TFPT scientific source excerpt Original repository path: experiments/theory-contracts/canonical-seam-fermionization-20261025/ORIGINAL-SPIN-AND-SEAM-DICTIONARY.txt Original file SHA-256: dfaef864428375b018d0eddcb77a4a5ccd085e996eb1c1b3f1d8db526991cf24 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 | ORIGINAL SPIN AND SEAM DICTIONARY 000002 | ================================= 000003 | 000004 | 25 October 2026. Bounded original-source audit and exact action comparison. 000005 | No Atlas, index, ledger, paper, or existing contract was changed. 000006 | 000007 | Question 000008 | -------- 000009 | 000010 | Does the original TFPT seam data already contain the precise data needed for 000011 | the standard fermionization 000012 | 000013 | (E8)_1 -- choose nonanomalous Z2 with fixed algebra (D8)_1, 000014 | -- couple it to a spin structure, 000015 | -- gauge the diagonal Z2 000016 | -> 16 Majorana-Weyl fermions, 000017 | 000018 | and is that operation the same as one of the original seam involutions, 000019 | quarter clocks, or cover half-flux operations? 000020 | 000021 | Verdict 000022 | ------- 000023 | 000024 | The answer splits into one positive algebraic identification and two missing 000025 | source identifications. 000026 | 000027 | PROVED, inside the already marked D5+A3 subset E8 compiler frame: 000028 | 000029 | The order-four glue has a unique square character 000030 | 000031 | chi_can(E_r)=(-1)^deg(r), 000032 | deg(r)=sum_(i=0)^4 r_i mod 4 (1) 000033 | 000034 | on the actual charged E8 root fields. Its +1 roots are precisely the 112 000035 | roots of the canonical D8 intermediate, and its -1 roots are precisely the 000036 | 128 spinor roots. Thus chi_can is exactly the nonanomalous E8 Z2 whose 000037 | invariant current algebra is D8 in the standard fermionization theorem. 000038 | This is an action dictionary, not an identification by root count. 000039 | 000040 | CONDITIONAL POSITIVE RESULT: 000041 | 000042 | Once that marked E8 source and its D8 are selected, the existing inverse 000043 | construction reconstructs 000044 | 000045 | E8 bosonic space: H_0 + H_s, 000046 | g-twisted space: H_v + H_c, 000047 | fermionic NS space: H_0 + H_v, 000048 | fermionic R space: H_s + H_c. (2) 000049 | 000050 | It supplies the 16 Majorana CAR character, positive half-integer NS time, 000051 | the plane two-point function, and the cylinder kernel. The actual original 000052 | cocycle brackets and all 112 retained D8 root actions on the 16 vector 000053 | weights pass. This is a valid inverse inside a given marked conformal E8_1 000054 | source. It is not yet a derivation from the primitive P1 seam. 000055 | 000056 | NOT ESTABLISHED BY THE CITED ORIGINAL P1/P2 SOURCES: 000057 | 000058 | 1. The canonical compiler D8 is not the same embedded D8 as the already 000059 | supplied folded CAR source in the simultaneously fixed O/M frame. Both 000060 | have 112 roots, but their intersection contains only 48 roots, with 64 000061 | roots exclusive to each. Their charged-root characters and quarter-time 000062 | actions differ. A common marked local-net intertwiner is therefore still 000063 | required. 000064 | 000065 | 2. Fullv21 A4 does select the local cover spin structure if its wording is 000066 | read literally: the order-four lift is the lift of the same Moebius Z2 000067 | identification. Squaring that lift traverses the core loop of the 000068 | orientable double and gives spin holonomy -1, hence NS. The periodic R 000069 | comparison preserves A2/A3 winding data but violates A4. The remaining 000070 | gap is the full two-cycle quadratic refinement and defect gauging. There 000071 | is also a local source inconsistency to resolve: A3 prints the integer 000072 | spectrum n+phi/(2 pi), whereas the A4-selected NS operator has the 000073 | half-integer reference spectrum n+1/2. 000074 | 000075 | The first additional source datum in the actual TFPT chain is therefore not 000076 | an abstract choice among unnamed D8 subalgebras. The marked D5+A3 glue already 000077 | selects one canonical D8. The first missing datum is a source-derived 000078 | intertwiner that transports that particular charged-root character to the 000079 | actual folded CAR field net while transporting Cartans, time, cocycles, and 000080 | locality together. After that map, the global spin refinement and diagonal 000081 | gauging still have to be supplied or derived. 000082 | 000083 | 1. What the standard fermionization theorem requires 000084 | ----------------------------------------------------- 000085 | 000086 | Smith, Lin, Tachikawa and Zheng, "Classification of chiral fermionic CFTs of 000087 | central charge <= 16", arXiv:2303.16917v2, provide the relevant standard: 000088 | 000089 | * Section 2.1, equations (2.1) to (2.4): fermionization of a bosonic theory 000090 | T_B requires a nonanomalous internal Z2 symmetry g, a spin-invertible 000091 | factor Q with phase 000092 | 000093 | (-1)^q(a)=(-1)^[Arf(q+a)-Arf(q)], 000094 | 000095 | and diagonal Z2 gauging. The spin structure q is explicit data. 000096 | 000097 | * Section 4.2.1, equations (4.15) and (4.16): the E8 class represented by 000098 | x=omega_7/2 is nonanomalous and has invariant algebra D8. The other 000099 | conjugacy class has invariant algebra E7+A1 and is anomalous. The action 000100 | on every vertex operator is 000101 | 000102 | Gamma_alpha -> exp(2 pi i (x,alpha)) Gamma_alpha. (3) 000103 | 000104 | * Section 5.1, equations (5.3) to (5.6): both untwisted and defect-inserted 000105 | partition functions along the spatial and temporal cycles enter. The 000106 | result is the spin theory of 16 Majorana-Weyl fermions. 000107 | 000108 | * Appendix A: the E8 lattice splits as D8 root lattice plus one spinor 000109 | coset. The even sublattice is generated explicitly, and 000110 | 000111 | Z_E8=chi_1^D8+chi_S^D8. (4) 000112 | 000113 | Primary source: https://arxiv.org/abs/2303.16917 000114 | 000115 | Equations (3) and (4) give the decisive comparison criterion. A TFPT Z2 must 000116 | have the same action on the charged root fields, not merely the same order, 000117 | the same number of labels, or the same abstract group name. 000118 | 000119 | 2. Exact native character in the marked E8 000120 | -------------------------------------------- 000121 | 000122 | The active compiler constructs E8 from the marked lattice 000123 | 000124 | K=D5+A3, E8/K isomorphic to Z4. (5) 000125 | 000126 | In standard target coordinates, choose the displayed glue generator 000127 | 000128 | g=(1/2,...,1/2). 000129 | 000130 | The unique order-two intermediate is 000131 | 000132 | K union (K+2g)=D8. (6) 000133 | 000134 | For the actual doubled E8 roots, the four glue degrees contain 000135 | 000136 | degree 0: 52 roots, 000137 | degree 1: 64 roots, 000138 | degree 2: 60 roots, 000139 | degree 3: 64 roots. (7) 000140 | 000141 | The internal order-four glue acts by 000142 | 000143 | G(E_r)=i^deg(r) E_r. (8) 000144 | 000145 | Therefore 000146 | 000147 | G^2(E_r)=(-1)^deg(r) E_r. (9) 000148 | 000149 | Degrees 0 and 2 give 112 fixed roots, exactly the standard D8 roots 000150 | +/-e_i +/-e_j. Degrees 1 and 3 give the 128 half-integral E8 spinor roots and 000151 | receive sign -1. Equation (9) is precisely the E8 Z2 action (3), in the 000152 | standard D8 embedding selected by the marked glue. 000153 | 000154 | The native certificate independently reconstructs this character inside the 000155 | lift kernel. It finds 000156 | 000157 | D5_A3_halfway_glue_character_mask = 152, 000158 | glue_even_root_count = 112, 000159 | glue_odd_spinor_root_count = 128, 000160 | character_matches_all_roots = True. (10) 000161 | 000162 | It also finds the short closed-word witness 000163 | 000164 | r0 r1 r0 = r2 r1 r2 000165 | 000166 | on the naked action, while the two lifts differ by exactly character 152. 000167 | This proves that the charged fields retain the carry character even when the 000168 | Cartan action forgets it. 000169 | 000170 | The character is not fixed pointwise by the full G31 action: 000171 | 000172 | G31 orbit size = 15, 000173 | stabilizer order = 3072, 000174 | central under G31 = False. (11) 000175 | 000176 | Thus full unmarked G31 symmetry does not select this D8. The marked D5+A3 000177 | flag does. This is the exact point where the P2 marking matters. 000178 | 000179 | Native source: 000180 | 000181 | tfpt_explorer/cartan_source.py:651-743 000182 | SHA256 b3e7963152999ab098111bc9971f237cd63a57975aa2f1d29614d3c994cd30c0 000183 | 000184 | The active lattice/net statement is in 000185 | 000186 | paper-source:9ea51c45ef95aa398750f1c3 000187 | documents/tfpt_1_architecture_e8/sections/005-source.tex:55-70,166-228 000188 | SHA256 2de406bb3e912bb795eca44e8552d11040413587aba14b3924e07b145b91bdcd 000189 | 000190 | There the same mu4 glue is explicitly used in both the root lattice and the 000191 | chiral net, and the local spinor extension SO(16)_1 -> E8_1 is recorded. That 000192 | is the bosonic extension D8+s -> E8. Its converse becomes fermionization only 000193 | after the spin/defect data of Section 1 are added. 000194 | 000195 | 3. The exact inverse and its scope 000196 | ---------------------------------- 000197 | 000198 | The existing function 000199 | 000200 | tfpt_explorer/source_realization.py:420-531