TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/SPIN-RIGIDIFIED-THETA-MODULE.txt
Original file SHA-256: 4bd17ffbb2be4b31c4eb7a4efe9cd8a1191c9065655889ce189f3b1f6c2499d2
Inclusive original lines: 181-276
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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000181 | 
000182 | Thus neither a Pauli translation nor a central cube-root phase survives.
000183 | 
000184 | 
000185 | 4. THE QUARTER LIFT LANDS IN THE UNIQUE NATIVE JOINT CLASS
000186 | 
000187 | Transport alpha through the same T.  By (1) and (17), it normalizes N64 and
000188 | satisfies
000189 | 
000190 |     alpha_native^2=iI,
000191 |     alpha_native sigma alpha_native^(-1)=sigma^(-1).           (18)
000192 | 
000193 | Its quotient action Q_bar consequently obeys
000194 | 
000195 |     Q_bar^2=I,
000196 |     Q_bar Sigma Q_bar=Sigma^(-1).                              (19)
000197 | 
000198 | The exact native census in PAULI-SPIN-CENTRAL-LIFT enumerates all 720
000199 | elements of Sp(4,2), all nine quotient actions satisfying the marked plane
000200 | conditions and (19), and all 64 Pauli-center lifts over each action.  It
000201 | proves:
000202 | 
000203 |   * every solution of (18) lies in zeta8 G31;
000204 |   * each Q_bar has exactly two solutions, alpha_native and -alpha_native;
000205 |   * the nine sign classes form one orbit under C_G31(sigma);
000206 |   * one representative is alpha_native=zeta8 (XI) U.
000207 | 
000208 | The continuous U(1) kernel of the full unitary normalizer creates no omitted
000209 | solutions.  The first equation in (18) restricts its phase to the two signs
000210 | already present in the exhaustive fiber census.  Likewise, section 3 used
000211 | gamma^3=1 and the trace to eliminate its full scalar phase.  Thus the earlier
000212 | census really is exhaustive for the transported geometric operators, rather
000213 | than only for a preselected rational phase convention.
000214 | 
000215 | 
000216 | 5. EXACT SCOPE
000217 | 
000218 | The result closes the following mathematical arrow:
000219 | 
000220 |     source-selected (A[2],N,W_q,gamma,alpha)
000221 |        -> native (N64,C^4,sigma,alpha_native)                  (20)
000222 | 
000223 | up to Heisenberg frame, alpha -> -alpha, and the single native centralizer
000224 | orbit.  No new four-dimensional representation, continuous phase or Cartan
000225 | choice is required.
000226 | 
000227 | This is a complex representation statement.  Pulling the native Hermitian
000228 | form or Gaussian E8 lattice back through T certainly produces such a form or
000229 | lattice on W_q, but the finite theorem does not prove that the original curve
000230 | or boundary source selected that integral lattice, its metric, a W8 operator,
000231 | or a physical state rule.  Those structures cannot be transferred merely
000232 | because the complex N64 modules are isomorphic.
000233 | 
000234 | Equation (20) is conditional on using W_q, because the one-copy factor is
000235 | what changes alpha^2 from +I on the ordinary second-order theta space to
000236 | +iI.  The geometry constructs and rigidifies this factor canonically, but
000237 | the original physical sewing rule has not yet been proved to insert it.
000238 | Nothing here supplies that missing rule, identifies a local field or state,
000239 | or proves that all native centralizer conjugacies preserve the additional
000240 | physical markings.
000241 | 
000242 | 
000243 | EXACT REPLAY USED
000244 | 
000245 |     python3 research-20261026-real-boundary/check_pauli_spin_central_lift.py
000246 |     python3 -OO research-20261026-real-boundary/check_pauli_spin_central_lift.py
000247 | 
000248 | The two modes pass with byte-identical output.  They verify the nine quotient
000249 | actions, every 64-element lift fiber, the two surviving signs, the explicit
000250 | matrix relations and the transitive centralizer orbit.
000251 | 
000252 | 
000253 | SOURCES
000254 | 
000255 |   * D. Mumford, On the Equations Defining Abelian Varieties I, Inventiones
000256 |     Mathematicae 1 (1966), sections 1-2, for theta groups and their
000257 |     irreducible Schrödinger representation:
000258 |     https://pazuki.perso.math.cnrs.fr/index_fichiers/Mumford66.pdf
000259 | 
000260 |   * research-20261026-real-boundary/SOURCE-SELECTED-THETA-SPIN.txt:
000261 |     the selected S_0 and projective spin lifts.
000262 | 
000263 |   * research-20261026-real-boundary/MARKED-Q-DIVISOR-CENTRAL-TWIST.txt,
000264 |     SHA256 da1f75831a46a0c0662f931486b77e6e2085126df452b009d1a90351d413d904:
000265 |     the spin-normalized theta family and the one-copy q divisor character.
000266 | 
000267 |   * research-20261026-real-boundary/PAULI-VERSUS-CODE-SOURCE-TARGET.txt:
000268 |     the explicit symplectic Phi, fixed-free cubic Pauli action and nine
000269 |     compatible quotient Q actions.
000270 | 
000271 |   * research-20261026-real-boundary/PAULI-SPIN-CENTRAL-LIFT.txt:
000272 |     the exact full-fiber lift census and native centralizer orbit.
000273 | 
000274 |   * verification/v783_two_qubit_clifford.py and
000275 |     verification/v634_st31_structure.py in the TFPT repository:
000276 |     the native N64, Clifford and centralizer certificates used by that census.