TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/theta_lattice_closure_20261027/review/COHERENT-TWO-LATTICE-FIELD-REVIEW.txt Original file SHA-256: 46c897da691c6f1730cc71a8d243788289328fbeccd61812247919a8acf9c2b6 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 | INDEPENDENT REVIEW OF THE COHERENT TWO LATTICE FIELD GROUPOID 000002 | 27 October 2026 000003 | 000004 | VERDICT 000005 | 000006 | ACCEPTED in the stated scope. 000007 | 000008 | The two-object construction is a full lattice-VOA statement, not a root-shell 000009 | or finite-state approximation. Given the already existing faithful C12 lift 000010 | on V_(Lambda_0) and any unitary lattice-VOA isomorphism 000011 | 000012 | T:V_(Lambda_0) -> V_(Lambda_1) 000013 | 000014 | lifting the fixed lattice isometry alpha_01, the definitions 000015 | 000016 | beta_1=T beta_0 T^(-1), 000017 | sigma_1=T sigma_0^(-1) T^(-1), 000018 | alpha_01=T, 000019 | alpha_10=beta_0 T^(-1) 000020 | 000021 | give a coherent groupoid action with 000022 | 000023 | alpha^2=diag(beta_0,beta_1), 000024 | alpha^8=1, 000025 | alpha sigma alpha^(-1)=sigma^(-1). 000026 | 000027 | No involutivity of T, no commutation of T with beta_0 and no splitting of the 000028 | full G31 extension enters this argument. 000029 | 000030 | The optional fixed-frame C4 x S3 section is also correctly established for 000031 | the chosen standard cocycle, but it changes the zero-on-basis J lift. It is a 000032 | compatible mathematical section, not a replacement for a previously frozen 000033 | source J or C12 linearization. 000034 | 000035 | 000036 | 1. THE GROUPOID ALGEBRA IS TYPE CORRECT 000037 | 000038 | The arrows have the exact source and target types 000039 | 000040 | alpha_01=T:V_0 -> V_1, 000041 | alpha_10=beta_0 T^(-1):V_1 -> V_0. 000042 | 000043 | Therefore 000044 | 000045 | alpha_10 alpha_01=beta_0, 000046 | alpha_01 alpha_10=T beta_0 T^(-1)=beta_1. 000047 | 000048 | Since beta_0 has exact order four, so does beta_1, and alpha has exact order 000049 | eight on V_0 direct_sum V_1. 000050 | 000051 | The cubic relation is equally formal. From 000052 | 000053 | sigma_1=T sigma_0^(-1) T^(-1) 000054 | 000055 | one gets 000056 | 000057 | T sigma_0=sigma_1^(-1)T. 000058 | 000059 | For the return arrow, commutation of beta_0 with sigma_0 gives 000060 | 000061 | beta_0 T^(-1) sigma_1 000062 | =sigma_0^(-1) beta_0 T^(-1). 000063 | 000064 | These are precisely the two components of 000065 | 000066 | alpha sigma alpha^(-1)=sigma^(-1). 000067 | 000068 | Transport preserves the order of beta and sigma and their commutation. Thus 000069 | 000070 | c_j=beta_j sigma_j 000071 | 000072 | has exact order twelve on both branches, with c_j^4=sigma_j and 000073 | c_j^9=beta_j. 000074 | 000075 | The proof begins with the existing branch-zero lift 000076 | 000077 | beta_0=c_0^9, sigma_0=c_0^4. 000078 | 000079 | It does not derive or replace that source marking. This is the right logical 000080 | direction for preserving the earlier nonsplitting and clock results. 000081 | 000082 | 000083 | 2. THE MOD TWO CHECK IS AN ALL CHARGE PROOF 000084 | 000085 | For the integral theta-E8 basis, the fixed standard cocycle is 000086 | 000087 | epsilon(m,n)=(-1)^(m^T F n), 000088 | 000089 | where F+F^T equals the Gram matrix modulo two and the diagonal gives 000090 | epsilon(m,m)=(-1)^(/2). 000091 | 000092 | For an exact lattice isometry A, the cocycle quotient is controlled by 000093 | 000094 | D_A=A^T F A+F mod 2. 000095 | 000096 | It is symmetric with zero diagonal. The checker constructs q_A with 000097 | 000098 | q_A(m+n)+q_A(m)+q_A(n)=m^T D_A n mod 2. 000099 | 000100 | Hence 000101 | 000102 | e^m -> (-1)^(q_A(m)) e^(Am) 000103 | 000104 | is a lift of A on the twisted lattice group algebra. Both epsilon and the 000105 | sign correction depend only on m,n modulo 2. Exhausting the 256 residue 000106 | classes and all 65536 ordered pairs therefore proves the cocycle identity for 000107 | all lattice charges. It is not merely a finite root probe. 000108 | 000109 | The exact integer matrix isometries and their geometric relations are checked 000110 | separately in the imported theta-lattice certificate. Oscillator modes are 000111 | transported by the same orthogonal A, so the group-algebra calculation extends 000112 | to every finite-energy lattice-VOA field and its adjoint. 000113 | 000114 | 000115 | 3. T NEEDS NO FALSE SAME FRAME RELATIONS 000116 | 000117 | The focused certificate chooses the zero-on-basis standard-cocycle lift of V 000118 | as T. It verifies deliberately that 000119 | 000120 | T^2 is not 1, 000121 | T beta_0 is not beta_0 T. 000122 | 000123 | All groupoid identities still hold. This is the decisive negative control: 000124 | the construction does not hide either of the stronger assumptions that would 000125 | be false for a general cocycle lift. 000126 | 000127 | Any two lifts of the same lattice isometry differ by a sign character of the 000128 | lattice. Replacing T by chi T conjugates all branch-one marks and the return 000129 | arrow by the same object-one gauge. The resulting two-object systems are 000130 | isomorphic. The construction is therefore canonical up to ordinary groupoid 000131 | equivalence, not as one distinguished physical character section. 000132 | 000133 | 000134 | 4. THE OPTIONAL STRICT SECTION IS CORRECTLY SCOPED 000135 | 000136 | For each of J,V,sigma the checker enumerates all 256 sign characters of the 000137 | fixed standard cocycle. The power-normalized candidate counts are 000138 | 000139 | J order four: 256, 000140 | V order two: 16, 000141 | sigma order three: 16. 000142 | 000143 | Exactly 256 raw triples satisfy the complete same-frame relations. One 000144 | explicit witness has masks 000145 | 000146 | J=0x03, V=0x35, sigma=0x2a. 000147 | 000148 | Its generated group has order 24 and the resulting c=J sigma has exact order 000149 | twelve. These 256 triples are raw sections before character-gauge or 000150 | centralizer quotient. They are not 256 physical theories or inequivalent 000151 | choices. 000152 | 000153 | No solution has the zero-on-basis J character. Consequently this optional 000154 | split section cannot silently be substituted for the fixed original J. The 000155 | main groupoid theorem does not use the optional masks and remains valid for 000156 | the original faithful c_0. 000157 | 000158 | The result supplies individual isometry lifts and one small split subgroup. 000159 | It neither constructs nor assumes a simultaneous lift of every element of 000160 | G31. The established nonsplitting of the full extension is preserved. 000161 | 000162 | 000163 | 5. PHYSICAL AND LOCAL SCOPE 000164 | 000165 | The result constructs two lattice VOAs and coherent vertex-field transports 000166 | between them. It does not construct a single alpha-invariant lattice VOA or 000167 | a connected local extension containing both branches. It also does not 000168 | identify the groupoid with a primitive P1 source, its state, its local net or 000169 | its physical sewing functional. 000170 | 000171 | The result does close the previously open algebraic field step: once the 000172 | branch-zero C12 lift and one lattice-VOA arrow T are admitted, every branch 000173 | mark and the full alpha action are fixed by transport. No additional 000174 | full-field phase fit is required. 000175 | 000176 | 000177 | 6. FOCUSED REPLAY 000178 | 000179 | I ran 000180 | 000181 | python3 research-20261027-fundamental-next/fields/ 000182 | check_small_vertex_lifts.py 000183 | 000184 | once. It completed successfully. Its output was byte-identical to the 000185 | stored vertex-lifts.json and had SHA256 000186 | 000187 | 51a243b5a2674dc08e67edd279e2b2c262482d536a97d4112b8f4a997348dc31. 000188 | 000189 | No broader suite was repeated. 000190 | 000191 | 000192 | REVIEWED FILES 000193 | 000194 | research-20261027-fundamental-next/fields/ 000195 | TWO-BRANCH-VERTEX-GROUPOID.txt 000196 | SHA256 fe9459880a659895813853671b5959855d32826cc52c244ca167ff2453b6f8dc 000197 | 000198 | research-20261027-fundamental-next/fields/ 000199 | check_small_vertex_lifts.py 000200 | SHA256 5493a6b479c554897eac88a5eea987f1f31f9c462c143f8d4607ef999da0ac42