TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/PAULI-VERSUS-CODE-SOURCE-TARGET.txt Original file SHA-256: 618a48566f2a1ad1be9b7f09aa1b2892f43b52644d9d8883db94fd6a75be699d 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 | PAULI TRANSLATIONS VERSUS GAUSSIAN CODE CONTEXTS 000002 | The correctly typed target of the genus two theta source 000003 | 2026-10-26 000004 | 000005 | VERDICT 000006 | 000007 | PROVED at the finite symplectic and marked cubic action level. 000008 | 000009 | The previous comparison used the wrong native four bit object. The source 000010 | 000011 | Jac(C)[2] = V_q direct_sum V_p 000012 | 000013 | is a translation group with its Weil pairing. Its native counterpart is 000014 | 000015 | N64/mu4 = Pauli16, 000016 | 000017 | with the commutator pairing. It is not the Gaussian code quotient 000018 | 000019 | V_code = L/(1+i)L. 000020 | 000021 | The nonzero code classes label the fifteen Pauli contexts through the point 000022 | line duality of W(3,2). They do not label the fifteen Pauli translations. 000023 | Verification v783 proves that the two full Sp(4,2) modules have zero 000024 | intertwiner space. Thus the earlier code fixed space mismatch remains a true 000025 | statement about contexts, but it is not an obstruction to the theta 000026 | translation map. 000027 | 000028 | On the correct Pauli target the frozen native sigma has no nonzero fixed 000029 | vector, exactly like the source cubic c_src. More strongly, its four bit 000030 | module has a unique unordered decomposition into two orthogonal, nondegenerate, 000031 | sigma stable planes. These planes provide an explicit finite symplectic 000032 | carrier for V_q and V_p, up to their exchange. The actual Q action can also be 000033 | transported, but the present source data do not select one of the nine native 000034 | Q markings in a fixed frame. The central lift test proves that all nine lie 000035 | in one actual C_G31(sigma) conjugacy orbit. Hence the fake fixed space obstruction 000036 | is removed and a maximal 000037 | explicit transport component is obtained. A unique full theta module 000038 | dictionary is not yet obtained. 000039 | 000040 | 000041 | 1. THE TWO NATIVE FOUR BIT OBJECTS 000042 | 000043 | In the frozen v783 chart 000044 | 000045 | z_k = r_(2k) + i r_(2k+1), 000046 | 000047 | the lattice permutation 000048 | 000049 | PI_SIG=(4,5,0,1,2,3,6,7) 000050 | 000051 | acts on C^4 by 000052 | 000053 | [0 0 1 0] 000054 | [1 0 0 0] 000055 | sigma = [0 1 0 0]. (1) 000056 | [0 0 0 1] 000057 | 000058 | It cycles the first three basis vectors and fixes the fourth. Thus 000059 | tr_C(sigma)=1 and sigma^3=I. On the code coordinates (F1,F2,F3,A), the same 000060 | matrix over F2 fixes the plane 000061 | 000062 | span{F1+F2+F3, A}. (2) 000063 | 000064 | This is the rank two fixed space used by the earlier code comparison. 000065 | 000066 | The Pauli translation quotient uses different coordinates. With v783's bit 000067 | order 000068 | 000069 | (x1,z1,x2,z2) 000070 | 000071 | and its Hermitian I,X,Z,Y representatives, exact conjugation by (1) gives 000072 | 000073 | XI |-> +IX, 000074 | ZI |-> -ZZ, 000075 | IX |-> +XX, 000076 | IZ |-> +ZI. (3) 000077 | 000078 | After forgetting the signs, the resulting symplectic matrix is 000079 | 000080 | [0 0 1 0] 000081 | [0 1 0 1] 000082 | M_sigma = [1 0 1 0]. (4) 000083 | [0 1 0 0] 000084 | 000085 | It satisfies 000086 | 000087 | M_sigma^2+M_sigma+I=0, 000088 | chi_M(t)=(t^2+t+1)^2, 000089 | ker(M_sigma-I)=0. (5) 000090 | 000091 | The native compiler is c_comp=i sigma. The scalar i is invisible under 000092 | Pauli conjugation, so Ad(c_comp)=Ad(sigma). This is why the order twelve 000093 | compiler lift and the order three source action can be compared at this 000094 | quotient level without identifying their full orders. 000095 | 000096 | The original relative clock source states the same distinction directly: 000097 | sigma fixes three code classes, hence three Pauli contexts under the v783 000098 | duality, and fixes no nonidentity Pauli. Equation (5) is the explicit four 000099 | bit matrix form of that statement. 000100 | 000101 | 000102 | 2. THE UNIQUE ORTHOGONAL TWO PLANE DECOMPOSITION 000103 | 000104 | There are thirty five two dimensional subspaces of F2^4. Exhaustive exact 000105 | enumeration finds exactly five that are M_sigma stable. Three are isotropic: 000106 | 000107 | {0, IX, XI, XX}, 000108 | {0, IZ, ZI, ZZ}, 000109 | {0, XZ, YY, ZX}. (6) 000110 | 000111 | They are precisely the three fixed Pauli contexts from the original source. 000112 | The other two are nondegenerate symplectic planes: 000113 | 000114 | W_A = {0, ZY, XY, YI}, 000115 | W_B = {0, IY, YX, YZ}. (7) 000116 | 000117 | They are orthogonal complements. Since (6) and (7) exhaust all stable 000118 | two planes, {W_A,W_B} is the unique unordered orthogonal decomposition into 000119 | nondegenerate sigma stable planes. 000120 | 000121 | Use the ordered bases 000122 | 000123 | W_A: (ZY,YI), 000124 | W_B: (IY,YZ). (8) 000125 | 000126 | On both bases (4) restricts to 000127 | 000128 | [1 1] 000129 | C_src = [1 0]. (9) 000130 | 000131 | This is exactly the pullback action of the source cubic on either coefficient 000132 | packet. Indeed c_src^* L_a=L_(a-1), and in the basis 000133 | e0=L0,e1=L1,L2=e0+e1 one has c_src e0=e0+e1 and c_src e1=e0. 000134 | 000135 | The Weil pairing on V_q and V_p is nondegenerate, and the two source planes 000136 | are orthogonal. Pauli commutation gives the same symplectic relations on 000137 | (7). Therefore the displayed assignment 000138 | 000139 | V_q |-> W_A, V_p |-> W_B (10) 000140 | 000141 | is an explicit c equivariant symplectic isomorphism. Exchanging W_A and W_B 000142 | gives the other unlabelled assignment. 000143 | 000144 | 000145 | 3. THE ACTUAL Q ACTION AND THE RESIDUAL MULTIPLICITY 000146 | 000147 | On each source plane, in the basis used above, the previously derived Q action 000148 | is 000149 | 000150 | [1 0] 000151 | A_Q = [1 1]. (11) 000152 | 000153 | It obeys A_Q^2=I and A_Q C_src A_Q=C_src^2. Transporting the block diagonal 000154 | action diag(A_Q,A_Q) through (8) gives the concrete native Pauli action 000155 | 000156 | [1 0 1 0] 000157 | [0 1 1 0] 000158 | Q_native = [0 0 1 0], (12) 000159 | [1 1 1 1] 000160 | 000161 | which is symplectic, preserves W_A and W_B, and satisfies 000162 | 000163 | Q_native^2=I, 000164 | Q_native M_sigma Q_native=M_sigma^2. (13) 000165 | 000166 | The multiplicity is exact, rather than a vague basis freedom: 000167 | 000168 | * there are 18 c equivariant symplectic identifications between the source 000169 | block and the native Pauli module; 000170 | * there are 9 symplectic involutions preserving both planes and satisfying 000171 | (13), three reflections on each irreducible plane; 000172 | * after one Q_native is fixed, exactly 2 simultaneous (c,Q) equivariant 000173 | symplectic identifications remain. They exchange the two planes. 000174 | 000175 | Thus c alone canonically selects the unordered pair of native planes. The 000176 | source labels q and p do not yet select which native plane receives which 000177 | label, and the inspected native data do not select one of the nine Q_native 000178 | markings. Equality of dimensions or conjugacy classes is no longer being 000179 | used as a dictionary: equations (8) through (13) give the actual map and its 000180 | complete finite ambiguity. After the central lift and the actual native