TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/SIX-LINE-THETA-COMPLETION.txt Original file SHA-256: 99676497d1de1769807c1e34208f45e12e95b6e9460618b1d84aa3e51e64675f Inclusive original lines: 181-279 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. 000181 | 000182 | |N64|=64, 000183 | N64/Z(G31)=F2^4, 000184 | G31/N64=Sp4(2)=S6, 000185 | |Z(G31)|=4. (15) 000186 | 000187 | E8.CLIFFORD2Q.01 restates the same four dimensional carrier as 000188 | 000189 | |G31| = 4 * 16 * 720, 000190 | 000191 | with the 16 projective Pauli translations and the full symplectic action. 000192 | This has the abstract shape of a genus two theta or Heisenberg normalizer. 000193 | The new cover supplies only a group of the required abstract 16 type: by the 000194 | Weil pairing, its deck characters form a symplectic F2^4. No map to the 000195 | native Pauli 16 is implied by that statement. 000196 | 000197 | For a principally polarized genus two Jacobian, the standard theta group of 000198 | O(2 Theta), after the corresponding theta data are chosen, provides a 000199 | Heisenberg central extension and its four dimensional Schrodinger module. 000200 | That general construction is not the missing theorem. Two source-specific 000201 | identifications are still required: 000202 | 000203 | 1. a marked central extension of this Jac(C)[2] by the same mu4 as the 000204 | native N64, including the commutator pairing and its four dimensional 000205 | Schrodinger representation; 000206 | 000207 | 2. the full Sp4(2) action. The old curve automorphisms used here generate 000208 | only congruent C6 on the six marked objects. Even the full 000209 | automorphism action of this fixed hexagonal branch set lies in its 000210 | dihedral D12, far below Sp4(2) congruent S6. 000211 | 000212 | The Atlas contains two related results, but neither supplies these missing 000213 | maps for the present cover. CURVE.CODE.2TORSION.01 finds only a bare 000214 | symplectic F2^4 comparison for the separate CM abelian surface A cut from 000215 | y^3=x^4-1; its inner equivariant and Arf comparisons fail. In particular, 000216 | its frozen tau invariant polarization is type (1,3), while Jac(C) has its 000217 | canonical principal polarization preserved by c. CURVE.CODE.OUTERSPIN.01 000218 | then constructs a canonical outer S6 bridge after an anchor pin, explicitly 000219 | as a combinatorial duality rather than an inner identification. 000220 | 000221 | Consequently no old source found in the targeted Atlas and paper search 000222 | identifies the standard theta group attached to the present marked Jacobian 000223 | with the native N64, selects the required theta structure and central phase, 000224 | or identifies the present fixed-curve action with the full native Sp4(2) 000225 | quotient. Importing those identifications from the shared numbers 16 and 000226 | 720 would repeat the exact inner-identification error excluded by the Atlas. 000227 | 000228 | 000229 | 6. EXACT CONSEQUENCE AND NEXT SOURCE RELATION 000230 | 000231 | The old source symmetry forces one new and useful completion: 000232 | 000233 | accepted q packet 000234 | + required s covariance 000235 | = q+p six line system 000236 | -> maximal unramified (Z/2)^4 cover of C. (16) 000237 | 000238 | The old T=sc point hexagon lifts to this coefficient cover, and its 000239 | M direct-sum (-M) decomposition is exact. This closes the first field 000240 | transport question at the flat coefficient level. 000241 | 000242 | It does not alter the distinct Q marking. The q packet remains the selected 000243 | v117 (4,3,2) source, Q remains its marked order-four coefficient clock, and 000244 | the p packet is added only to close the coefficient system under s. 000245 | 000246 | The smallest remaining G31 test is now sharply typed. One must construct, 000247 | from the original source rather than from an arbitrary symplectic basis, a 000248 | map 000249 | 000250 | Jac(C)[2] -> V=L/(1+i)L (17) 000251 | 000252 | which carries the Weil pairing, the chosen mu4 theta extension, the q/p 000253 | marking, and the actual clock actions to the native Pauli/G31 data. The bare 000254 | existence of 720 symplectic maps is already known and is not sufficient. 000255 | The old inner comparison gives zero at the action and Arf layers; the outer 000256 | bridge is a separate pinned duality. Until (17) is supplied with its central 000257 | lift, the exact result is the six line coefficient completion, not a curve 000258 | derivation of full G31 or of the charged field system. 000259 | 000260 | 000261 | SOURCE ANCHORS 000262 | 000263 | [OMITTED: non-scientific conversation or execution metadata] 000264 | [OMITTED: non-scientific conversation or execution metadata] 000265 | [OMITTED: non-scientific conversation or execution metadata] 000266 | [OMITTED: non-scientific conversation or execution metadata] 000267 | [OMITTED: non-scientific conversation or execution metadata] 000268 | [OMITTED: non-scientific conversation or execution metadata] 000269 | [OMITTED: non-scientific conversation or execution metadata] 000270 | [OMITTED: non-scientific conversation or execution metadata] 000271 | [OMITTED: non-scientific conversation or execution metadata] 000272 | [OMITTED: non-scientific conversation or execution metadata] 000273 | [OMITTED: non-scientific conversation or execution metadata] 000274 | [OMITTED: non-scientific conversation or execution metadata] 000275 | [OMITTED: non-scientific conversation or execution metadata] 000276 | [OMITTED: non-scientific conversation or execution metadata] 000277 | [OMITTED: non-scientific conversation or execution metadata] 000278 | 000279 | No Atlas, ledger, paper, or prior report was modified.