TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/SPIN2-ANNULUS-WHOLE-SOURCE-REVIEW.txt Original file SHA-256: 7f6b63b83be1e440fb870885f17a54aea799084be96c7b784d6f7c30774354f7 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 | GEOMETRIC SPIN(2) ANNULUS MAP ON THE WHOLE F5 SOURCE 000002 | Independent bounded operator review, 30 October 2026. 000003 | No Atlas, original or registered report mutation. 000004 | 000005 | VERDICT 000006 | 000007 | The pure geometric Spin(2) lift is an exact and simpler whole source map on 000008 | the smooth product annulus. With the actual F5 normal matrices, put 000009 | 000010 | Theta_spin=P_r R_x tensor(i_old Z) tensor I_96. (1) 000011 | 000012 | Define its target from the given q source itself by 000013 | 000014 | A_p=-R_x A_q R_x^-1, 000015 | Phi_p=R_x Phi_q R_x^-1=Phi_q composed with inv. (2) 000016 | 000017 | Then Theta_spin carries the complete collar operator D_q to D_p. Its 000018 | boundary part sends B_q to -B_p. After exchanging the two radial faces it 000019 | preserves the oriented selfdual real structure and carries the complete 000020 | graph covariance G(B_q) to G(B_p), including both crossing blocks. 000021 | 000022 | This map is OLD complex linear and NEW complex linear. It preserves Gamma 000023 | and Ical, reverses C0 on one face, and squares to -I. In the charged chart 000024 | its exact field map is T=i_old gamma_F R_x. It sends h_q to -h_p and 000025 | therefore reverses the actual circle parameter: 000026 | 000027 | Theta_spin U_q(t) Theta_spin^-1=U_p(-t). (3) 000028 | 000029 | [OMITTED: non-scientific conversation or execution metadata] 000030 | [OMITTED: non-scientific conversation or execution metadata] 000031 | [OMITTED: non-scientific conversation or execution metadata] 000032 | [OMITTED: non-scientific conversation or execution metadata] 000033 | [OMITTED: non-scientific conversation or execution metadata] 000034 | [OMITTED: non-scientific conversation or execution metadata] 000035 | [OMITTED: non-scientific conversation or execution metadata] 000036 | [OMITTED: non-scientific conversation or execution metadata] 000037 | [OMITTED: non-scientific conversation or execution metadata] 000038 | 000039 | The remaining scope is geometric and original source specific. The given 000040 | spin pullback must act on the actual quarter domain and Fuchs reference by 000041 | the stated P_r R_x map. The smooth constant frame identities below do not 000042 | by themselves close the corner domain or select this source family as the 000043 | physical P1 execution. 000044 | 000045 | 1. THE SPIN(2) FACTOR IS THE EXISTING NORMAL CLIFFORD PRODUCT 000046 | 000047 | Retain exactly 000048 | 000049 | N=[[0,-1],[1,0]], Z=diag(1,-1), S=N Z, 000050 | D=N partial_r+S A+Z Phi. (4) 000051 | 000052 | Writing A=-i_old partial_x plus its even connection, the two Clifford 000053 | coefficients of the two oriented derivatives are 000054 | 000055 | c_r=N, c_x=-i_old S. 000056 | 000057 | They square to -I and anticommute. Their even Clifford product is 000058 | 000059 | c_r c_x=N(-i_old S)=i_old Z. (5) 000060 | 000061 | Thus the normal factor in (1) is the ordinary Spin(2) lift of the joint 000062 | pi rotation of the two collar directions. It obeys 000063 | 000064 | (i_old Z)^2=-I, 000065 | (i_old Z)N(i_old Z)^-1=-N, 000066 | (i_old Z)S(i_old Z)^-1=-S, 000067 | (i_old Z)Z(i_old Z)^-1=Z. (6) 000068 | 000069 | The two pullbacks reverse partial_r and send A_q to -A_p. Equations 000070 | (2) and (6) therefore give term by term 000071 | 000072 | N partial_r -> N partial_r, 000073 | S A_q -> S A_p, 000074 | Z Phi_q -> Z Phi_p, 000075 | 000076 | and hence 000077 | 000078 | Theta_spin D_q Theta_spin^-1=D_p. (7) 000079 | 000080 | Phi_p in (2) is the functorial pullback of the already given coefficient. 000081 | It is not another freely supplied Yukawa or Majorana matrix. 000082 | 000083 | 2. ACTUAL F5 TYPES AND THE ORIENTED TWO FACE REAL STRUCTURE 000084 | 000085 | Use the actual v252 grading and the existing F5 definitions 000086 | 000087 | Gamma=Z tensor gamma_F, 000088 | Ical=i_old Gamma, 000089 | C0=i_old N tensor I_96. (8) 000090 | 000091 | For the boundary part W=R_x tensor(i_old Z) tensor I_96 one has 000092 | 000093 | [W,Gamma]=[W,Ical]=0, 000094 | W C0 W^-1=-C0, 000095 | W^2=-I. (9) 000096 | 000097 | Consequently W is NEW complex linear. On the two faces set 000098 | 000099 | C_faces=diag(C0,-C0), 000100 | W_faces=face_swap tensor W. 000101 | 000102 | The face exchange supplies the second sign, so 000103 | 000104 | W_faces C_faces W_faces^-1=C_faces. (10) 000105 | 000106 | This is the required selfdual CAR reality identity. The induced CAR map is 000107 | a complex linear star automorphism. Its square on one particle fields is 000108 | -I and therefore its square on the CAR algebra is fermion parity. 000109 | 000110 | The finite algebra is not exchanged. W acts by the identity on all96 finite 000111 | coefficients. In the charged chart the factor gamma_F below commutes with 000112 | the actual v252 representation, so this route also preserves the displayed 000113 | marked finite algebra instead of transporting the particle and antiparticle 000114 | bimodule actions through Kswap. 000115 | 000116 | The finite antiunitary J_F=Kswap kappa_split anticommutes with W when it is 000117 | extended trivially over the normal slots, because it conjugates i_old. The 000118 | charged operator T below commutes with J_F: conjugating i_old and reversing 000119 | gamma_F give two signs. These are compatible chart identities. J_F is not 000120 | C_faces, and no additional commutation axiom with the whole normal spin lift 000121 | is imposed here. 000122 | 000123 | 3. CHARGED FIELD CHART AND THE MINUS TIME RULE 000124 | 000125 | Let 000126 | 000127 | P_plus/minus=(I plus/minus gamma_F)/2, 000128 | E v=(P_plus v,i_old P_minus v), 000129 | h(A,Phi)=A gamma_F-i_old gamma_F Phi. (11) 000130 | 000131 | Direct multiplication gives the exact intertwiner 000132 | 000133 | W E=E T, T=i_old gamma_F R_x. (12) 000134 | 000135 | Because R_x sends A_q to -A_p and gamma_F anticommutes with the odd 000136 | coefficient Phi, one obtains 000137 | 000138 | T h(A_q,Phi_q) T^-1=-h(A_p,Phi_p). (13) 000139 | 000140 | Also T^2=-I. In the oriented two face chart 000141 | 000142 | T_faces(v0,v1,bar(w0),bar(w1)) 000143 | =(E v0+C0 E w0, E v1-C0 E w1), 000144 | 000145 | the complete face swapped map is 000146 | 000147 | (v0,v1,bar(w0),bar(w1)) 000148 | ->(T v1,T v0,overline(T w1),overline(T w0)). (14) 000149 | 000150 | Thus fields remain fields and adjoints remain adjoints. There is no hidden 000151 | Bogoliubov exchange. Equation (14) also shows explicitly that every ordered 000152 | CAR product is transported without introducing a new word reversal. 000153 | 000154 | For the actual circle unitary U(t)=exp(t Ical B), equations (9) and the 000155 | boundary identity in the next section give 000156 | 000157 | W_faces(Ical B_q)W_faces^-1=-Ical B_p, 000158 | W_faces U_q(t)W_faces^-1=U_p(-t). (15) 000159 | 000160 | The same result follows from (13) in the charged chart. The parameter is 000161 | therefore minus t. Calling this plus time would silently replace the present 000162 | Spin(2) lift by the different NEW antilinear K_F construction. 000163 | 000164 | 4. COMPLETE BOUNDARY GRAPH AND STATE TRANSPORT 000165 | 000166 | The boundary operator is 000167 | 000168 | B(A,Phi)=Z tensor A-S tensor Phi. (16) 000169 | 000170 | Equations (2) and (6) give 000171 | 000172 | W B_q W^-1=-B_p. (17) 000173 | 000174 | For the existing graph functions 000175 | 000176 | a(-x)=I-a(x), c(-x)=c(x). 000177 | 000178 | Therefore G(-B_p) is exactly G(B_p) with its two face labels exchanged. 000179 | The explicit face swap in W_faces restores the target's standard order: 000180 | 000181 | W_faces G(B_q)W_faces^-1=G(B_p). (18) 000182 | 000183 | This includes both diagonal occupations and both crossing blocks. It 000184 | transports the already given graph covariance and its CAR state premise. 000185 | It does not choose another covariance, another Gaussian state or another p 000186 | background. 000187 | 000188 | 5. QUARTER DOMAIN, CLUTCH AND REFERENCE 000189 | 000190 | The tangential pullback still sends 000191 | 000192 | alpha_p=-alpha_q modulo 1, U_p=U_q^-1. (19) 000193 | 000194 | Hence the periodic 1/4 chart is transported to the 3/4 chart. The radial 000195 | pullback reverses the annulus clutch orientation. For a constant frame the 000196 | same displayed scalar clutch is recovered when 000197 | 000198 | R_x u_q R_x^-1=u_q^-1. (20) 000199 | 000200 | In general the target domain is the functorial image Theta_spin Dom(D_q),