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: 1-180 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 | CENTRAL LIFT CENSUS FOR THE SOURCE SPIN QUARTER 000002 | Exact central lift census in the frozen v783 chart 000003 | 2026-10-26 000004 | 000005 | VERDICT 000006 | 000007 | PROVED at the finite Clifford level, with the source interpretation stated 000008 | conditionally below. 000009 | 000010 | For every one of the nine native Pauli actions compatible with the source 000011 | relations 000012 | 000013 | Q_bar^2=I, 000014 | Q_bar c_bar Q_bar=c_bar^(-1), 000015 | 000016 | the full fiber of 64 lifts inside G31 contains no operator whose square is 000017 | +iI or -iI. Its scalar square census is 000018 | 000019 | square +I: 8, 000020 | square -I: 8, 000021 | nonscalar: 48. (1) 000022 | 000023 | The corresponding fiber in the other two qubit Clifford coset 000024 | 000025 | zeta8 G31, zeta8=exp(pi i/4), 000026 | 000027 | has the uniform census 000028 | 000029 | square +iI: 8, 000030 | square -iI: 8, 000031 | nonscalar: 48. (2) 000032 | 000033 | If the sought source to native theta extension realizes 000034 | 000035 | beta |-> J=+iI, 000036 | alpha^2=beta, 000037 | 000038 | then alpha cannot land in G31 and does land in zeta8 G31. Under this explicit 000039 | central identification, the nontrivial index two Galois parity of v783 is 000040 | forced. The original spin source itself proves beta^2=-I and fixes the +i 000041 | phase on the marked fiber at q0. It does not yet construct a global four 000042 | dimensional theta module in which beta is the scalar +iI. That global 000043 | identification remains part of the missing extension dictionary. In 000044 | particular, the marked fiber phase does not by itself prove that the functor 000045 | S0 -> Theta -> H0(2Theta) carries beta to scalar +iI: the determinant of 000046 | cohomology has Euler characteristic zero, so cancellation of that central 000047 | fiber phase is a live possibility which the actual functor must decide. No statement 000048 | here identifies the v783 parity with physical chirality or with complex 000049 | conjugation. 000050 | 000051 | Imposing the exact lifted cubic relation removes the residual Pauli ambiguity. 000052 | For every one of the nine Q actions, exactly two lifts satisfy 000053 | 000054 | alpha sigma alpha^(-1)=sigma^(-1). (3) 000055 | 000056 | They are alpha and -alpha. Thus each Q action has a unique joint lift modulo 000057 | sign once the frozen native sigma and this lifted relation are required. The 000058 | nine resulting sign classes form one conjugacy orbit under the actual native 000059 | centralizer C_G31(sigma). Whether that conjugacy is a physical gauge 000060 | equivalence depends on the additional P1, P2, q versus p, anchor, state, and 000061 | readout markings. The result constructs one native conjugacy class of finite 000062 | clock subgroup candidates, not a fully marked theta module isomorphism. 000063 | 000064 | 000065 | 1. WHY THE 64 ELEMENT FIBER IS THE ACTUAL G31 FIBER 000066 | 000067 | The exact v783 calculation proves 000068 | 000069 | ker(G31 -> Sp(4,2)) = mu4 times Pauli16, (4) 000070 | 000071 | and proves that G31 is the Q(i) rational index two half of the matrix Clifford 000072 | group C2. The other half is zeta8 G31. Thus, for any Q(i) rational Clifford 000073 | representative U of a specified symplectic action, the actual G31 fiber is 000074 | 000075 | {epsilon P U : epsilon in mu4, P in Pauli16}. (5) 000076 | 000077 | It has 4 times 16 = 64 distinct elements. Multiplying all of (5) by zeta8 000078 | gives the other Clifford fiber over the same symplectic action. 000079 | 000080 | The checker constructs all 720 symplectic actions from the same standard 000081 | two qubit Clifford generators used in v783. It then selects exactly the nine 000082 | involutions that preserve the two source matched nondegenerate planes and 000083 | reverse the frozen cubic action. Every selected BFS representative is 000084 | Q(i) rational, so (5) applies without an extra phase adjustment. 000085 | 000086 | 000087 | 2. THE CENTRAL SQUARE OBSTRUCTION INSIDE G31 000088 | 000089 | For each of the nine representatives U, the checker multiplies by all four 000090 | central phases and all sixteen Hermitian Paulis. It squares every resulting 000091 | matrix exactly over Q(i,sqrt(2)). The census is (1) in all nine cases. 000092 | 000093 | Multiplication of a lift A by the central phase zeta8 changes its square by 000094 | 000095 | (zeta8 A)^2=zeta8^2 A^2=i A^2. (6) 000096 | 000097 | Consequently (1) becomes (2). This is not a numerical phase fit. Equation 000098 | (6) and the exhaustive 64 element fiber census prove both the obstruction in 000099 | G31 and the positive lift in its unique nontrivial Clifford coset. 000100 | 000101 | The source spin result fixes the +i branch on the marked fiber at q0. A 000102 | compatible native central realization is 000103 | 000104 | beta |-> J=iI, (7) 000105 | 000106 | Under (7), alpha must have odd v783 Galois parity. Equation (7) is not yet a 000107 | source proved global scalar identity because the source four dimensional theta 000108 | module has not been identified with the native module. 000109 | 000110 | The square condition alone leaves four Pauli factors modulo sign in each 000111 | fiber. At the quotient level the original Q action inverts the cubic source 000112 | action. To test the stronger native lift, fix the actual compiler matrix 000113 | sigma and impose the entrywise matrix identity (3). This is the required 000114 | lifted compatibility condition; SOURCE-SELECTED-THETA-SPIN does not itself 000115 | prove the corresponding cubic spin linearization or cocycle. The exact filter 000116 | leaves two elements in every fiber, related by overall minus sign. The 000117 | resulting native candidate obeys 000118 | 000119 | sigma^3=I, 000120 | alpha^2=J, 000121 | J=iI central, 000122 | alpha sigma alpha^(-1)=sigma^(-1). (8) 000123 | 000124 | The generated matrix group has order 24. It is C3 semidirect C8, with the C8 000125 | action on C3 factoring through inversion and alpha^2 generating the central 000126 | C4. This is an exact compatible realization of the source quotient and spin 000127 | relations after (7) and (3) are imposed. 000128 | 000129 | 000130 | 3. THE NINE CANDIDATES ARE ONE NATIVE CENTRALIZER CONJUGACY ORBIT 000131 | 000132 | Exact enumeration gives 000133 | 000134 | |C_Sp4(2)(M_sigma)|=18. (9) 000135 | 000136 | This centralizer acts transitively by conjugation on all nine Q actions. 000137 | The native v634 calculation independently proves 000138 | 000139 | |C_G31(c_comp)|=72. (10) 000140 | 000141 | Since J is central and c_comp=J sigma, this is also C_G31(sigma). Its kernel 000142 | in N64 is exactly mu4: any kernel element commuting with sigma would define a 000143 | fixed Pauli translation, while M_sigma has no nonzero fixed vector. Therefore 000144 | 000145 | C_G31(sigma)/mu4 = C_Sp4(2)(M_sigma). (11) 000146 | 000147 | In particular, every conjugating element in (9) has an actual lift commuting 000148 | with sigma and J. Such a lift sends a joint alpha class over one Q action to 000149 | a joint alpha class over the conjugate Q action. Section 2 proved that each 000150 | target fiber contains only alpha and -alpha. It follows that the nine sign 000151 | classes form one orbit under C_G31(sigma). 000152 | 000153 | Thus the nine are one mathematical conjugacy class for the specified native 000154 | pair. Calling this conjugacy a gauge equivalence requires the additional 000155 | claim that the allowed gauge group is the full centralizer. In particular, 000156 | the calculation does not show that every centralizer transformation preserves 000157 | P1, P2, q versus p, anchor, state, or readout markings. Those constraints were 000158 | not included and must be tested in a fully marked physical dictionary. 000159 | 000160 | 000161 | 4. ONE FULLY EXPLICIT LIFT 000162 | 000163 | Use the source matched native action from 000164 | PAULI-VERSUS-CODE-SOURCE-TARGET.txt: 000165 | 000166 | [1 0 1 0] 000167 | [0 1 1 0] 000168 | Q_native = [0 0 1 0]. (12) 000169 | [1 1 1 1] 000170 | 000171 | A Q(i) rational G31 representative is 000172 | 000173 | U=H1 H2 CNOT H1 CNOT H2 000174 | 000175 | [1 0 0 0] 000176 | [0 0 0 -1] 000177 | = [0 0 1 0]. (13) 000178 | [0 1 0 0] 000179 | 000180 | Its Pauli adjoint action is (12). Define