TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/theta_lattice_closure_20261027/eta/P1-ETA-SEWING-DUAL-LINE-TEST.txt Original file SHA-256: 350063b81d790219be7d19b1f13d0b3b76a6513930b82f45d5bb97cafc10711a 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 | P1 ETA AND SEWING DUAL LINE TEST 000002 | The dual defect module is canonical; the physical P1 readout is not yet typed as that dual 000003 | 2026-10-27 000004 | 000005 | Atlas audit frame: ab02252b5bdcfccd5fb5cf3d 000006 | 000007 | 000008 | VERDICT 000009 | ======= 000010 | 000011 | There are two different questions, and the old wording made the second one a 000012 | condition for the first. 000013 | 000014 | 1. Abstract module construction. Once the original negative-sea convention 000015 | 000016 | C=I-P_H, U C U^*=C+Pi_V, 000017 | 000018 | has produced the three-dimensional charged defect V, the two relative 000019 | vacuum lines 000020 | 000021 | K=det(V), K^*=det(V)^(-1) (1) 000022 | 000023 | and their perfect CAR pairing are functorial. No eta term and no separately 000024 | selected physical state are needed to construct K^*. For the existing 000025 | marks, 000026 | 000027 | beta|K=-i, beta|K^*=+i, 000028 | c|K=c|K^*=1. (2) 000029 | 000030 | The Q character includes the already proved q_1,q_2 transposition, or, 000031 | equivalently, the B_O ordering factor in the family formula. Thus the 000032 | dual line required by the native complex module already exists as a 000033 | consequence of the CAR defect and adjunction. Asking the P1 eta term to 000034 | choose it is an unnecessary extra condition for the abstract module 000035 | reconstruction. 000036 | 000037 | 2. Physical source readout. In the strongest original formulas checked here, 000038 | the P1 eta and sewing data are scalars, phases, positivity forms, or index 000039 | data. None is written as a complex-linear outgoing boundary covector with 000040 | values in K^*. Therefore those formulas do not yet prove that the physical 000041 | P1 response evaluates the charged ket through the dual line. This is a 000042 | typing gap for the physical amplitude, not a gap in the construction of the 000043 | dual representation. 000044 | 000045 | The exact positive conditional statement is small. If the original outgoing 000046 | seam amplitude is a covector on the shifted negative-sea sector, then it lies 000047 | in K^*, pairs with the charged ket in K to a scalar, and has beta character +i. 000048 | No reversal of the occupied projection is needed. The sources read in this 000049 | audit do not state that outgoing-covector typing. 000050 | 000051 | 000052 | 1. THE CAR ADJOINT CONVENTION AND THE DUAL LINE 000053 | =============================================== 000054 | 000055 | Use a complex Hilbert product conjugate-linear in the first variable and 000056 | linear in the second. Creation a^*(v) is linear in v and annihilation a(v) is 000057 | conjugate-linear in v. For an orthonormal frame v_1,v_2,v_3 of V, 000058 | 000059 | Xi=a^*(v_1)a^*(v_2)a^*(v_3)Omega 000060 | 000061 | spans K=det(V). The Hilbert adjoint gives the bra 000062 | 000063 | R(Xi): K -> C, R(Xi)(xi)=. (3) 000064 | 000065 | For fixed Xi, (3) is complex-linear in xi, so R(Xi) belongs to K^*. The Riesz 000066 | map R:K->K^* is conjugate-linear. Under a unitary symmetry g, 000067 | 000068 | g|K=det(g|V), g|K^*=det(g|V)^(-1). (4) 000069 | 000070 | This is the precise reason that beta=i I_V gives -i on K and +i on K^*. 000071 | 000072 | The often used annihilation expression 000073 | 000074 | (5) 000075 | 000076 | is conjugate-linear in the vector arguments w_j with the convention above. 000077 | It is the Hermitian adjoint form of the pairing. It should not silently be 000078 | called a complex-linear element of det(V)^*. The fully algebraic 000079 | complex-linear statement uses covectors phi_j in V^* and the canonical 000080 | contractions 000081 | 000082 | det(V)^* tensor det(V) -> C, 000083 | (phi_1 wedge phi_2 wedge phi_3, v_1 wedge v_2 wedge v_3) 000084 | |-> det(phi_i(v_j)). (6) 000085 | 000086 | Equations (3) and (6) agree after the anti-linear Riesz identification 000087 | V ~= V^*. They prove the existence and character of the dual line. They do 000088 | not select a preferred nonzero section of that line. A ray or positive state 000089 | also cannot select such a section, because its overall phase is absent from 000090 | all state expectations. This phase fact is irrelevant to the abstract module, 000091 | which needs the line and its group action rather than a distinguished vector. 000092 | 000093 | The full family version is the already checked identity 000094 | 000095 | K=B(E)=N_q(S_0) tensor B_O, 000096 | K^*=B(E)^(-1)=N_q(S_0)^(-1) tensor B_O^(-1). (7) 000097 | 000098 | Both are odd superlines. Dualization reverses the central complex character 000099 | but preserves the odd parity. Dropping B_O is permitted only in the already 000100 | scoped ungraded module comparison where its effect is the residual alpha sign; 000101 | it is not an equality of graded determinant lines. 000102 | 000103 | 000104 | 2. WHAT THE ACTUAL P1 FORMULAS PROVIDE 000105 | ====================================== 000106 | 000107 | 2.1 The old fullv21 seam gives the winding, not a covector 000108 | ---------------------------------------------------------------- 000109 | 000110 | The old source cuts the double cover, equips the two seam faces with a unitary 000111 | matching operator, and then chooses 000112 | 000113 | D_Gamma=i d/dtheta, U_Gamma(theta)=exp(i theta). 000114 | 000115 | The interpolation of the boundary condition from phi=0 to 2pi has spectral 000116 | flow one. This fixes the directed one-step shift on the same field space. It 000117 | does not define a Fock boundary ket, an outgoing bra, or a determinant-line 000118 | valued sewing map. 000119 | 000120 | Source: 000121 | 000122 | Atlas paper-source:1fe2ddfabc74ac0e94a4686d 000123 | documents/archive--paper-latex--tfpt-theory-fullv21/sections/ 000124 | 001-section-introduction-the-problem-of-parameters.tex:198-255 000125 | SHA256 85bb11511be05385a7cadf6071dbede216b12befeb9d6767bc4460748cbec5d7 000126 | 000127 | 2.2 The primitive eta contribution is a scalar action factor 000128 | ------------------------------------------------------------ 000129 | 000130 | The later primitive action is 000131 | 000132 | S_prim=Tr f(D_rel/chi)-Tr f(D_ref/chi)+(i pi/2) Delta eta. 000133 | 000134 | Its contribution to exp(-S_prim) is therefore 000135 | 000136 | a=exp(-i pi Delta eta/2). (8) 000137 | 000138 | With the already checked Dai-Freed conventions and vanishing zero modes, 000139 | 000140 | a^2=tau_rel^(-1), (9) 000141 | 000142 | not a=tau_rel. An actual Pfaffian or square-root determinant interpretation 000143 | would additionally need the real structure, gluing orientation and kernel 000144 | correction. The source writes (8) as a complex scalar in the action. It does 000145 | not type it as a section of K or K^*, and it contains none of the beta,c,Q 000146 | actions needed to make that identification. 000147 | 000148 | The scalar is compatible with a hidden evaluation pairing K^* tensor K to C, 000149 | as every correctly typed closed sewing amplitude must be. Compatibility is 000150 | not selection: from the closed scalar alone one cannot reconstruct which cut 000151 | face carries K, which carries K^*, or how either face carries beta,c,Q. The 000152 | Gamma_+,Gamma_- labels in fullv21 are therefore the right candidate 000153 | incoming-outgoing orientation, but the source does not extend them to a 000154 | line-valued open amplitude. 000155 | 000156 | Source: 000157 | 000158 | Atlas paper-source:8027564739689dd02e02f11c 000159 | documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/ 000160 | sections/001-section-primitive-core-and-carrier-interface-handoff.tex:302-318 000161 | SHA256 7d6e673d99c29d950dc601283cd51c1cc0b113567eeb79c8396fec9c82a02c79 000162 | 000163 | Convention audit used: 000164 | 000165 | experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/ 000166 | boundary_origin_20261027/ETA-CONVENTION-AND-BOUNDARY-FACTOR.txt 000167 | SHA256 198cc720a955d7f6ece859ee324e73fc6e2a2a7fcfc9d51f8eeca22142faeab1 000168 | 000169 | 2.3 The relative generating functional is scalar and positivity is phase blind 000170 | ----------------------------------------------------------------------------- 000171 | 000172 | The original closure source normalizes 000173 | 000174 | Z_rel[J,eta,bar eta]=Z_adm[J,eta,bar eta]/Z_ref[0,0,0]. (10) 000175 | 000176 | The denominator is declared to be a fixed positive number. Fermionic 000177 | reflection positivity is then a nonnegative Pfaffian or determinant quadratic 000178 | form on positive-time test spinors. Equation (10) is a scalar normalized 000179 | functional. The positivity statement fixes neither a complex phase nor the 000180 | contravariant orientation of a charged boundary line. It can coexist with 000181 | the ket K, the dual K^*, and their invariant pairing. 000182 | 000183 | Source: 000184 | 000185 | Atlas paper-source:032c510c1e9343d3edda1ae5 000186 | documents/archive--paper-latex--tfpt-42/sections/ 000187 | 022-section-closure-theorems-from-the-admissibility-complex.tex:258-315 000188 | SHA256 fe626eb8126375f8e0028914a19e045802478f76f0c9b8e6655b8331b70848b2 000189 | 000190 | The same section states at 219-245 that the harmonic determinant line is 000191 | parallel and hence has SU(3) family holonomy. Parallelity does not choose 000192 | between a line and its dual and does not define an outgoing seam covector. 000193 | 000194 | 2.4 The named determinant formulas still do not provide the typing 000195 | ----------------------------------------------------------------- 000196 | 000197 | The strong-CP source defines 000198 | 000199 | a_Sigma=arg det_adm U_Sigma in S^1_det. 000200 |