TFPT scientific source excerpt Original repository path: experiments/theory-contracts/atlas-defect-synthesis-20261024/DOILY-QUARTIC.txt Original file SHA-256: 666355194ef46ca3e8768dcb05a6c4158c7f3f9d2a5f64aa10e5eaa94dc6c853 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 | DOILY QUARTIC BRIDGE 000002 | ==================== 000003 | 000004 | Statement 000005 | --------- 000006 | 000007 | Let L be the Construction A E8 lattice used by the original compiler and let 000008 | J be its fixed integral complex structure. Put 000009 | 000010 | A = L/(1-J)L. 000011 | 000012 | The Bakalov Kac commutator form makes A a four dimensional symplectic F2 000013 | space. Its fifteen nonzero vectors and fifteen isotropic planes form the 000014 | doily W(3,2). Let B be the 15 by 15 point line incidence matrix. 000015 | 000016 | The original native quartic five is G31 equivariantly isomorphic to the 000017 | point side kernel ker(B^T). This kernel is the centered permutation module 000018 | on the six ovoids. It is not the centered module on the six spreads, and no 000019 | sign twist of either module works. 000020 | 000021 | This is an exact finite representation theorem. It does not identify the 000022 | native four with the J twisted ground space, does not identify conformal 000023 | states, and does not identify this five with a P2 or D5 carrier by dimension. 000024 | 000025 | Exact construction 000026 | ------------------ 000027 | 000028 | The checker imports only the tracked constructors 000029 | 000030 | tfpt_explorer/process.py 000031 | tfpt_explorer/twisted_source.py 000032 | 000033 | and reconstructs every further object. 000034 | 000035 | The deterministic Darboux order is (e1,f1,e2,f2). The JSON certificate lists 000036 | all fifteen points, all fifteen lines, all six ovoids, all six spreads, the 000037 | five quartic word supports, and every one of the sixty roots and action 000038 | matrices. Therefore the displayed intertwiner has no hidden relabeling. 000039 | 000040 | For an ovoid O define the point vector 000041 | 000042 | v_O = 3 1_O - 1. 000043 | 000044 | The six such vectors obey 000045 | 000046 | B^T v_O = 0, 000047 | sum_O v_O = 0, 000048 | (v_O,v_O') = 36 delta_O,O' - 6. 000049 | 000050 | They have rank five and span ker(B^T). The incidence calculation is 000051 | 000052 | rank(B) = 10, 000053 | spectrum(B B^T) = 9^1, 4^9, 0^5. 000054 | 000055 | Thus the orthogonal projector onto the ovoid five is both 000056 | 000057 | P5 = (1/36) sum_O v_O v_O^T 000058 | 000059 | and the incidence polynomial 000060 | 000061 | P5 = ((B B^T - 9 I)(B B^T - 4 I))/36. 000062 | 000063 | All sixty marked reflections 000064 | ----------------------------- 000065 | 000066 | For every Gaussian ray z from process._hamming_rays(), the checker converts z 000067 | back to its actual eight dimensional E8 root. twisted_source then computes 000068 | the induced action on A. Every action is a symplectic transvection. 000069 | 000070 | On the six ovoids every root reflection is one transposition. Hence its 000071 | centered five dimensional trace is 3. On the six spreads it is a product of 000072 | three transpositions and the centered trace is -1. 000073 | 000074 | The native quartic action is rebuilt without importing another checker. If 000075 | r is the native four dimensional reflection, the checker applies (2r)^tensor4 000076 | to the five unnormalised equal coefficient word sums. The result is an 000077 | integral numerator N_r, with quartic action N_r/16. For all sixty roots, 000078 | 000079 | N_r^2 = 256 I, 000080 | tr(N_r/16) = 3. 000081 | 000082 | Solving all simultaneous intertwiner equations over Q gives 000083 | 000084 | candidate constraint rank Hom dimension 000085 | ovoid standard 24 1 000086 | ovoid sign twist 25 0 000087 | spread standard 25 0 000088 | spread sign twist 25 0. 000089 | 000090 | In the centered ovoid basis f_j=e_j-e_5 and the listed unnormalised quartic 000091 | word basis, the primitive generator of the one dimensional Hom space is 000092 | 000093 | [ 3 3 3 0 3] 000094 | [-1 1 1 0 -1] 000095 | T = [-1 -1 1 0 1] 000096 | [-1 1 -1 0 1] 000097 | [ 1 1 1 2 1]. 000098 | 000099 | Its determinant is -96. The checker verifies separately for every root that 000100 | 000101 | N_r T = 16 T D_r, 000102 | 000103 | where D_r is the centered ovoid permutation. Since the sixty reflections 000104 | generate G31, this is the full marked equivariance statement. 000105 | 000106 | The interface is precisely 000107 | 000108 | G31/N64 = Sp(4,2) = S6, 000109 | 000110 | where 000111 | 000112 | N64 = Z4 central-product 2^(1+4). 000113 | 000114 | Here 2^(1+4) is the extraspecial two-qubit Pauli group of order 32 and 000115 | its central involution is identified with the involution in Z4. Thus N64 000116 | has order 64. It is not a direct product of mu4 with a sixteen-element 000117 | Pauli label set; those sixteen labels describe the projective quotient. 000118 | 000119 | Both A and the native quartic five kill the kernel. The quartic five is the 000120 | standard S6 representation supplied by the six actual A ovoids. 000121 | 000122 | Why the four dimensional identification remains false 000123 | ------------------------------------------------------ 000124 | 000125 | The checker also reconstructs the native Clifford conjugation action on the 000126 | four Pauli labels. Over F2 it solves 000127 | 000128 | X rho_A(g) = phi_native(g) X 000129 | 000130 | for all sixty roots. The Hom space has dimension zero, so there is no 000131 | invertible marked identification between A and the native Pauli label module. 000132 | They remain related by the S6 outer duality. 000133 | 000134 | The independent spectral obstruction is retained as well. For a twisted 000135 | ground transvection, the sixteen Clifford lifts have squared trace magnitude 000136 | 8 twice and 0 fourteen times. A native complex reflection has squared trace 000137 | magnitude 4. All 60 times 16 comparisons therefore fail exactly. The new 000138 | five dimensional theorem does not repair or reinterpret the failed native 000139 | four bridge. 000140 | 000141 | What this supplies for state and selection 000142 | ------------------------------------------ 000143 | 000144 | The ovoid construction does supply canonical representation level selection 000145 | data: 000146 | 000147 | 1. P5 is a basis independent rational projector selecting the unique point 000148 | side incidence kernel. 000149 | 2. The six operators 000150 | 000151 | E_O = v_O v_O^T / 36 000152 | 000153 | are positive rank one operators, are permuted by G31, and sum to P5. They 000154 | form a canonical six outcome covariant resolution on the selected channel. 000155 | 3. The invariant positive form on the native quartic five is unique up to a 000156 | positive scalar. In the unnormalised word basis its primitive matrix is 000157 | 000158 | diag(1,3,3,3,6). 000159 | 000160 | This is also the form inherited from the tensor Hilbert space because the 000161 | five word support sizes are 4,12,12,12,24. 000162 | 000163 | Transporting the six centered ovoid rays through T gives the following six 000164 | primitive rays directly in the native quartic word basis: 000165 | 000166 | ( 1,-1,-1,-1, 0) 000167 | ( 1, 1,-1, 1, 0) 000168 | ( 1, 1, 1,-1, 0) 000169 | (-2, 0, 0, 0, 1) 000170 | ( 1,-1, 1, 1, 0) 000171 | (-2, 0, 0, 0,-1). 000172 | 000173 | Each has squared norm 10 for G=diag(1,3,3,3,6). The native operators 000174 | 000175 | E_a = q_a q_a^T G / 12 000176 | 000177 | are positive with respect to G, are permuted exactly by all sixty native 000178 | quartic actions, and sum to I_5. Thus the bridge supplies a concrete 000179 | six-outcome covariant POVM on the original quartic five, not merely an 000180 | abstract dimension match. 000181 | 000182 | The exact symmetric form calculation has dimension one. Representation 000183 | theory therefore leaves only its overall positive scale. The equivariant 000184 | intertwiner is likewise unique up to a nonzero scalar. Requiring it to be an 000185 | isometry fixes the modulus, leaving a complex phase, or only a sign over R. 000186 | 000187 | The full commutant is one dimensional, and there is no nonzero vector fixed 000188 | by all sixty reflections. Consequently the 000189 | geometry does not select one pure state or one ovoid. After choosing the 000190 | invariant form and trace normalization, the only fully invariant density 000191 | operator is the maximally mixed state I_V5/5. Selecting one of the six rank 000192 | one effects needs an additional symmetry breaking datum, such as an anchored 000193 | quadratic refinement. That datum is not supplied by A, J, and incidence 000194 | alone. 000195 | 000196 | Reproduction 000197 | ------------ 000198 | 000199 | From the repository root: 000200 |