TFPT scientific source excerpt Original repository path: experiments/theory-contracts/native-octic-seam-origin-20261025/GEOMETRIC-SOURCE-JET.txt Original file SHA-256: aa409fd5dbce2d3b32639aeb45be5666f0627172a73c933e1c07121a99b59058 Inclusive original lines: 1-130 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 | THE TRANSPORTED GEOMETRIC SOURCE JET AND ITS EIGHTH ORDER OCTIC RESPONSE 000002 | ======================================================================== 000003 | 000004 | Scope: exact for the already supplied quarter CAR source, its eightfold fold, 000005 | and fixed native current dictionary. CAR-OCTIC-DICTIONARY.txt realizes the 000006 | neutral octic directly in the even CAR current algebra, without requiring 000007 | a full E8 root extension. The P1 admissibility conclusion is 000008 | conditional only at the existing interface that identifies this source net, 000009 | state, reflection and time with an object of the primitive P1 seed category. 000010 | No matrix potential, F8 coupling or detector is added. 000011 | 000012 | 1. The actual source diffeomorphism and its one-particle jet 000013 | ------------------------------------------------------------- 000014 | 000015 | Let phi_t be a smooth orientation-preserving diffeomorphism of the original 000016 | cover circle, with lifted periodicity 000017 | 000018 | phi_t(theta+2 pi)=phi_t(theta)+2 pi, 000019 | phi_0(theta)=theta, 000020 | dot phi_0(theta)=v(theta). 000021 | 000022 | For a real smooth profile g on the fold circle, the actual eight-preimage 000023 | lift is 000024 | 000025 | v(theta)=g(8 theta)/8. 000026 | 000027 | Thus a properly supported fold interval lifts to eight separated intervals 000028 | on the original one-channel circle. It is local in the quotient c=8 fold 000029 | net, not point-local in the original c=1 localization. 000030 | 000031 | In the quarter-twisted chart the canonical half-density transport is 000032 | 000033 | (U_t^tw f)(theta)=sqrt(phi_t'(theta)) f(phi_t(theta)). (1) 000034 | 000035 | It is unitary on L2(S1,dtheta), preserves the original quarter boundary 000036 | condition, and maps H1_alpha onto itself. With 000037 | 000038 | A_v^tw=-i(v partial_theta+v'/2), 000039 | 000040 | equation (1) has dot U_0^tw=i A_v^tw. Hence, for the active convention 000041 | 000042 | B_t^tw=U_t^tw B^tw U_t^tw*, B^tw=-i partial_theta, 000043 | 000044 | the exact first jet is 000045 | 000046 | dot B_0^tw=i[A_v^tw,B^tw] 000047 | =i(v' partial_theta+v''/2). (2) 000048 | 000049 | For v=g(8theta)/8 this is 000050 | 000051 | dot B_0^tw=i(g'(8theta) partial_theta+4 g''(8theta)). (3) 000052 | 000053 | The periodic chart is related to the twisted chart by 000054 | 000055 | (Q_alpha f)(theta)=exp(i alpha theta) f(theta), 000056 | B^per=Q_alpha* B^tw Q_alpha=-i partial_theta+alpha. 000057 | 000058 | The transport itself must be gauge conjugated. It is therefore 000059 | 000060 | U_t^per=Q_alpha* U_t^tw Q_alpha, 000061 | (U_t^per f)(theta) 000062 | =exp(i alpha(phi_t(theta)-theta)) 000063 | sqrt(phi_t'(theta)) f(phi_t(theta)), (4) 000064 | 000065 | not the plain half-density pullback on periodic functions. Its generator is 000066 | 000067 | A_v^per=Q_alpha* A_v^tw Q_alpha 000068 | =-i(v partial_theta+v'/2)+alpha v, 000069 | dot U_0^per=i A_v^per. 000070 | 000071 | Thus the same transported operator in the periodic chart has first jet 000072 | 000073 | dot B_0^per=i[A_v^per,B^per] 000074 | =i(v' partial_theta+v''/2)-alpha v'. (5) 000075 | 000076 | For v=g(8theta)/8 equation (5) is 000077 | 000078 | dot B_0^per 000079 | =i(g'(8theta) partial_theta+4 g''(8theta)) 000080 | -alpha g'(8theta). (6) 000081 | 000082 | The signs in (2) and (5) follow from dot U_0=iA_v in their respective 000083 | charts. Choosing instead U_t=exp(-itA_v) reverses both the diffeomorphism 000084 | convention and these first-jet signs. 000085 | 000086 | The finite operator is the already classified scalar first-order source 000087 | fiber. Put x=phi_t and a_t=1/phi_t'. In the twisted chart, 000088 | 000089 | B_t^tw=-i(a_t partial_theta+a_t'/2). (7) 000090 | 000091 | Gauge conjugation gives the equivalent periodic-chart formula 000092 | 000093 | B_t^per=Q_alpha* B_t^tw Q_alpha 000094 | =-i(a_t partial_theta+a_t'/2)+alpha a_t. (8) 000095 | 000096 | which is exactly the half-density normal form in 000097 | SCALAR-FIRST-ORDER-SOURCE-FIBER-AND-SEAM-WARD.txt. Thus this is a canonical 000098 | transport of the original source, not a fitted scalar or matrix deformation. 000099 | 000100 | 2. All source and P1 comparison data must move together 000101 | -------------------------------------------------------- 000102 | 000103 | Functional calculus and unitary covariance give 000104 | 000105 | C_t=U_t C_alpha U_t*, 000106 | Dom(B_t)=U_t Dom(B_alpha), 000107 | 000108 | and the same transport must be applied to the bulk/collar operator, 000109 | reflection, reference, relative operator and their domains. Here U_t and 000110 | B_t are taken together in either the twisted chart or the gauge-equivalent 000111 | periodic chart above. Mixing plain periodic pullback with the twisted-chart 000112 | operator would describe a different transport. 000113 | 000114 | Denote the compatible collar/bulk lift of U_t by U_tilde: 000115 | 000116 | D_t=U_tilde D U_tilde*, R_t=U_tilde R U_tilde*, 000117 | D_ref,t=U_tilde D_ref U_tilde*, D_rel,t=U_tilde D_rel U_tilde*. 000118 | 000119 | On Fock space use the projective implementer Gamma(U_t), including the fixed 000120 | normal-ordering and Schwarzian conventions. Transport the net, time and 000121 | state by 000122 | 000123 | A_loc,t=Ad_Gamma(U_t)(A_loc), 000124 | tau_s,t=Ad_Gamma(U_t) o tau_s o Ad_Gamma(U_t)*, 000125 | omega_t=omega o Ad_Gamma(U_t)*, 000126 | 000127 | and transport Theta and every detector in the same way. 000128 | 000129 | Under the existing premise that the supplied CAR/fold source is an object of 000130 | the P1 operational seed category and that its circle transport has this