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: 32-159 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. 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 | 000102 | 4. Keep the WHOLE cyclic compiler, including sigma 000103 | 000104 | Conjugating only J does not entitle us to freeze the old sigma. Indeed 000105 | kappa does not commute with the old sigma: the root e4+e5 in zero-based 000106 | coordinates has kappa eigenvalue-1, while sigma sends it to e0+e1 with 000107 | eigenvalue+1. All candidate sigma field phases leave this witness intact. 000108 | 000109 | [OMITTED: non-scientific conversation or execution metadata] 000110 | [OMITTED: non-scientific conversation or execution metadata] 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 | 000116 | (22,20,19,24,19,20,22,20,19,24,19,20). (4.1) 000117 | 000118 | After toralization, use Section2 to carry the ninth power to kappa. 000119 | All remaining toral compiler possibilities are 000120 | 000121 | c_t=kappa sigma_t, sigma_t=exp(2pi i w(0)), 000122 | w=b/3, b in E8/3E8. (4.2) 000123 | 000124 | This is exhaustive: for any c_t with c_t^9=kappa and c_t^12=1, its 000125 | fourth power is sigma_t and c_t=kappa sigma_t. There are exactly3^8=6561 000126 | toral b choices. The certificate checks ALL of them against the full 000127 | character(4.1). Exactly480 match. Reflections in the52 actual U roots 000128 | generate ONE orbit containing all480 matches. Hence the entire original 000129 | cyclic compiler is conjugate to one explicit representative(4.2), with 000130 | 000131 | b=(0,0,0,0,1,0,1,0) modulo3 000132 | 000133 | in the ordered integral basis. The standard coordinates w are printed 000134 | in native-clock-conjugacy-results.json. Its exact powers are 000135 | 000136 | c_t^9=kappa, c_t^4=sigma_t, c_t^12=1. 000137 | 000138 | [OMITTED: non-scientific conversation or execution metadata] 000139 | [OMITTED: non-scientific conversation or execution metadata] 000140 | [OMITTED: non-scientific conversation or execution metadata] 000141 | [OMITTED: non-scientific conversation or execution metadata] 000142 | 000143 | 5. What has been removed, and what remains original source work 000144 | 000145 | Within the full E8 field-automorphism category, a second unrelated quarter 000146 | clock class and a second unrelated cyclic compiler class are unnecessary: 000147 | the native class is exactly the carrier/glue class above. Its primitive 000148 | toral orientations are the same7560 flags already occurring in the 000149 | tetrahedral incidence geometry. The fixed-J local compiler extension is 000150 | the canonical C4 extension over its intrinsic fixed subnet. 000151 | 000152 | The theorem gives existence of a conjugator, not a preferred one. It 000153 | does not construct a microscopic P1/QWZ-to-VOA bosonization B_N, choose a 000154 | state/physical time, or prove that the original scalar seam and carrier 000155 | parity are interchangeable microscopic operations. It does not keep the 000156 | Gaussian lattice Cartan and all named marks fixed: a full conjugacy must 000157 | transport them. Full marked-tuple equivalence requires its additional 000158 | intertwining conditions. The source gap found in 000159 | SCALAR-QUARTER-REPRODUCTION.txt remains the next physical obligation.