TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/PAULI-SPIN-CENTRAL-LIFT-INDEPENDENT-REVIEW.txt Original file SHA-256: 81b238c1ff40cbd5804fc13841d0762fcb4ece53a8cc0189719aea4f3fc3632f Inclusive original lines: 1-115 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. 000001 | INDEPENDENT REVIEW: PAULI SPIN CENTRAL LIFT 000002 | 2026-10-26 000003 | 000004 | Reviewed sources 000005 | 000006 | PAULI-SPIN-CENTRAL-LIFT.txt 000007 | SHA-256 e5ef50c899eba80dca03e1b7023edf816df3b45a669534fccf2fef215b0ae8e6 000008 | 000009 | check_pauli_spin_central_lift.py 000010 | SHA-256 984143297e98ebcc91185f642fb3946c31f53da7035fb6e5c8b4dc589d3d832b 000011 | 000012 | VERDICT 000013 | 000014 | ACCEPTED AT THE FINITE CLIFFORD LEVEL, WITH THE CENTRAL SOURCE MAP 000015 | CONDITIONAL AS STATED IN THE REVIEWED REPORT. 000016 | 000017 | 1. FIBER AND CENTRAL-SQUARE CENSUS 000018 | 000019 | The checker uses the exact v783 Clifford generators and enumerates all 720 000020 | symplectic actions. It finds the nine involutions preserving both native 000021 | source-matched symplectic planes and conjugating the frozen cubic action to 000022 | its inverse. Each chosen Q(i)-rational representative has the full 64-element 000023 | G31 fiber mu4 times Pauli16. 000024 | 000025 | For every one of the nine fibers the square census is 000026 | 000027 | +I: 8, -I: 8, nonscalar: 48. 000028 | 000029 | After multiplication by zeta8 it becomes 000030 | 000031 | +iI: 8, -iI: 8, nonscalar: 48. 000032 | 000033 | If the sought global theta-extension dictionary maps beta to J=+iI, then a 000034 | corresponding alpha cannot lie in G31 and does lie in the odd zeta8 G31 000035 | Clifford coset. The selected local +i phase at q0 makes this identification a 000036 | compatible candidate, but does not prove it globally on H0(2Theta). The 000037 | actual theta functor must still decide whether the fiber phase survives or 000038 | cancels in cohomology. 000039 | 000040 | 2. FULL CUBIC-QUARTER RELATION 000041 | 000042 | The square condition alone leaves four Pauli factors modulo overall sign. 000043 | The corrected checker additionally applies the exact matrix filter 000044 | 000045 | alpha sigma alpha^(-1)=sigma^(-1). 000046 | 000047 | In every one of the nine Q fibers exactly two elements survive. They are 000048 | alpha and -alpha, so the lift is unique modulo the source sign once the 000049 | quotient Q action is fixed. 000050 | 000051 | For the displayed Q action the square-only factors are ZI, ZZ, XI and XZ. 000052 | Only XI satisfies the cubic conjugation relation. The explicit matrix 000053 | 000054 | alpha=zeta8 (XI) U 000055 | 000056 | obeys alpha^2=iI, alpha^4=-I, alpha^8=I and 000057 | alpha sigma alpha^(-1)=sigma^(-1). This corrects the earlier square-only ZI 000058 | candidate. 000059 | 000060 | The generated group has order 24. Indeed alpha has order eight, sigma has 000061 | order three, their cyclic subgroups have trivial intersection, and conjugation 000062 | by alpha acts on by inversion. It is C3 semidirect C8, with the C8 000063 | action factoring through C2 and alpha^2 generating the central C4. 000064 | 000065 | 3. CENTRALIZER ORBIT 000066 | 000067 | The exact Sp(4,2) enumeration gives a centralizer of order 18 for the frozen 000068 | Pauli action of sigma, and conjugation by this centralizer is transitive on all 000069 | nine quotient Q actions. 000070 | 000071 | The independent v634 group census gives |C_G31(J sigma)|=72. Since J is 000072 | central, this is also |C_G31(sigma)|. Its kernel over the symplectic 000073 | centralizer is exactly mu4: a kernel element commuting with sigma would define 000074 | a sigma-fixed Pauli translation, and the fixed space on N64/mu4 is zero. 000075 | Thus 000076 | 000077 | C_G31(sigma)/mu4 = C_Sp4(2)(Sigma), 000078 | 000079 | and every quotient conjugator has an actual lift commuting with sigma and J. 000080 | Conjugation transports the unique sign class over one Q action to the unique 000081 | sign class over the conjugate action. Hence the nine alpha sign classes form 000082 | one conjugacy orbit under the actual native centralizer. Calling that orbit a 000083 | physical gauge class would additionally require the full centralizer to be an 000084 | allowed gauge group. 000085 | 000086 | 4. REPLAY 000087 | 000088 | Both commands pass: 000089 | 000090 | python3 check_pauli_spin_central_lift.py 000091 | python3 -OO check_pauli_spin_central_lift.py 000092 | 000093 | Their complete standard outputs are byte-identical with SHA-256 000094 | 000095 | 823edd5914789f231ebb765f0f7e685ba55c09a96b3629176897e5c857532977. 000096 | 000097 | 5. PRECISE SCOPE 000098 | 000099 | The result constructs a finite four-dimensional Clifford candidate for the 000100 | central phase and the joint cubic-quarter relations after beta->J and the 000101 | lifted cubic relation are imposed. It removes the 000102 | fourfold Pauli-factor ambiguity for each fixed quotient Q action. The nine 000103 | quotient representatives are one native centralizer conjugacy orbit when only 000104 | sigma, J and the Pauli translation structure are held fixed. 000105 | 000106 | This orbit statement does not identify a preferred representative after adding 000107 | P1, P2, ordered q-versus-p, anchor, state, or readout data. Those markings are 000108 | outside the centralizer contract checked here. 000109 | 000110 | This is not yet the geometric isomorphism from the source theta or 000111 | Schrodinger module to the native Gaussian lattice module. That conclusion 000112 | still requires a marked isomorphism of the complete theta and N64 central 000113 | extensions and the induced four-dimensional representations. No physical 000114 | state, local field, or P1 source identification follows from the finite lift 000115 | census alone.