TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/QP-PULLBACK-F5-OPERATOR-REVIEW.txt Original file SHA-256: 85f5d6a0f9975b09ef31b9f68d2b4304b8aad1767775ce9c039565e0a26d33c7 Inclusive original lines: 1-180 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 | Q TO P PULLBACK ON THE ACTUAL F5 SOURCE 000002 | Independent narrow operator review, 30 October 2026. 000003 | No Atlas, original or prior-report mutation. 000004 | 000005 | VERDICT 000006 | 000007 | The proposed formula is correctly typed and exact as a covariance between a 000008 | q source and its explicitly transformed p source. In the actual v252 chart, 000009 | 000010 | K_F=J_F kappa_split=Kswap 000011 | 000012 | is a real complex-linear unitary involution. It reverses gamma_F and sends 000013 | the complete full_D coefficient, including its symmetric Majorana block, to 000014 | its entrywise conjugate. If the old-linear spatial pullback R_x satisfies 000015 | 000016 | R_x A_q R_x^-1=-A_p, 000017 | 000018 | then T=K_F R_x obeys 000019 | 000020 | T h(A_q,Phi_q) T^-1=h(A_p,-bar(Phi_q) composed with inv). (1) 000021 | 000022 | The F5 lift 000023 | 000024 | Ucal=(i N) tensor T (2) 000025 | 000026 | intertwines the actual charged chart, preserves Gamma, Ical and C0, carries 000027 | the complete boundary operator and therefore carries its whole two-face 000028 | graph state. Since T and Ucal are linear, circle evolution uses the SAME 000029 | parameter t. This is not the old E0/C_raw transport in another frame: its 000030 | finite factor, coefficient target and time rule are different. 000031 | 000032 | The result is a source-family covariance. It neither proves that the p tuple 000033 | is the same source as the q tuple nor derives either tuple as the operational 000034 | P1 source. In a periodic quarter chart the pullback sends alpha to -alpha 000035 | modulo one and the holonomy to its inverse. Thus 1/4 is carried to 3/4; there 000036 | is no fixed-quarter conclusion. 000037 | 000038 | 1. ACTUAL FINITE COEFFICIENT 000039 | 000040 | In verification/v252_full_finite_triple.py the finite Real structure is 000041 | 000042 | J_F=Kswap kappa_split, 000043 | 000044 | where kappa_split is ordinary coefficient conjugation in the real signed 000045 | wedge chart and Kswap exchanges the particle and antiparticle 48-blocks. 000046 | Consequently composition with the same kappa_split gives the linear matrix 000047 | 000048 | K_F=J_F kappa_split=Kswap, K_F^2=I. (3) 000049 | 000050 | The actual grading has opposite signs on the two 48-blocks, hence 000051 | 000052 | K_F gamma_F K_F^-1=-gamma_F. (4) 000053 | 000054 | Write the complete allowed coefficient as 000055 | 000056 | Phi=[[S,T_M],[T_M^*,bar(S)]], 000057 | 000058 | where T_M is the symmetric nu_R Majorana block. Direct block exchange gives 000059 | 000060 | K_F Phi K_F^-1=[[bar(S),T_M^*],[T_M,S]]=bar(Phi). (5) 000061 | 000062 | The last equality uses the original physical condition T_M^T=T_M. It is not 000063 | an arbitrary-matrix identity after dropping Majorana symmetry. Equations 000064 | (3)-(5) were also replayed with generic complex Yukawas and a generic complex 000065 | symmetric Majorana input through the actual full_D constructor; both matrix 000066 | residuals vanished exactly in floating arithmetic. 000067 | 000068 | K_F is not a symmetry of the displayed finite algebra representation in 000069 | general. The actual rep() uses the weak/quaternion action on the particle 000070 | block and the colour action on the antiparticle block. Kswap exchanges those 000071 | two bimodule actions. The precise family statement therefore also transports 000072 | 000073 | pi_q(a) -> pi_p(a)=K_F pi_q(a) K_F^-1. (5a) 000074 | 000075 | Calling the map a symmetry of one marked SM source would additionally require 000076 | an original algebra automorphism identifying these two actions. No such 000077 | identification is used below; the proved statement concerns the operator, 000078 | selfdual chart, state graph and time of the transported source family. 000079 | 000080 | 2. CHARGED OPERATOR AND THE FIRST DECISIVE IDENTITY 000081 | 000082 | Use the already established charged generator 000083 | 000084 | h(A,Phi)=A gamma_F-i gamma_F Phi. (6) 000085 | 000086 | The spatial pullback R_x acts on the base/domain and commutes with the finite 000087 | gamma_F and K_F. Let Phi_q composed with inv denote its coefficient pullback. 000088 | Combining the two sign reversals in A and gamma_F with (5) yields 000089 | 000090 | T(A_q gamma_F)T^-1=A_p gamma_F, 000091 | T(-i gamma_F Phi_q)T^-1=+i gamma_F bar(Phi_q composed inv). 000092 | 000093 | This is exactly (1), with target data 000094 | 000095 | Phi_p=-bar(Phi_q) composed with inv. (7) 000096 | 000097 | The minus sign in (7) is part of the transported source. Omitting it would 000098 | break (1); inserting it as an independently chosen p coupling would reverse 000099 | the logical direction of the result. 000100 | 000101 | 3. F5 LIFT, CHARGED CHART AND WHOLE GRAPH 000102 | 000103 | Retain the actual matrices 000104 | 000105 | Z=diag(1,-1), N=[[0,-1],[1,0]], S=N Z, 000106 | Gamma=Z tensor gamma_F, 000107 | Ical=i Gamma, C0=i N tensor I, 000108 | B(A,Phi)=Z tensor A-S tensor Phi. (8) 000109 | 000110 | Let P_plus/minus=(I plus/minus gamma_F)/2 and 000111 | 000112 | E v=(P_plus v, i P_minus v). (9) 000113 | 000114 | Because T reverses gamma_F, while iN exchanges the two normal components, 000115 | direct multiplication gives 000116 | 000117 | Ucal E=E T. (10) 000118 | 000119 | The normal factor iN reverses both Z and S. Together with the A sign and 000120 | (5), it gives 000121 | 000122 | Ucal B(A_q,Phi_q) Ucal^-1=B(A_p,Phi_p). (11) 000123 | 000124 | The two sign reversals also imply 000125 | 000126 | [Ucal,Gamma]=[Ucal,Ical]=[Ucal,C0]=0. (12) 000127 | 000128 | Thus Ucal is NEW-complex-linear and respects the existing selfdual real 000129 | form. On both faces use Ucal, and on the charged T_faces chart use T on the 000130 | two field entries and bar(T) on the two conjugate entries. Equations 000131 | (10)-(12) include the adjoint slots and preserve their fixed relative minus. 000132 | Functional calculus then gives the complete result, including crossing 000133 | blocks, 000134 | 000135 | (Ucal direct-sum Ucal) G(B_q) (Ucal direct-sum Ucal)^-1=G(B_p). (13) 000136 | 000137 | Therefore the pure graph covariance and every CAR word defined from it are 000138 | transported to the p-family member. No new Gaussian covariance is selected. 000139 | 000140 | A targeted actual96 replay used two spatial source points, generic distinct 000141 | complex full_D coefficients and a nonfixed A_q. It obtained zero residuals 000142 | for (3)-(5) and (10)-(12), zero residual for (1) and (11), graph residual 000143 | 4.82e-14, and same-parameter time residual 3.21e-16. 000144 | 000145 | 4. DOMAIN, QUARTER AND TIME 000146 | 000147 | For old-linear reflection/inversion R_x f(x)=f(-x), 000148 | 000149 | R_x(-i partial_x+alpha)R_x^-1 000150 | =-(-i partial_x-alpha). (14) 000151 | 000152 | The domain holonomy is simultaneously inverted. Hence the transported 000153 | periodic chart is 000154 | 000155 | alpha_p=-alpha_q mod 1, U_p=U_q^-1. (15) 000156 | 000157 | In particular alpha_q=1/4 goes to alpha_p=3/4. A statement that the same 000158 | quarter is fixed would require a separate gauge carry and its action on all 000159 | other marked data. It does not follow from R_x. 000160 | 000161 | Because T is complex linear, (1) gives 000162 | 000163 | T exp(i t h_q) T^-1=exp(i t h_p). (16) 000164 | 000165 | The parameter is not reversed. The graph covariance (13) is therefore a 000166 | whole state/time family transport. It becomes a symmetry of one fixed source 000167 | only if the original source data separately identify the p target with the q 000168 | object, including their domains, coefficients and marks. 000169 | 000170 | 5. COMPARISON WITH THE OLD E0 AND C_RAW MAPS 000171 | 000172 | The normal factor iN also occurs when earlier geometric reflections are 000173 | composed, but that does not identify the operations. The old E0 map uses an 000174 | antiunitary finite coefficient D_E kappa P_E, sends Phi to Phi^E and reverses 000175 | the circle parameter. Its C_raw composition remains targeted at Phi^K or, 000176 | after the stated compensator, bar(Phi), and has the previously checked 000177 | minus-time rule. 000178 | 000179 | The present map instead uses the linear Kswap R_x, reverses gamma_F, sends 000180 | the source to (7), and obeys the plus-time rule (16). Equality with an old