TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/theta_lattice_closure_20261027/fields/TWO-BRANCH-VERTEX-GROUPOID.txt
Original file SHA-256: fe9459880a659895813853671b5959855d32826cc52c244ca167ff2453b6f8dc
Inclusive original lines: 201-297
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.

000201 | 
000202 | Exactly 256 triples additionally satisfy
000203 | 
000204 |   [V,J]=1,
000205 |   [sigma,J]=1,
000206 |   V sigma V^(-1)=sigma^(-1).                                 (20)
000207 | 
000208 | The lexicographically first witness has character masks
000209 | 
000210 |   t_J     =0x03,       support {0,1},
000211 |   t_V     =0x35,       support {0,2,4,5},
000212 |   t_sigma =0x2a,       support {1,3,5}.                       (21)
000213 | 
000214 | The generated group of vertex automorphisms has exactly 24 elements.  For
000215 | 
000216 |   c=J sigma
000217 | 
000218 | the checker proves
000219 | 
000220 |   order(c)=12,             c^4=sigma,          c^9=J.         (22)
000221 | 
000222 | No triple in the complete census has t_J=0.  Therefore the zero-on-basis
000223 | standard J lift does not extend to this strict same-frame C4 x S3 section.
000224 | This is not an obstruction to the groupoid construction in section 2.  It is
000225 | the exact reason the transported two-object construction is the safer result
000226 | when an earlier J or C12 lift must remain fixed.
000227 | 
000228 | The number 256 counts raw sign-character sections of the fixed standard
000229 | cocycle.  It is not a count of physical choices or inequivalent theories.
000230 | Character gauge and further marked centralizers can identify sections.  No
000231 | quotient census is needed for the existence theorem proved here.
000232 | 
000233 | 
000234 | 4. SCOPE
000235 | ========
000236 | 
000237 | PROVED:
000238 | 
000239 |   * the actual standard cocycle on the theta E8 basis admits a unitary lift of
000240 |     the lattice isomorphism Lambda -> zeta_8 Lambda;
000241 | 
000242 |   * given the existing faithful branch-zero C12 lift, transport along that one
000243 |     arrow produces coherent beta, sigma and C12 actions on branch one;
000244 | 
000245 |   * alpha has exact order eight on the direct sum groupoid representation,
000246 |     squares to the branchwise beta and reverses sigma on all vertex fields;
000247 | 
000248 |   * the optional same-frame subgroup C4 x S3 also splits after a compatible
000249 |     sign-character choice and has exact order 24.
000250 | 
000251 | NOT CLAIMED:
000252 | 
000253 |   * no splitting of the full G31 lattice extension;
000254 | 
000255 |   * no alteration of the established nonsplitting result for that larger
000256 |     extension;
000257 | 
000258 |   * no single alpha-invariant lattice or single local VOA containing both
000259 |     branches;
000260 | 
000261 |   * no physical P1 sewing, state or field-measure identification;
000262 | 
000263 |   * no claim that the optional masks (21) equal a previously frozen source
000264 |     linearization.  They are a concrete compatible section of the fixed
000265 |     cocycle extension.
000266 | 
000267 | The groupoid result needs none of the optional masks.  It preserves any given
000268 | faithful C12 lift on branch zero and defines every branch-one mark by transport.
000269 | The executable certificate instantiates beta_0,sigma_0 with the displayed
000270 | strict witness so that every identity can be replayed entrywise.  The theorem
000271 | it certifies is stronger in the relevant direction: for any already fixed
000272 | faithful c_0, set beta_0=c_0^9 and sigma_0=c_0^4, and equations (2),(3) prove
000273 | the same groupoid identities for every cocycle lift T.  The instantiation is
000274 | not used to replace the original c_0 or J linearization.
000275 | 
000276 | Individual lattice-isometry lifts exist by (8)-(11).  Nothing here constructs
000277 | or assumes a simultaneous lift of every element of G31.
000278 | 
000279 | 
000280 | 5. CERTIFICATE
000281 | ==============
000282 | 
000283 | Input lattice checker:
000284 | 
000285 |   research-20261027-fundamental-next/lattice/check_theta_lattice.py
000286 |   SHA256 a45fcf5c66ebd208a050c30b9297dfca6dfccc28a68398d75cd91285f9e18b5e
000287 | 
000288 | New exact checker and outputs:
000289 | 
000290 |   research-20261027-fundamental-next/fields/check_small_vertex_lifts.py
000291 |   research-20261027-fundamental-next/fields/vertex-lifts.json
000292 |   research-20261027-fundamental-next/fields/vertex-lifts-optimized.json
000293 | 
000294 | The checker runs normally and with python3 -OO.  The two JSON outputs are byte
000295 | identical.  It checks the cocycle polarization identity, all lift relations on
000296 | all 256 classes of Lambda/2Lambda, the complete sign-character census, exact
000297 | group order 24, both C12 clocks and the two branch alpha identities.