TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/CUBIC-NATIVE-MARKING-TEST.txt Original file SHA-256: 361a7c61680ae2450a3b62a34a71cfe7525d65433f123f0ffea3668d9f44de48 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 SOURCE CUBIC MATCHES THE NATIVE PAULI SIGMA, NOT THE CODE LABEL SIGMA 000002 | The first exact native intertwiner test after spin selection 000003 | 2026-10-26 000004 | 000005 | VERDICT 000006 | 000007 | PROVED, with a material target correction. The source object Jac(C)[2] is a 000008 | theta translation space. Its native counterpart is the projective Pauli 000009 | translation module 000010 | 000011 | P_nat=N64/mu4, 000012 | 000013 | not the Gaussian code label module V_code=L/(1+i)L. Verification v783 proves 000014 | that these are two different Sp(4,2) modules: the Gaussian classes correspond 000015 | to Pauli contexts by point to line duality, and there is no linear module 000016 | isomorphism between code labels and Pauli translations. 000017 | 000018 | On the correctly typed module P_nat, the original native sigma is fixed point 000019 | free and has the same characteristic polynomial as the source cubic c_src: 000020 | 000021 | chi(x)=(x^2+x+1)^2, fixed dimension 0. (1) 000022 | 000023 | An explicit symplectic intertwiner is given below. Moreover, the native sigma 000024 | action canonically determines an unordered pair of orthogonal invariant 000025 | symplectic planes. This is exactly the type of the source pair {V_q,V_p}. 000026 | 000027 | The earlier fixed dimension mismatch remains true only for the wrongly typed 000028 | comparison with V_code, where native sigma has fixed dimension two. It is a 000029 | diagnostic that forces the target correction, not an obstruction to the 000030 | theta to Pauli bridge. 000031 | 000032 | 000033 | 1. THE SOURCE ACTION 000034 | 000035 | The accepted source result gives 000036 | 000037 | Jac(C)[2]=V_q direct_sum V_p, (2) 000038 | 000039 | an orthogonal sum of symplectic two dimensional F2 planes. On either plane, 000040 | c_src cycles the three nonzero elements. In a symplectic basis 000041 | 000042 | e_0=L_0, e_1=L_1, L_2=e_0+e_1, 000043 | 000044 | the pullback convention c_src^*L_a=L_(a-1) gives 000045 | 000046 | [1 1] 000047 | C_0 = [1 0]. (3) 000048 | 000049 | Therefore 000050 | 000051 | C_src=diag(C_0,C_0), 000052 | chi_Csrc(x)=(x^2+x+1)^2, 000053 | ker(C_src-I)=0. (4) 000054 | 000055 | The inverse pullback convention replaces C_0 by C_0^2 and leaves all 000056 | conjugacy data unchanged. 000057 | 000058 | 000059 | 2. THE TWO NATIVE FOUR BIT MODULES MUST NOT BE IDENTIFIED 000060 | 000061 | The frozen Gaussian chart pairs the four real coordinate pairs into C^4. The 000062 | native element sigma cycles the first three complex coordinates and fixes the 000063 | fourth: 000064 | 000065 | (z_1,z_2,z_3,z_4) |-> (z_3,z_1,z_2,z_4). (5) 000066 | 000067 | On the Gaussian code label quotient V_code, v689 proves that this becomes the 000068 | family plus anchor action 000069 | 000070 | F_1 |-> F_2 |-> F_3 |-> F_1, A |-> A. (6) 000071 | 000072 | Thus V_code^sigma has dimension two. Its characteristic polynomial is 000073 | 000074 | (x+1)^2 (x^2+x+1). (7) 000075 | 000076 | So there is no c_src to sigma intertwiner from Jac(C)[2] to V_code. This 000077 | calculation is correct but it answers the wrong typed question. 000078 | 000079 | Verification v783 gives the decisive distinction. The projective Pauli group 000080 | P_nat=N64/mu4 acts trivially on the sixteen Gaussian code labels. The induced 000081 | G31 action on V_code and its conjugation action on the Pauli bits have 000082 | 000083 | dim Hom_F2[G31](V_code,P_nat)=0, 000084 | 000085 | with zero invertible intertwiners. The fifteen nonzero code labels map to the 000086 | fifteen Pauli contexts, meaning maximal commuting triples, through the outer 000087 | point to line duality of W(3,2). They do not map linearly to the fifteen 000088 | nonidentity Pauli translations. 000089 | 000090 | The theta group of a genus two Jacobian is instead a central extension of 000091 | Jac(C)[2] by phases. Therefore P_nat, the translation quotient of the native 000092 | Heisenberg group N64, is the first type correct target for the source test. 000093 | 000094 | 000095 | 3. ACTUAL NATIVE SIGMA ACTION ON PAULI TRANSLATIONS 000096 | 000097 | Use the v783 Pauli bit order 000098 | 000099 | (x_1,z_1,x_2,z_2). 000100 | 000101 | In the frozen computational chart, conjugation by the coordinate permutation 000102 | (5) sends the Pauli generators, with irrelevant signs suppressed, as follows: 000103 | 000104 | X_1 |-> X_2, 000105 | Z_1 |-> Z_1 Z_2, 000106 | X_2 |-> X_1 X_2, 000107 | Z_2 |-> Z_1. (8) 000108 | 000109 | Hence its exact symplectic matrix is 000110 | 000111 | [0 0 1 0] 000112 | [0 1 0 1] 000113 | Sigma_Pauli = [1 0 1 0] 000114 | [0 1 0 0]. (9) 000115 | 000116 | Direct multiplication gives 000117 | 000118 | Sigma_Pauli^3=I, 000119 | ker(Sigma_Pauli-I)=0, 000120 | chi_Sigma(x)=(x^2+x+1)^2. (10) 000121 | 000122 | This also follows from the exact Clifford conjugation map in v783, but (8) 000123 | and (9) make the relevant marked element explicit rather than relying on a 000124 | group order census. 000125 | 000126 | 000127 | 4. AN EXPLICIT SYMPLECTIC INTERTWINER 000128 | 000129 | Take the source basis 000130 | 000131 | (e_0^q,e_1^q,e_0^p,e_1^p) 000132 | 000133 | with symplectic Gram matrix diag([[0,1],[1,0]],[[0,1],[1,0]]). In the native 000134 | Pauli basis, define Phi by the following columns: 000135 | 000136 | Phi(e_0^q)=IY =(0,0,1,1), 000137 | Phi(e_1^q)=YZ =(1,1,0,1), 000138 | Phi(e_0^p)=ZY =(0,1,1,1), 000139 | Phi(e_1^p)=YI =(1,1,0,0). (11) 000140 | 000141 | Thus 000142 | 000143 | [0 1 0 1] 000144 | [0 1 1 1] 000145 | Phi = [1 0 1 0] 000146 | [1 1 1 0]. (12) 000147 | 000148 | With Omega the standard Pauli commutator form, exact F2 multiplication gives 000149 | 000150 | Phi^T Omega Phi=diag(Omega_2,Omega_2), 000151 | Sigma_Pauli Phi=Phi C_src. (13) 000152 | 000153 | Therefore Phi is an invertible symplectic c_src to sigma intertwiner. The 000154 | first required marked native test passes without a search among 720 frames. 000155 | 000156 | 000157 | 5. THE q|p PLANE SPLIT IS VISIBLE NATIVELY 000158 | 000159 | Sigma_Pauli has exactly five invariant two dimensional F2 subspaces. Three 000160 | are isotropic. Exactly two are symplectic: 000161 | 000162 | W_q={0,IY,YZ,YX}, 000163 | W_p={0,ZY,XY,YI}. (14) 000164 | 000165 | They are orthogonal complements. Consequently the native marked sigma 000166 | action itself selects the unordered pair 000167 | 000168 | {W_q,W_p} (15) 000169 | 000170 | as its unique pair of invariant symplectic planes. Formula (11) sends the 000171 | source unordered pair {V_q,V_p} onto (15). 000172 | 000173 | The selection is not yet an ordered or fully framed identification. There 000174 | are three c equivariant symplectic bases on each plane and two ways to exchange 000175 | the planes, leaving 000176 | 000177 | 2 * 3 * 3 = 18 (16) 000178 | 000179 | symplectic c_src to sigma intertwiners. The remaining source operations s, 000180 | Q, and the selected spin lift beta must decide whether one of these eighteen