TFPT scientific source excerpt Original repository path: experiments/theory-contracts/e-series-joint-source-20261023/COMPILER-CLOSURE.txt Original file SHA-256: de73510dc40b968940c68662ab07252995f96a98363e6cb1e9436937ecf62b3c Inclusive original lines: 1-200 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.JOINT.05 000002 | 23 October 2026 000003 | 000004 | Minimal joint closure of the native D5+A3 compiler cuts 000005 | 000006 | 1. Result 000007 | 000008 | Let Phi be the existing E8 root system in the standard real eight coordinate 000009 | chart. Let 000010 | 000011 | H = W(D5) x W(D3), 000012 | 000013 | where D5 uses coordinates 0,...,4 and D3=A3 uses coordinates 5,6,7. Retain the 000014 | actual native quarter rotation 000015 | 000016 | J(x0,x1,...,x7)=(-x1,x0,-x3,x2,-x5,x4,-x7,x6). 000017 | 000018 | The two exact transition group closures are 000019 | 000020 | = W(D8), |W(D8)| = 5,160,960, 000021 | 000022 | and, for the original 60 G31 pair reflections, 000023 | 000024 | = W(E8), |W(E8)| = 696,729,600. (1.1) 000025 | 000026 | The first equality closes the four original D5 compatible E6 cuts to exactly 000027 | 224 integer A2 normal E6 cuts. They form 56 coordinate three block frames, 000028 | with four tetrahedral cuts in every frame. On the entire 128 dimensional half 000029 | root current module, every frame gives four literal orthogonal projectors of 000030 | rank 32 whose sum is the identity. For each projector P, 000031 | 000032 | rank P = 32, rank(1-P) = 96, 000033 | Tr(I128)/Tr(1-P) = 128/96 = 4/3. (1.2) 000034 | 000035 | This is the chirality free form of the previous rank 16 plus rank 48 result. 000036 | It uses both native chiral blocks together and therefore introduces no chosen 000037 | 64 dimensional block. 000038 | 000039 | The second equality enlarges the orbit to all 1120 regular A2 normal E6 cuts. 000040 | The fixed 128 dimensional half root sector is not invariant under this larger 000041 | group. The W(E8) covariant replacement is the complete unordered family of 000042 | 1120 E6 projectors on the full E8 root or adjoint carrier. 000043 | 000044 | These are transition groups on transported compiler flags. They are not 000045 | stabilizers of one fixed object carrying the original D5+A3 split and J. 000046 | 000047 | 2. Why H and J close to W(D8) 000048 | 000049 | Every element of H is an even signed coordinate permutation. J is also an 000050 | even signed coordinate permutation, so is contained in W(D8). 000051 | 000052 | The coordinate permutation of J is 000053 | 000054 | (01)(23)(45)(67). 000055 | 000056 | The product (01)(23)(67) already lies in S5 x S3 inside H. Therefore the 000057 | permutation image of contains the crossing transposition (45). Together 000058 | with the adjacent transpositions internal to the two original blocks, this 000059 | generates all of S8. 000060 | 000061 | H already contains a sign reversal on any pair within either original block. 000062 | Conjugating one such pair reversal by the now available S8 image gives every 000063 | two coordinate sign reversal. These vectors span the seven dimensional even 000064 | sign space. Hence the kernel is the complete group 2^7 of even sign changes. 000065 | Thus 000066 | 000067 | = 2^7 : S8 = W(D8). 000068 | 000069 | The checker verifies the crossing transposition explicitly, verifies rank 7 000070 | of the pair sign span over F2, and verifies that the resulting action has the 000071 | two expected root orbits: 112 integer D8 roots and 128 half roots. 000072 | 000073 | 3. The 224 cut closure and the 56 projector frames 000074 | 000075 | A regular A2 inside D8 is supported on exactly three coordinate axes. A fixed 000076 | triple carries a D3=A3 root system, and that D3 contains four A2 subsystems. 000077 | Therefore 000078 | 000079 | number of integer A2 subsystems = binomial(8,3) times 4 = 56 times 4 = 224. 000080 | 000081 | The checker constructs these subsystems from the exact root relation 000082 | 000083 | A(alpha,beta)={plus or minus alpha, plus or minus beta, 000084 | plus or minus(alpha+beta)}, 000085 | (alpha,beta)=-1, 000086 | 000087 | and proves that one W(D8) orbit contains all 224. This is exactly the union of 000088 | the four integer normal orbit rows in the previous H classification: 000089 | 000090 | 40 + 120 + 60 + 4 = 224. 000091 | 000092 | For an integer A2 subsystem A, define a diagonal operator on the literal half 000093 | root basis M128 by 000094 | 000095 | P_A e_lambda = e_lambda if (lambda,A)=0, 000096 | 0 otherwise. (3.1) 000097 | 000098 | Every P_A has rank 32. If A1,...,A4 are the four A2 faces in one coordinate 000099 | D3 frame, then 000100 | 000101 | P_Ai P_Aj = delta_ij P_Ai, 000102 | P_A1 + P_A2 + P_A3 + P_A4 = I128. (3.2) 000103 | 000104 | The reason is literal. Restrict a half root sign vector to the selected three 000105 | coordinates. Its projective sign axis is exactly one of the four tetrahedral 000106 | axes. Equation (3.2) follows without a fitted embedding. 000107 | 000108 | The complement of one face is the sum of the other three projectors, so it 000109 | has rank 96. Equation (1.2) is therefore structural and is independent of a 000110 | chosen face. Relative to each transported five plus three coordinate split, 000111 | every half root also has the unchanged squared projection norms 000112 | 000113 | ||lambda_D5||^2 = 5/4, ||lambda_A3||^2 = 3/4. (3.3) 000114 | 000115 | Thus the projector ranks, the ratio 4/3, and the glue norms 5/4 and 3/4 belong 000116 | to one common carrier dictionary. 000117 | 000118 | J acts by exact conjugation, 000119 | 000120 | J P_A J^-1 = P_(JA). 000121 | 000122 | It permutes the 56 frames and their four faces. In particular it sends the 000123 | original coordinate frame {5,6,7} to {4,6,7}. It sends all four original 000124 | D5 compatible cuts outside the originally fixed family. The closure retains 000125 | the entire transported dictionary and supplies no preferred face. 000126 | 000127 | 4. Why the original G31 pair operations force W(E8) 000128 | 000129 | The preceding E7 pair certificate proves that every original G31 reflection 000130 | is literally 000131 | 000132 | g_alpha = s_alpha s_(J alpha), (4.1) 000133 | 000134 | where alpha and J alpha are orthogonal E8 roots. Hence every g_alpha belongs 000135 | to W(E8). The native v634 certificate also proves that J itself belongs to the 000136 | group generated by the 60 reflections. Consequently it is enough to determine 000137 | what one actual pair reflection does to the two W(D8) root orbits. 000138 | 000139 | In doubled integral coordinates, take 000140 | 000141 | 2 alpha = (-1,-1,-1,-1,-1,-1,-1,-1), 000142 | 2 beta = (-2,-2,0,0,0,0,0,0). 000143 | 000144 | The exact original operation (4.1) gives 000145 | 000146 | 2 g_alpha(beta) = (-1,-1,1,1,1,1,1,1). (4.2) 000147 | 000148 | Thus an actual G31 pair reflection maps an integer D8 root to an E8 half root. 000149 | Since W(D8) is transitive on each of the 112 and 128 root sectors, adding this 000150 | one operation makes the joint group transitive on all 240 E8 roots. 000151 | 000152 | Now W(D8) already contains the root reflection s_delta for every integer D8 000153 | root delta. For any E8 root gamma, choose a joint group element k with 000154 | k delta=gamma. Then 000155 | 000156 | k s_delta k^-1 = s_gamma. 000157 | 000158 | The joint group therefore contains all 240 E8 root reflections. Those 000159 | reflections generate W(E8). Conversely H, J, and every operation (4.1) lie in 000160 | W(E8). This proves both inclusions and hence the second equality in (1.1). 000161 | No enumeration of the 696,729,600 group elements is used. 000162 | 000163 | 5. The larger closure scope 000164 | 000165 | W(E8) is transitive on the 1120 regular A2 subsystems. Therefore the minimal 000166 | full closure of any original E6 cut is the complete family 000167 | 000168 | { C_A = Phi intersect span(A)^perp : A is a regular A2 in E8 }, 000169 | |{C_A}| = 1120. (5.1) 000170 | 000171 | On the 240 root current basis, the literal cut projector has rank 72. Adding 000172 | the six dimensional Cartan complement of the normal A2 plane gives the usual 000173 | rank 78 E6 adjoint projector. This complete 1120 member family is exactly 000174 | W(E8) covariant. 000175 | 000176 | The coordinate tetrahedra also enlarge. The 56 disjoint D3 frames close to 000177 | 7560 regular A3=D3 root subsystems. Each contains four A2 faces. The incidence 000178 | identity is 000179 | 000180 | 7560 times 4 = 1120 times 27 = 30240. (5.2) 000181 | 000182 | Thus every E6 cut belongs to 27 transported A3 tetrahedra. Unlike the 56 000183 | coordinate frames, the 7560 full frames overlap and do not partition the cut 000184 | family. 000185 | 000186 | The witness (4.2) proves at the same time that the integer plus half root 000187 | polarization is not W(E8) invariant. Therefore the uniform rank 32 projector 000188 | description on a fixed M128 is the exact intermediate W(D8) structure. It 000189 | must not be advertised as a fixed W(E8) module. Full W(E8) covariance uses 000190 | the 240 root current or 248 adjoint carrier and transports the complete flag. 000191 | 000192 | 6. Canonical content and retained mark 000193 | 000194 | The mathematical canonical objects are: 000195 | 000196 | * under W(D8), the unordered 224 cuts, their 56 four projector resolutions, 000197 | and the transported five plus three polarizations; 000198 | * under W(E8), the unordered 1120 E6 cuts and 7560 overlapping A3 frames. 000199 | 000200 | An individual E6 cut, one tetrahedral vertex, one chiral 64 block, and the