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: 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 | THE SPIN-RIGIDIFIED THETA MODULE IS THE NATIVE N64 MODULE 000002 | Finite Heisenberg and Clifford closure after the marked q divisor twist 000003 | 2026-10-26 000004 | 000005 | VERDICT 000006 | 000007 | PROVED at the finite geometric module level, conditional only on the already 000008 | isolated physical insertion rule. 000009 | 000010 | Let A=Jac(C), let L=O_A(2 Theta), and use the source-selected spin 000011 | normalization and the marked negative-divisor line from 000012 | MARKED-Q-DIVISOR-CENTRAL-TWIST. If the physical source uses the one-copy 000013 | factor N_q(S_0)^(-1), then 000014 | 000015 | W_q=H^0(A,L) tensor N_q(S_0)^(-1) 000016 | 000017 | carries the exact source relations 000018 | 000019 | gamma^3=I, 000020 | alpha^2=iI, 000021 | alpha gamma alpha^(-1)=gamma^(-1). (1) 000022 | 000023 | Here gamma is the normalized cubic lift and alpha is the spin-rigidified 000024 | quarter lift. The exact normalization in (1) does not add a phase fit. If 000025 | an order-three lift gamma_0 initially obeys 000026 | 000027 | alpha gamma_0 alpha^(-1)=kappa gamma_0^(-1), 000028 | 000029 | then kappa^3=1. There is a unique cube root lambda with 000030 | kappa lambda^2=1, and gamma=lambda gamma_0 still has order three. This 000031 | normalization is forced by the actual geometric relation QcQ^(-1)=c^(-1). 000032 | It does not change the q norm line, where a cube-root rescaling enters through 000033 | its third power. This fixes the representative of the initially projective 000034 | theta linearization; it is precisely the normalization used to define the 000035 | source gamma in (1), not a later change of an independently fixed operator. 000036 | 000037 | The finite theta group of L contains a canonical 64-element 000038 | Heisenberg subgroup N with center mu4. Its action on W_q is the unique 000039 | four-dimensional irreducible representation with central character 000040 | zeta |-> zeta. Every explicit symplectic identification 000041 | 000042 | Phi:A[2] -> N64/mu4 000043 | 000044 | lifts to an isomorphism N -> N64 fixing mu4, and finite Stone-von Neumann 000045 | then supplies an intertwiner W_q -> C^4, unique up to scalar. 000046 | 000047 | For the already matched fixed-free cubic action, this intertwiner can be 000048 | chosen so that gamma maps to the frozen native sigma itself. No residual 000049 | cube-root phase remains. The transported alpha then lies in the single 000050 | joint native class already proved by the exact Pauli-spin census: for each 000051 | of the nine compatible quotient actions there are exactly alpha and -alpha, 000052 | and the nine sign classes form one orbit under the actual native centralizer. 000053 | 000054 | Thus the finite geometric theta module, its mu4 center, its cubic lift and 000055 | its spin quarter lift are the native N64/Clifford package up to Heisenberg 000056 | frame, the unavoidable sign of alpha, and the stated centralizer orbit. 000057 | Calling every such frame change a physical gauge transformation would still 000058 | require the additional P1, P2, anchor, state and readout markings. 000059 | 000060 | The remaining source question is no longer a missing four-dimensional module 000061 | intertwiner. It is the earlier, sharply typed physical question whether the 000062 | original boundary, sewing or Calderon construction inserts one inverse copy 000063 | of N_q(S_0). Without that insertion the ordinary theta functor has central 000064 | phase +1 and equation (1) does not hold. 000065 | 000066 | 000067 | 1. THE FOURTH-ROOT SUBGROUP IS CLOSED AND HAS ORDER 64 000068 | 000069 | The theta group of L fits into 000070 | 000071 | 1 -> C* -> G(L) -> K(L)=A[2] -> 1. (2) 000072 | 000073 | For x,y over A[2], their commutator is the Weil pairing e_2(x,y), hence is 000074 | +1 or -1 and is central. Define 000075 | 000076 | N={g in G(L):g^4=1}. (3) 000077 | 000078 | This is a subgroup. Indeed, in this class-two group, 000079 | 000080 | (gh)^4=g^4 h^4 [h,g]^6=1 (4) 000081 | 000082 | for g,h in N, because [h,g]^2=1. Also g^(-1) belongs to N. Over each of 000083 | the sixteen elements of A[2], any theta-group lift can be multiplied by a 000084 | scalar so that its fourth power is one. Exactly four scalar multiples do 000085 | so. Therefore 000086 | 000087 | |N|=16*4=64, ker(N -> A[2])=mu4. (5) 000088 | 000089 | The commutator pairing on A[2] is nondegenerate. Hence an element central 000090 | in N has trivial image in A[2], and 000091 | 000092 | Z(N)=mu4. (6) 000093 | 000094 | This also shows directly that (3) is the finite theta Heisenberg group, not 000095 | an arbitrary 64-element subgroup selected from G(L). 000096 | 000097 | 000098 | 2. EVERY SYMPLECTIC PHI LIFTS, AND THE MODULE IS UNIQUE 000099 | 000100 | Choose a symplectic basis e_1,f_1,e_2,f_2 of A[2]. Choose lifts in N. 000101 | Each basis lift has square +1 or -1. Multiplication by i reverses that 000102 | square, so the four lifts may be normalized to square +1. Their only 000103 | nontrivial commutators are 000104 | 000105 | [E_j,F_j]=-1. (7) 000106 | 000107 | These are exactly the standard two-qubit Pauli relations with center mu4. 000108 | Consequently every specified symplectic map Phi lifts to a center-preserving 000109 | isomorphism 000110 | 000111 | Phi_tilde:N -> N64. (8) 000112 | 000113 | Different lift choices differ by a character of A[2], equivalently by an 000114 | inner Pauli change of Heisenberg frame. There is no further extension-class 000115 | obstruction hidden behind Phi. 000116 | 000117 | Riemann-Roch gives 000118 | 000119 | dim H^0(A,O(2 Theta))=2^2=4. (9) 000120 | 000121 | The theta-group action has central weight one and is irreducible. The finite 000122 | Stone-von Neumann theorem says that N has exactly one irreducible 000123 | representation with this faithful central character, of dimension 000124 | 000125 | sqrt(|N/Z(N)|)=4. (10) 000126 | 000127 | The constant line N_q(S_0)^(-1) changes the alpha and beta characters proved 000128 | in the preceding report, but it does not alter the N action. Equations 000129 | (8)-(10) therefore produce an intertwiner 000130 | 000131 | T:W_q -> C^4, T rho_N(g) T^(-1)=rho_native(Phi_tilde(g)), 000132 | 000133 | unique up to one nonzero scalar. (11) 000134 | 000135 | 000136 | 3. THE CUBIC OPERATOR HAS NO RESIDUAL PAULI OR SCALAR SHIFT 000137 | 000138 | The source action of c on A[2] is fixed-free and Phi identifies it with the 000139 | native Pauli action Sigma of the frozen sigma. Transport gamma through T. 000140 | The unitary normalizer of N64 maps onto Sp(4,2); the kernel is U(1) times the 000141 | Pauli translations. Hence initially 000142 | 000143 | T gamma T^(-1)=lambda p sigma (12) 000144 | 000145 | for a scalar lambda and a Pauli label p. 000146 | 000147 | Conjugating (12) by a Pauli q changes p by (1-Sigma)q. Since Sigma has no 000148 | nonzero fixed vector on the four-dimensional F2 Pauli quotient and has odd 000149 | order, 1-Sigma is invertible. There is therefore a unique q that removes p. 000150 | After this Heisenberg-frame change, 000151 | 000152 | T gamma T^(-1)=lambda sigma. (13) 000153 | 000154 | Both gamma and sigma have exact order three, so lambda^3=1. It remains to 000155 | exclude the two nontrivial cube roots without choosing a trace convention. 000156 | 000157 | The source relation in (1), together with unitarity, gives 000158 | 000159 | tr(gamma)=tr(gamma^(-1))=conjugate(tr(gamma)), (14) 000160 | 000161 | so tr(gamma) is real. On End(W_q), conjugation by gamma permutes the Pauli 000162 | basis through the fixed-free action of c. Only the identity Pauli is fixed; 000163 | the fifteen nonzero labels form five three-cycles. Therefore 000164 | 000165 | tr(Ad gamma)=1=|tr(gamma)|^2. (15) 000166 | 000167 | No physical metric is assumed here. The finite group generated by N, alpha 000168 | and gamma admits an invariant positive Hermitian form by averaging, so the 000169 | finite complex representation may be made unitary for this trace argument. 000170 | 000171 | An order-three operator on a four-dimensional space has multiplicities 000172 | (n_0,n_1,n_2) for eigenvalues 1,omega,omega^2. Reality in (14) gives 000173 | n_1=n_2. Together with (15) and n_0+n_1+n_2=4, this forces 000174 | 000175 | (n_0,n_1,n_2)=(2,1,1), tr(gamma)=1. (16) 000176 | 000177 | The frozen native sigma has the same spectrum and trace one. Equations 000178 | (13) and (16) therefore force 000179 | 000180 | lambda=1, T gamma T^(-1)=sigma. (17)