TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/CLAUDE29-NATIVE-THETA-COMPOSITION-REVIEW.txt Original file SHA-256: da497fe88766278e77cebfbd4c344a9f91b8664f4a5811500450bdd449e10bc3 Inclusive original lines: 1-195 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 | 000004 | No Atlas or original source was changed. 000005 | 000006 | VERDICT 000007 | 000008 | The existing native product reflection does not turn the new chiral E0 map 000009 | into a same-Phi symmetry for generic complex Yukawa and Majorana controls. 000010 | The two antiunitary maps compose to a NEW-complex-linear Bogoliubov map, but 000011 | its target is still the transported source: 000012 | 000013 | raw chiral map: G(B_Phi) -> G(B_(Phi^K)), 000014 | torus-compensated map: G(B_Phi) -> G(B_(bar(Phi))). 000015 | 000016 | The native reflection is a symmetry inside every already fixed graph. It is 000017 | not an additional coefficientwise conjugation Phi' -> Phi. Therefore a 000018 | second antilinearity does not cancel the source transport. 000019 | 000020 | The composition also reverses the circle parameter between the two source 000021 | families and squares to minus one on the one-particle selfdual space. Its 000022 | square is fermion parity on the CAR algebra and the identity only after 000023 | restriction to even observables. No CPT name is justified by these facts. 000024 | 000025 | 1. ORIGINAL THETA, READ IN ITS ACTUAL TYPE 000026 | 000027 | The existing source uses 000028 | 000029 | F=S_normal tensor gamma_F, 000030 | R_geo=face_swap tensor F, 000031 | C_faces=diag(C0,-C0). 000032 | 000033 | On the NEW complex one-particle space R_geo is antiunitary, commutes with 000034 | C_faces and obeys, for every allowed selfadjoint odd Phi, 000035 | 000036 | F B_Phi F=-B_Phi, 000037 | R_geo G(B_Phi) R_geo^-1=G(B_Phi). (1) 000038 | 000039 | The corresponding cyclic antiunitary in the thermal Hilbert-Schmidt 000040 | representation is 000041 | 000042 | J_R=Q_Gamma^* J_HS. 000043 | 000044 | It implements the same geometric reflection on the bounded CAR fields and 000045 | fixes the graph state. It is not J_F and it is not a map which replaces the 000046 | external constructor parameters by their complex conjugates. In particular, 000047 | (1) remains a same-Phi identity even when Y_f and M_R are generic complex 000048 | matrices. 000049 | 000050 | The GNS formula is source specific: J_R acts in the Hilbert-Schmidt 000051 | representation built from h_Phi. Thus the clean composition test is first 000052 | performed with R_geo on the common one-particle selfdual chart. The existing 000053 | graph-to-HS intertwiner then gives the same conclusion for J_R. 000054 | 000055 | 2. SMALLEST COEFFICIENT BLOCK TEST 000056 | 000057 | Let Jcal_E be the chiral E0 antiunitary from the preceding review. In OLD 000058 | complex coordinates it has the form 000059 | 000060 | Jcal_E=Ical D_K kappa, 000061 | 000062 | with the base pullback understood. It satisfies 000063 | 000064 | Jcal_E B_Phi Jcal_E^-1=B_(Phi^K), 000065 | 000066 | where the actual v252 constructor gives 000067 | 000068 | Y'_f=omega S_f bar(Y_f) S_f, 000069 | M'_R=omega^2 S_f bar(M_R) S_f. (2) 000070 | 000071 | The first decisive block is already enough. Since the native F in (1) 000072 | changes B to minus B without changing its coefficient, 000073 | 000074 | (F Jcal_E) B_Phi (F Jcal_E)^-1=-B_(Phi^K). (3) 000075 | 000076 | Equation (3) is not -B_Phi for generic complex controls. The native 000077 | reflection has preserved Phi^K exactly as it preserves every other fixed 000078 | coefficient. It has not applied a second copy of bar(.) to (2). 000079 | 000080 | 3. ALL FACES AND THE TORUS-COMPENSATED MAP 000081 | 000082 | Extend Jcal_E diagonally to the two faces and put 000083 | 000084 | C_raw=R_geo Jcal_E,faces. 000085 | 000086 | Both factors are NEW antiunitary, hence C_raw is NEW complex linear. Both 000087 | commute with C_faces, so C_raw is a correctly typed selfdual Bogoliubov map. 000088 | Functional calculus and (3) give the complete graph identity 000089 | 000090 | C_raw G(B_Phi) C_raw^-1=G(B_(Phi^K)). (4) 000091 | 000092 | This includes both diagonal face blocks and both crossing blocks. 000093 | 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 | 000098 | U_K Phi^K U_K^*=bar(Phi). 000099 | 000100 | The compensated antiunitary 000101 | 000102 | Jhat_E=U_K Jcal_E 000103 | 000104 | is again an involution and transports B_Phi to B_(bar(Phi)). Its composition 000105 | with the native reflection is 000106 | 000107 | C_hat=R_geo Jhat_E,faces, 000108 | C_hat G(B_Phi) C_hat^-1=G(B_(bar(Phi))). (5) 000109 | 000110 | Thus the new torus absorption is fully retained. It sharpens the target from 000111 | Phi^K to bar(Phi), but it does not make (5) a same-Phi identity. Such an 000112 | identity requires Phi=bar(Phi), or a separately allowed equivalence which 000113 | identifies those two fixed backgrounds. The existence of the native Theta 000114 | does not supply that equivalence. 000115 | 000116 | The same conclusion holds in either order. Applying the native reflection 000117 | first leaves G(B_Phi) fixed; applying it second leaves the transported target 000118 | fixed. 000119 | 000120 | 4. SQUARE AND TIME PARAMETER 000121 | 000122 | Direct matrix multiplication gives for both versions 000123 | 000124 | C_raw^2=C_hat^2=-I (6) 000125 | 000126 | on the two-face one-particle space. Therefore the induced CAR map squares to 000127 | the sign automorphism c(u)->c(-u), namely fermion parity. It becomes an 000128 | involution on the even algebra used by the proven reflection-positive form, 000129 | but not on odd fields. 000130 | 000131 | The two antiunitary time rules must also be kept separate. The existing J_R 000132 | obeys 000133 | 000134 | J_R L_HS J_R^-1=-L_HS, 000135 | J_R exp(i t L_HS) J_R^-1=exp(i t L_HS). 000136 | 000137 | The equality with the same parameter in the second line uses the 000138 | antilinearity of J_R. The chiral E0 antiunitary transports h_Phi to its 000139 | target and consequently sends its one-particle circle group to the target at 000140 | -t. Their linear composition therefore obeys 000141 | 000142 | C_raw U_Phi(t) C_raw^-1=U_(Phi^K)(-t), 000143 | C_hat U_Phi(t) C_hat^-1=U_(bar(Phi))(-t). (7) 000144 | 000145 | The composition reverses the parameter as an ordinary complex-linear 000146 | intertwiner. This is not the same statement as the source-internal native 000147 | Theta relation and does not identify circle time with radial OS time. 000148 | 000149 | 5. TARGETED ACTUAL96 REPLAY 000150 | 000151 | A targeted replay imported the original v252 full_D constructor and used 000152 | generic complex Gaussian-integer Yukawa matrices and a generic complex 000153 | symmetric Majorana matrix, seed 2931. It constructed the full 192 dimensional 000154 | B operator and the complete 384 dimensional two-face graph. The results were 000155 | 000156 | ||Phi^K-Phi(equation (2))|| 2.35e-14, 000157 | ||U_K Phi^K U_K^*-bar(Phi)|| 2.14e-14, 000158 | ||F B_Phi F+B_Phi|| 0, 000159 | ||C_raw B_Phi C_raw^-1+B_(Phi^K)|| 1.54e-15, 000160 | raw same-Phi B failure norm 81.16, 000161 | ||C_hat B_Phi C_hat^-1+B_(bar(Phi))|| 2.96e-14, 000162 | compensated same-Phi B failure norm 65.73, 000163 | ||C_raw G_Phi C_raw^-1-G_(Phi^K)|| 3.45e-14, 000164 | raw same-Phi graph failure norm 9.73, 000165 | ||C_hat G_Phi C_hat^-1-G_(bar(Phi))|| 3.41e-14, 000166 | compensated same-Phi graph failure norm 7.89. 000167 | 000168 | The NEW-complex-linearity and C_faces commutators vanished exactly. The 000169 | residuals of (6) against -I were below 6.5e-15. Normal and Python -O output 000170 | were byte identical, SHA256 000171 | 000172 | 0ed44dc485de745defcf321bc9fc61a33efef9dd8c1113bb69512304f17f64a0. 000173 | 000174 | 6. EXACT SCOPE 000175 | 000176 | This calculation proves a complete operator, face, state-graph and circle 000177 | intertwiner between the displayed source family members. It does not select 000178 | the complex controls, prove that the marked original K action admits U_K, 000179 | identify the cyclic circle with physical OS time, or establish a same-source 000180 | symmetry for the physical CKM background. Physical Theta, finite J_F, the F5 000181 | real structure C0 and the E0 family antiunitary remain distinct operations. 000182 | 000183 | Sources 000184 | 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 | [OMITTED: non-scientific conversation or execution metadata] 000194 | [OMITTED: non-scientific conversation or execution metadata] 000195 |