove that the physical 000183 | source reads that dual rather than the ket. 000184 | 000185 | 000186 | SOURCES CHECKED 000187 | 000188 | * experiments/theory-contracts/native-arc-charge-source-interface-20261026/ 000189 | PROJECTOR-STATE-RIGIDITY.txt 000190 | 000191 | * experiments/theory-contracts/original-admissible-source-closure-20261025/ 000192 | FOCK-DEFECT-REVIEW.txt 000193 | 000194 | * experiments/theory-contracts/original-admissible-source-closure-20261025/ 000195 | DIRECT-DEFECT-CLOSURE.txt 000196 | 000197 | * research-20261027-boundary-origin/uniqueness/ 000198 | SAME-SPIN-REFERENCE-SIMPLIFICATION.txt 000199 | 000200 | * research-20261027-boundary-origin/source/