TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/CLAUDE33-FACE-GRAPH-AND-TIME-REVIEW.txt Original file SHA-256: c106777f365804a4f3a20d880eb88560f4f5e567b2713040b0fa7aff88b77a87 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 | [OMITTED: non-scientific conversation or execution metadata] 000002 | [OMITTED: non-scientific conversation or execution metadata] 000003 | [OMITTED: non-scientific conversation or execution metadata] 000004 | 000005 | VERDICT 000006 | 000007 | The proposed annulus operator is an exact whole-source covariance to the 000008 | already derived q-to-p image source. Put 000009 | 000010 | K_F=Kswap, T=K_F R_x, 000011 | Theta_-=P_r R_x tensor Z tensor gamma_F K_F. (1) 000012 | 000013 | On the smooth constant-frame annulus it carries 000014 | 000015 | D_q=N partial_r+S A_q+Z Phi_q 000016 | 000017 | to D_p with A_p=-R_x A_q R_x^-1 and 000018 | Phi_p=-bar(Phi_q) composed with inv. It needs no second freely chosen 000019 | background. On the boundary it sends B_q to -B_p; the radial face swap then 000020 | turns the whole two-face graph into the standard target graph G(B_p). 000021 | 000022 | [OMITTED: non-scientific conversation or execution metadata] 000023 | [OMITTED: non-scientific conversation or execution metadata] 000024 | [OMITTED: non-scientific conversation or execution metadata] 000025 | [OMITTED: non-scientific conversation or execution metadata] 000026 | [OMITTED: non-scientific conversation or execution metadata] 000027 | [OMITTED: non-scientific conversation or execution metadata] 000028 | 000029 | Consequently the geometric spin condition r^6=-I selects Theta_- only after 000030 | an actual spinframe-to-F5 field intertwiner fixes the remaining i_old phase. 000031 | The operator and graph identities alone do not fix it. Exact commutation with 000032 | the displayed J_F would select i_old Theta_-, whereas allowing J_F to go to 000033 | -J_F retains Theta_-; the two signs implement the same opposite-algebra 000034 | action but differ as fixed reference operators. C_faces does not distinguish 000035 | the two because i_old is OLD-complex-linear and commutes with C_faces. 000036 | 000037 | The actual circle evolution is transported with the SAME parameter t. The 000038 | quarter domain is carried from alpha to -alpha modulo one. The smooth annulus 000039 | gluing is preserved when the radial and tangential inversions act together, 000040 | but the original Fuchs corner/reference domain still requires its explicit 000041 | endpoint transport. No P1 execution or physical source choice follows. 000042 | 000043 | 1. WHOLE COLLAR OPERATOR 000044 | 000045 | Use the actual normal matrices 000046 | 000047 | N=[[0,-1],[1,0]], Z=diag(1,-1), S=N Z. (2) 000048 | 000049 | The radial pullback P_r reverses partial_r. The tangential pullback obeys 000050 | 000051 | R_x A_q R_x^-1=-A_p. (3) 000052 | 000053 | Conjugation by the normal Z reverses N and S and preserves Z. On the finite 000054 | coefficient, the actual v252 matrices give 000055 | 000056 | K_F gamma_F K_F^-1=-gamma_F, 000057 | K_F Phi_q K_F^-1=bar(Phi_q), 000058 | (gamma_F K_F) Phi_q (gamma_F K_F)^-1=-bar(Phi_q). (4) 000059 | 000060 | The last equality uses oddness of Phi_q. The three terms therefore transform 000061 | as 000062 | 000063 | N partial_r -> N partial_r, 000064 | S A_q -> S A_p, 000065 | Z Phi_q -> Z[-bar(Phi_q) composed with inv]. (5) 000066 | 000067 | This proves 000068 | 000069 | Theta_- D_q Theta_-^-1=D_p. (6) 000070 | 000071 | The plus lift with finite factor K_F would instead target +bar(Phi_q). Both 000072 | are algebraically valid family maps, but only (1) reaches the negative target 000073 | already independently derived from Ucal. That agreement fixes the target 000074 | sign for this source map; it does not by itself fix the field-lift phase. 000075 | 000076 | 2. F5 TYPES AND THE TWO REAL STRUCTURES 000077 | 000078 | On one boundary face retain 000079 | 000080 | Gamma=Z tensor gamma_F, Ical=i_old Gamma, 000081 | C0=i_old N tensor I. (7) 000082 | 000083 | Let W=R_x tensor Z tensor gamma_F K_F denote the boundary part of Theta_-. 000084 | Direct multiplication gives 000085 | 000086 | W Gamma W^-1=-Gamma, 000087 | W Ical W^-1=-Ical, 000088 | W C0 W^-1=-C0, 000089 | W^2=-I. (8) 000090 | 000091 | Thus W is NEW-antilinear. On both faces use 000092 | 000093 | C_faces=diag(C0,-C0), 000094 | Theta_faces=face_swap tensor W. (9) 000095 | 000096 | The face swap reverses the two diagonal signs while W reverses C0, hence 000097 | 000098 | Theta_faces C_faces Theta_faces^-1=C_faces. (10) 000099 | 000100 | This is the relevant selfdual CAR condition. It proves that Theta_faces 000101 | defines a star-preserving order-preserving map on the complete selfdual field 000102 | algebra. C0 alone is not preserved on one face, so calling Theta_- a 000103 | one-face Majorana symmetry would be wrong. 000104 | 000105 | The finite KO6 antiunitary is J_F=K_F kappa_split. All matrices in (1) are 000106 | real in the checked chart. Since K_F anticommutes with gamma_F, 000107 | 000108 | J_F Theta_- J_F^-1=-Theta_-, 000109 | J_F(i_old Theta_-)J_F^-1=i_old Theta_-. (11) 000110 | 000111 | [OMITTED: non-scientific conversation or execution metadata] 000112 | [OMITTED: non-scientific conversation or execution metadata] 000113 | [OMITTED: non-scientific conversation or execution metadata] 000114 | [OMITTED: non-scientific conversation or execution metadata] 000115 | [OMITTED: non-scientific conversation or execution metadata] 000116 | [OMITTED: non-scientific conversation or execution metadata] 000117 | 000118 | As in the preceding q-to-p review, K_F also transports the displayed finite 000119 | algebra representation to pi_p(a)=K_F pi_q(a)K_F^-1; it does not commute with 000120 | generic pi_q(a). Thus (6) is a covariance of finite source triples. A 000121 | same-marked-algebra symmetry still needs the original algebra automorphism 000122 | which identifies those two bimodule actions. 000123 | 000124 | 3. RELATION TO R_GEO Ucal AND THE FIELD/ADJOINT MAP 000125 | 000126 | The actual normal identity is 000127 | 000128 | S(i_old N)=i_old Z. (12) 000129 | 000130 | With R_geo=face_swap tensor S tensor gamma_F and 000131 | Ucal=i_old N tensor K_F R_x, it follows exactly that 000132 | 000133 | R_geo Ucal=i_old Theta_faces. (13) 000134 | 000135 | The old scalar i_old cancels from operator conjugation, which explains why 000136 | both sides give the same D, B and graph transport. It does not cancel from 000137 | the one-particle field map or its square. 000138 | 000139 | This can be seen in the actual charged chart. Put 000140 | 000141 | P_plus/minus=(I plus/minus gamma_F)/2, 000142 | E v=(P_plus v,i_old P_minus v), 000143 | T=K_F R_x. (14) 000144 | 000145 | Then 000146 | 000147 | W E=-C0 E T, 000148 | W C0 E=E T. (15) 000149 | 000150 | Hence W exchanges field and adjoint slots. In representative notation the 000151 | complete T_faces map is 000152 | 000153 | (v0,v1,bar(w0),bar(w1)) 000154 | ->(-T w1, T w0, overline(-T v1), overline(T v0)). (16) 000155 | 000156 | The signs in (16) include the fixed opposite C0 convention on the second 000157 | face. This is a genuine Bogoliubov particle/adjoint exchange, not Ucal acting 000158 | separately on two unchanged charged entries. 000159 | 000160 | Because Theta_faces commutes with C_faces, its induced CAR map sends a field 000161 | c(u) to c(Theta_faces u), preserves the star and preserves product order. 000162 | Reordering the images of two odd fields into a preferred q-before-p order 000163 | still produces the forced Koszul minus from the preceding CAR/U6 review. 000164 | Moreover 000165 | 000166 | alpha_Theta^2=fermion parity, 000167 | 000168 | whereas the lift i_old Theta has square +I on one-particle fields. Thus the 000169 | phase ambiguity is invisible to B and G but visible on odd fields and their 000170 | ordered products. An explicit spinframe/F5 field intertwiner is required 000171 | before the geometric r^6=-I can decide between them. 000172 | 000173 | 4. FACE-SWAPPED GRAPH AND STATE 000174 | 000175 | The boundary operator is 000176 | 000177 | B(A,Phi)=Z tensor A-S tensor Phi. (17) 000178 | 000179 | Equations (3)-(4) give 000180 | 000181 | W B_q W^-1=-B_p. (18) 000182 | 000183 | For the graph functions a(-x)=I-a(x) and c(-x)=c(x). Therefore W on both 000184 | face fibers changes G(B_q) into the face-swapped matrix of G(B_p). The 000185 | explicit face swap in Theta_faces changes it back to the target's standard 000186 | label order: 000187 | 000188 | Theta_faces G(B_q) Theta_faces^-1=G(B_p). (19) 000189 | 000190 | Geometrically the two annulus boundaries have been exchanged; algebraically 000191 | (19) is written in the target source's ordinary face order. The already 000192 | defined pure graph covariance is therefore transported, including both 000193 | crossing blocks. No second covariance or state is added. 000194 | 000195 | A targeted actual96 replay with generic distinct complex full_D values at 000196 | two tangential points gave zero residual for (6), (8), (10), (11), (15) and 000197 | (18). The full graph residual in (19) was 4.88e-14. 000198 | 000199 | 5. DOMAIN, REFERENCE AND ACTUAL TIME 000200 |