TFPT scientific source excerpt Original repository path: experiments/theory-contracts/e-series-source-reconstruction-20261024/NATIVE-CLOCK-CONJUGACY.txt Original file SHA-256: 5f208d8d32fdc77e774ec77b5a320f378cfc9e2c8793d4d3d439983aae583a45 Inclusive original lines: 1-101 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 | MATH.SEAM.ESERIES.SOURCE-RECONSTRUCTION.06 / native field-clock conjugacy 000002 | 24 October 2026 000003 | 000004 | 1. The precise common structure 000005 | 000006 | Keep the actual E8 lattice VOA, original cocycle and native full-field 000007 | clock Jhat. Isomorphisms in THIS theorem are full unitary inner E8 field 000008 | automorphisms. They need not preserve the originally named Heisenberg 000009 | Cartan, Gaussian V4, D5+A3 chart, Y or quartic mark. This is deliberately 000010 | a broader equivalence category than original sign-only lattice lifts. 000011 | It does not split any previously nonsplit G31 extension. 000012 | 000013 | Define on the same field source 000014 | 000015 | v=(1/2,1/2,1/2,1/2,1/2,0,0,0), 000016 | kappa=exp(2pi i v(0)). 000017 | 000018 | This is the parity flip of the five carrier channels on elementary 000019 | fermions. On the four E8 glue sectors over U=V_D5 tensor V_A3 it acts 000020 | by(1,i,-1,-i). Its square is gamma, the sign of the D8 spinor sector, 000021 | so kappa has exact order4 on V_E8. Its fixed charge lattice is precisely 000022 | L0=D5+D3, identified with D5+A3 by the explicit isometry T of 000023 | SCALAR-QUARTER-REPRODUCTION.txt; therefore 000024 | 000025 | (V_E8)^kappa=U. (1.1) 000026 | 000027 | This gives one quarter clock from the same carrier/glue data. On the 000028 | unextended elementary CAR it has order2, not the original uniform scalar 000029 | quarter action. Its order4 arises on the spinor extension. Its toral 000030 | conjugacy class can now be compared to the native Jhat class exactly. 000031 | 000032 | 2. Complete toral four-torsion calculation 000033 | 000034 | The compact unitary automorphism group of the level-one E8 VOA is the 000035 | compact connected E8 group. Indeed V_E8 is generated by its weight-one 000036 | affine E8 algebra, and E8 has no diagram outer automorphisms. Every compact 000037 | group element is conjugate into a maximal torus. Its period lattice here 000038 | is E8*=E8, so EVERY element of order dividing4 is torally represented by 000039 | 000040 | a/4, a in E8/4E8, |E8/4E8|=4^8=65536. (2.1) 000041 | 000042 | Use the original ordered integral E8 basis b_j=B_code,j/2; the checker 000043 | stores doubled standard coordinates B_code and uses G=B_code^T B_code/4. 000044 | On the 000045 | root current e^alpha the toral action is i^(a.alpha); all eight Cartan 000046 | currents are fixed. The exact integer certificate evaluates ALL65536 000047 | characters on ALL240 roots. There are exactly15120 with weight-one 000048 | multiplicities 000049 | 000050 | (dim1,dimi,dim-1,dim-i)=(60,64,60,64). (2.2) 000051 | 000052 | In the original integral coordinates, 4v has residue 000053 | 000054 | (0,1,3,2,0,2,0,0) modulo4. 000055 | 000056 | The ACTUAL eight simple-root reflections generate an orbit of15120 000057 | residues, and that orbit equals the ENTIRE matching set in(2.2). 000058 | This is not an inference from one matching dimension or trace: every 000059 | possible toral representative has been classified for this character. 000060 | 000061 | The original Jhat has precisely the same character. Its240 roots form 000062 | sixty four-cycles, with phase product+1 because the actual lift has fourth 000063 | powerI. Each cycle contributes one of each fourth root of unity. Its 000064 | Cartan action has four+i and four-i eigenvectors. Thus(2.2) holds for the 000065 | actual original operator, including its lattice-cocycle phases. 000066 | 000067 | Toralize Jhat by compact conjugacy. Its toral representative belongs to 000068 | the complete matching set(2.2), hence a Weyl element carries it to v. 000069 | Consequently there EXISTS a unitary inner field automorphism C with 000070 | 000071 | C kappa C^-1=Jhat. (2.3) 000072 | 000073 | The affine weight-one algebra generates the whole VOA, so this is a 000074 | full-field statement, not a comparison of a240/248 dimensional truncation. 000075 | In particular (V_E8)^Jhat=C U C^-1. The native quarter clock itself 000076 | determines an intrinsic copy of the compiler subnet in this category. 000077 | 000078 | 3. The15120 clocks are oriented versions of the7560 tetrahedral frames 000079 | 000080 | The certificate transports the original A3 factor with EVERY simple-root 000081 | reflection during the clock-orbit construction. Reaching one residue by 000082 | different words always gives the same A3. For every residue its fixed root 000083 | kernel is exactly the transported A3 normal plus its orthogonal D5 core. 000084 | 000085 | All15120 primitive clock characters have exactly7560 distinct fixed root 000086 | kernels, each with precisely two inverse clock characters. The7560 A3 000087 | factors are all distinct and form the actual full Weyl orbit. Thus 000088 | 000089 | oriented primitive clocks --2:1--> tetrahedron/compiler flags, 000090 | 15120 7560. (3.1) 000091 | 000092 | There is also a direct proof of the two-to-one result. Each transported 000093 | fixed root lattice has determinant16, and a primitive clock kernel has 000094 | index4 in E8, hence determinant16 as well. They are equal. E8/L0=C4 has 000095 | precisely two faithful characters, inverse to each other. The two clocks 000096 | therefore carry the actual orientation of the C4 glue over that flag. 000097 | 000098 | This connects the previous incidence geometry with an actual field-clock 000099 | action on the same roots. It does not identify every native J-preserving 000100 | mark with an originally fixed toral mark. 000101 |