TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/PAULI-SPIN-CENTRAL-LIFT.txt
Original file SHA-256: e5ef50c899eba80dca03e1b7023edf816df3b45a669534fccf2fef215b0ae8e6
Inclusive original lines: 181-262
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 |     alpha_native = zeta8 (XI) U.                             (14)
000183 | 
000184 | Direct exact multiplication gives
000185 | 
000186 |     alpha_native^2=iI=J,
000187 |     alpha_native^4=-I,
000188 |     alpha_native^8=I,
000189 |     alpha_native sigma alpha_native^(-1)=sigma^(-1).         (15)
000190 | 
000191 | Thus (14) is an exact native realization of the desired spin order, central
000192 | square, and lifted cubic conjugation law.  It lies in
000193 | zeta8 G31, while its square lies in the mu4 center of G31.
000194 | 
000195 | For this displayed Q_native, the square condition alone leaves the four Pauli
000196 | factors
000197 | 
000198 |     ZI, ZZ, XI, XZ.                                          (16)
000199 | 
000200 | The full conjugation law (3) retains only XI.  Therefore (14) is unique modulo
000201 | alpha maps to -alpha for the displayed Q action.
000202 | 
000203 | 
000204 | 5. WHAT THIS ADDS, AND WHAT IT DOES NOT
000205 | 
000206 | The finite result supplies the following compatible dictionary candidate:
000207 | 
000208 |     source spin lift, if beta is identified globally with iI,
000209 |       maps to the odd zeta8 Clifford coset,
000210 |     source beta
000211 |       maps to the native center J=iI,
000212 |     lifted cubic and quarter relation, if imposed on frozen sigma,
000213 |       maps to the exact order 24 subgroup (8).                (17)
000214 | 
000215 | It repairs the earlier wrong target obstruction as an existence test.  The old statement that a
000216 | central scalar cannot exchange J with -J remains unchanged.  The present
000217 | statement concerns the square class of a lifted Q action, not conjugation of
000218 | the center.
000219 | 
000220 | The remaining gap is now smaller and concrete.  One still needs to construct
000221 | the source theta Heisenberg extension, prove how its global beta and cubic
000222 | linearization map to J and frozen sigma, test the additional markings, and
000223 | identify the complete four dimensional Schrodinger representation.
000224 | The finite Stone von Neumann theorem would identify representations after an
000225 | extension isomorphism is supplied; it does not select that isomorphism.
000226 | 
000227 | 
000228 | 6. EXACT CHECK
000229 | 
000230 | Run
000231 | 
000232 |     python3 research-20261026-real-boundary/check_pauli_spin_central_lift.py
000233 |     python3 -OO research-20261026-real-boundary/check_pauli_spin_central_lift.py
000234 | 
000235 | Both runs pass.  They enumerate Sp(4,2), all nine Q actions, each actual
000236 | 64 element G31 fiber, and each zeta8 coset fiber.  They also verify the explicit
000237 | matrix (13), the full relations (15), and the transitive centralizer orbit.
000238 | 
000239 | 
000240 | SOURCE ANCHORS
000241 | 
000242 | Atlas audit frame: revision 966491c8570e1f00b7db8adf, read only.
000243 | 
000244 |   * verification/v783_two_qubit_clifford.py, SHA256
000245 |     8f4851634b83f61671b04f3a6211059c40758d21caaf1f779302c799e1e9d6c4:
000246 |     lines 496--519 fix the Pauli chart and commutator form; lines 588--640 fix
000247 |     exact Clifford conjugation; lines 698--743 prove (4); lines 745--780 prove
000248 |     the mu8 scalar group, the index two decomposition, and its Q(i) rationality
000249 |     character.
000250 |   * verification/v634_st31_structure.py, SHA256
000251 |     5ff53ed031ca03103187ac22edf723aabce88bee3df1085c52486ecb0044d37c:
000252 |     lines 971--1011 compute the native compiler centralizer, in particular
000253 |     |C_G31(c_comp)|=72.  Because J is central, the same order applies to
000254 |     C_G31(sigma).
000255 |   * research-20261026-real-boundary/PAULI-VERSUS-CODE-SOURCE-TARGET.txt:
000256 |     equations (7) through (13) construct the nine element comparison problem
000257 |     and the displayed Q_native.
000258 |   * research-20261026-real-boundary/SOURCE-SELECTED-THETA-SPIN.txt, SHA256
000259 |     cf4478191ccd75d62a2f049a54193541009bed24877df9cbe0ed9f942d07d717:
000260 |     lines 97--145 prove alpha^4=-I, beta=alpha^2, beta^2=-I, and select the +i
000261 |     local branch from the original marked orientation.
000262 |