TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/e-series-joint-source-20261023/JOINT-CLOCK.txt
Original file SHA-256: 02a15050dc0d8f7ad69ad1d867318d7a9abfa4bee274e14f1f06a12cf16c9628
Inclusive original lines: 191-281
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.

000191 | 
000192 | [OMITTED: non-scientific conversation or execution metadata]
000193 | [OMITTED: non-scientific conversation or execution metadata]
000194 | 
000195 | Using the diagonal relation (2.1), equations (3.3), and the exact cover
000196 | powers gives
000197 | 
000198 | [OMITTED: non-scientific conversation or execution metadata]
000199 | [OMITTED: non-scientific conversation or execution metadata]
000200 | [OMITTED: non-scientific conversation or execution metadata]
000201 | 
000202 | [OMITTED: non-scientific conversation or execution metadata]
000203 | [OMITTED: non-scientific conversation or execution metadata]
000204 | 
000205 | [OMITTED: non-scientific conversation or execution metadata]
000206 | 
000207 | Equation (4.2) retains the oriented sigma deck motion in the marked C12
000208 | presentation.  It becomes vertical sigmahat on the associated local system
000209 | because Delta12^4 lies in the kernel of its peripheral C4 representation.
000210 | Equation (4.3) is the joint compiler J: it covers the nontrivial base quarter
000211 | rotation and acts by Jhat on the fibre.
000212 | 
000213 | The peripheral J is instead
000214 | 
000215 |   Jper[p,v]=[Delta12 p,v]=[p,Jhat v].                 (4.5)
000216 | 
000217 | It covers the identity on B°.  Therefore
000218 | 
000219 | [OMITTED: non-scientific conversation or execution metadata]
000220 | [OMITTED: non-scientific conversation or execution metadata]
000221 | [OMITTED: non-scientific conversation or execution metadata]
000222 | 
000223 | This is the precise resolution of the two roles called J.  They share the
000224 | same fibre lift because the associated local system was built from the
000225 | peripheral J monodromy.  They remain different total automorphisms because
000226 | their base components differ.  No phase or character gauge can remove that
000227 | base-component distinction.
000228 | 
000229 | The compiler factorization also holds on the joint object:
000230 | 
000231 | [OMITTED: non-scientific conversation or execution metadata]
000232 | 
000233 | where Jcomp is (4.3) and Sigma is (4.2).  This is the existing c=J sigma
000234 | relation with cover, base and fibre roles all retained.
000235 | 
000236 | 5. What this decides and what it does not identify
000237 | 
000238 | The restriction of the original character-valued lattice-lift extension to
000239 | the cyclic compiler subgroup <c> splits.  Fixing Jhat leaves one character
000240 | gauge class, represented by (3.4).  This does not split the full G31
000241 | extension.  The predecessor's explicit nonsplitting certificate remains
000242 | unchanged.
000243 | 
000244 | No Burau lattice is identified with geometric H1 or with the E8 lattice.  The
000245 | existing integral Phi12 intertwiner remains an exact unpolarized comparison
000246 | between its two stated modules.  Its complementary spectra and alternating
000247 | form types remain different.  The present construction instead puts the
000248 | existing cover and the existing V_E8 fibre into their standard associated
000249 | local system, so no carrier identification is needed.
000250 | 
000251 | No physical seam, state, time, coupling or field-selection rule is asserted.
000252 | This is a complete mathematical joint action for the supplied marked cover,
000253 | the unchanged lattice cocycle and the cyclic compiler subgroup.
000254 | 
000255 | 6. Reproducible certificate and references
000256 | 
000257 | Run
000258 | 
000259 |   python3 -B experiments/theory-contracts/e-series-joint-source-20261023/joint_clock.py
000260 |   python3 -OO experiments/theory-contracts/e-series-joint-source-20261023/joint_clock.py
000261 | 
000262 | Both modes return the same affine rank, four representatives, one gauge orbit,
000263 | and zero representatives with separately generator-normalized sigmahat.
000264 | 
000265 | The lift formalism is consistent with the standard lattice-VOA fact that a
000266 | lattice isometry lifts with character freedom, while group relations can
000267 | acquire cocycle obstructions:
000268 | 
000269 |   Dong and Nagatomo, Automorphism Groups and Twisted Modules for Lattice
000270 |   Vertex Operator Algebras, arXiv:math/9808088.
000271 | 
000272 |   Bakalov and Kac, Twisted Modules over Lattice Vertex Algebras,
000273 |   arXiv:math/0402315.
000274 | 
000275 |   Okada, On the Symmetry of Odd Leech Lattice CFT,
000276 |   arXiv:2412.19430, especially the lift-extension method.
000277 | 
000278 | These general references do not supply the coefficients above.  The masks,
000279 | affine rank, gauge orbit and joint cover relations are computed from the
000280 | actual E8 basis, epsilon, J, sigma and forced C12 cover retained in the cited
000281 | predecessor contracts.