TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/source_time_compatibility_20261027/GSO-SOURCE-AUDIT.txt Original file SHA-256: c9dc02cfdcdae20c50baf39518d91e4e3302bb765c6ccb59a8ed9cdc151ff4b0 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 | P1 SPIN STRUCTURE, GSO, AND LOCAL E8 SOURCE AUDIT 000002 | 27 October 2026 000003 | 000004 | Scope 000005 | 000006 | This note asks one narrow question. Does the original P1 source already contain 000007 | an operation that selects the local bosonic E8 extension of the D8 current net? 000008 | The qualifying operation could be either 000009 | 000010 | (a) a sum over spin structures, equivalently fermion parity gauging with its 000011 | twisted sectors and relative Arf phases, or 000012 | (b) a source derived local maximality or multi interval Haag duality rule 000013 | that forces the holomorphic extension. 000014 | 000015 | The audit keeps the known marked D5+A3 lattice and its C4 glue, the finite CAR 000016 | collar, and the compiler maps as established inputs. It does not ask whether an 000017 | E8 extension can be constructed once its sector support is chosen. That 000018 | conditional extension theorem is already available. 000019 | 000020 | DECISION 000021 | 000022 | SCOPED NEGATIVE. 000023 | 000024 | The original P1 operational seed does not specify a spin structure functor, a 000025 | sum over the four torus spin structures, a fermion parity orbifold including 000026 | its twisted sector, an Arf phase, a localized Ramond spin field, or a source 000027 | rule of bosonic maximality. It also does not prove the continuum local net 000028 | hypotheses from which multi interval Haag duality with mu=1 would follow. 000029 | Therefore it does not force 000030 | 000031 | (A+B+C+D)/2 = chi_0 + chi_s 000032 | 000033 | or 000034 | 000035 | (A+B+C-D)/2 = chi_0 + chi_c 000036 | 000037 | for the four spin torus blocks in the standard D8 convention. 000038 | 000039 | There is, however, a genuine positive P1 descendant that should be retained. 000040 | The v522 analysis derives the only reflection compatible subgroup of the 000041 | finite NS clock tower: its two torsion {1,(-1)^F}. It also proves reflection 000042 | positivity after restriction to the even observable algebra. This derives the 000043 | fermion parity gauge datum and the D8 or even CAR side. It does not perform the 000044 | full gauging operation that adds twisted sectors and chooses their relative 000045 | phase. Calling both operations “GSO” hides the remaining step. 000046 | 000047 | The precise remaining datum is small and constructive. One must derive from 000048 | P1 either 000049 | 000050 | 1. four compatible source amplitudes on a common spin torus together with 000051 | the parity gauging and Arf weight, or 000052 | 2. one normalized local spin or simple current field satisfying the energy, 000053 | relative locality, braiding, and Q system closure conditions, or 000054 | 3. the actual P1 scaling limit as a completely rational local net with a 000055 | source derived holomorphic completion, mu=1. 000056 | 000057 | Any one of these closes the already classified extension to one of the two 000058 | labelled E8 embeddings. The original h=(1,1) marking selects the plus embedding 000059 | only after the marked field identification is transported into that local 000060 | extension. 000061 | 000062 | 1. WHAT A FULL SPIN SUM DOES 000063 | 000064 | Let the four fermionic torus blocks be written, in one standard convention, as 000065 | 000066 | A = Z_NS^NS = chi_0 + chi_v, 000067 | B = Z_NS^R = chi_0 - chi_v, 000068 | C = Z_R^NS = chi_s + chi_c, 000069 | D = Z_R^R = chi_s - chi_c. 000070 | 000071 | Then 000072 | 000073 | (A+B+C+D)/2 = chi_0 + chi_s, 000074 | (A+B+C-D)/2 = chi_0 + chi_c. 000075 | 000076 | These are the two labelled E8 extensions of D8. The sign is the Arf or spin 000077 | SPT choice, equivalently the labelled s versus c chirality after all field 000078 | labels are retained. The unrefined q series cannot distinguish chi_s and 000079 | chi_c. 000080 | 000081 | This formula contains two logically separate operations: 000082 | 000083 | i. projection in an untwisted sector, for example (A+B)/2=chi_0; 000084 | ii. addition of the twisted Ramond sector with a specified relative phase. 000085 | 000086 | The first operation alone produces the parity invariant observable algebra. 000087 | It does not produce chi_0+chi_s or chi_0+chi_c. 000088 | 000089 | This distinction is explicit in the primary literature. Smith, Lin, 000090 | Tachikawa, and Zheng state that a bosonic theory is obtained by summing spin 000091 | structures, T_B=T_F/(-1)^F, and that the inverse fermionization carries an 000092 | Arf quadratic refinement. Their torus formula (5.1) displays all four defect 000093 | blocks and the independent Arf sign. In their E8 example the untwisted blocks 000094 | are chi_0 plus or minus chi_s and the twisted blocks are chi_c plus or minus 000095 | chi_v. Thus the existence or parity of four sectors is necessary input. It is 000096 | not itself the gauging map. 000097 | 000098 | Primary source: 000099 | 000100 | Philip Boyle Smith, Ying-Hsuan Lin, Yuji Tachikawa, Yunqin Zheng, 000101 | Classification of chiral fermionic CFTs of central charge <=16, 000102 | arXiv:2303.16917v2. 000103 | https://arxiv.org/html/2303.16917v2 000104 | Relevant passages: equations (2.1), (2.2), and (5.1), plus equations 000105 | (5.3) and (5.4) for the E8 to D8 decomposition. 000106 | 000107 | 2. THE STRONGEST POSITIVE RESULT ALREADY IN THE ORIGINAL DEVELOPMENT 000108 | 000109 | The v522 result is stronger than a generic claim that parity exists. It proves 000110 | on the finite seam system that 000111 | 000112 | alpha^4 = full wrap = 2 pi rotation = (-1)^F, 000113 | 000114 | and that the only reflection compatible, hence gaugeable, part of the Z8 NS 000115 | clock tower is its two torsion {1,(-1)^F}. On the parity projected bond cut 000116 | algebra it finds an exactly positive even block; odd rows and columns vanish. 000117 | It also records that the Ramond control with the alternative wrap fails state 000118 | invariance and that the verdict of the enclosing OS twistor contract remains 000119 | UNDECIDED. 000120 | 000121 | Exact original location: 000122 | 000123 | Atlas section: section:e8fb60dc4323384272d7 000124 | Atlas source: paper-source:545334c7ca9708d2eba823bf 000125 | documents/changelog/sections/003-source.tex:352-375 000126 | 000127 | What this establishes: 000128 | 000129 | * the distinguished Z2 is not freely chosen; 000130 | * parity projection is compatible with the actual reflection; 000131 | * the finite even observable algebra has the required reflection positive 000132 | form in the tested bond cut realization. 000133 | 000134 | What it does not establish: 000135 | 000136 | * a family of amplitudes functorial in all four spin structures; 000137 | * a sum over spin structures; 000138 | * the Arf sign; 000139 | * a local Ramond spin field in the continuum net; 000140 | * a holomorphic E8 vacuum net. 000141 | 000142 | The correct reading is therefore “P1 descendants derive the parity gauge 000143 | datum” rather than “P1 already performs the E8 GSO extension.” 000144 | 000145 | 3. THE FOUR SPIN STRUCTURE CENSUS DOES NOT SUPPLY THE SUM 000146 | 000147 | The v490 calculation resolves the ground state parity in the four torus spin 000148 | structures (AA,AP,PA,PP). Per Majorana layer it obtains (+,+,+,-). For sixteen 000149 | layers all four parities become positive. The resulting gauged torus ground 000150 | state degeneracy is four, matching mu(SO(16)_1)=det(D8)=4. The same source says 000151 | that the later mu:4 to 1 condensation is a cited conditional theorem, that the 000152 | condensed model is interacting, and that SEAM.EQUIV.01 remains open. 000153 | 000154 | Exact original location: 000155 | 000156 | Atlas section: section:67f92d5b57079948ad55 000157 | Atlas source: paper-source:627e6e199139a9f83ce2c7a2 000158 | documents/changelog/sections/008-source.tex:208-224 000159 | 000160 | This census proves anomaly compatibility of the sixteen layer system and 000161 | identifies the pre condensation sector count. It supplies no linear functional 000162 | (A,B,C,D) -> (A+B+C+/-D)/2 and no relative phase. In fact its directly measured 000163 | endpoint is mu_pre=4, the D8 side, rather than mu=1. 000164 | 000165 | 4. WHY v469 DOES NOT TURN ADMISSIBILITY INTO SELECTION 000166 | 000167 | The v469 crossed product route is mathematically useful and remains valid in 000168 | its stated scope. For the chosen D8 spinor simple current s, 000169 | 000170 | h_s = 16/16 = 1, 000171 | 000172 | so the statistics phase is +1. Longo and Rehren locality and the simple current 000173 | extension package then give a local index two extension 000174 | 000175 | SO(16)_1 crossed with {1,s}, 000176 | 000177 | and KLM gives mu=4/2^2=1. The analogous marked C4 glue has integer weights and 000178 | mu=16/4^2=1. 000179 | 000180 | Exact original location: 000181 | 000182 | Atlas section: section:ce5afee193d0ce1ea07e 000183 | Atlas source: paper-source:83382c35e5fa2102c0c2845d 000184 | documents/changelog/sections/010-source.tex:108-129 000185 | 000186 | This proves that the selected current is condensable and that the resulting 000187 | local extension is E8. It does not prove that P1 selects or inserts that 000188 | current. The source itself leaves SEAM.EQUIV.01 open. Its statement that the 000189 | class D edge “admits” a bosonic E8 description is an existence or phase 000190 | classification statement. It is not a local field and state intertwiner from 000191 | the primitive seed. 000192 | 000193 | Primary mathematical machinery: 000194 | 000195 | Roberto Longo and Karl Henning Rehren, Nets of Subfactors, 000196 | arXiv:hep-th/9411077, 000197 | https://arxiv.org/abs/hep-th/9411077 000198 | 000199 | Jens Boeckenhauer, An Algebraic Formulation of Level One 000200 | Wess-Zumino-Witten Models,