TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/Q-NATIVE-CLOCK-TEST.txt Original file SHA-256: bf565a83ce5e7d8a70bd24ad77b97ae24610d37180b2e675c866aa6cb3578ce4 Inclusive original lines: 1-176 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 | Q VERSUS THE NATIVE GAUSSIAN QUARTER CLOCK 000002 | A smallest quotient action test 000003 | 2026-10-26 000004 | 000005 | VERDICT 000006 | 000007 | PROVED, with deliberately narrow scope. On the presently reconstructed 000008 | genus two coefficient object Jac(C)[2], the geometric quarter map Q has a 000009 | nontrivial involutive reduction. On the native Gaussian code object 000010 | 000011 | V_E8 = L/(1+i)L, 000012 | 000013 | the original quarter clock J:x |-> i x has trivial reduction. Consequently 000014 | there is no invertible F2 linear, and hence no symplectic, intertwiner between 000015 | these two chosen quotient objects that sends Q literally to J. This is a 000016 | two by two obstruction, before any search among the 720 symplectic markings. 000017 | 000018 | The old source identifies Q with the rank three coefficient clock U. It does 000019 | not state the additional bridge Q |-> J on the Gaussian quotient. The result 000020 | therefore rejects that added literal bridge premise. It is not a contradiction 000021 | of the proved q packet, the C4 triangle source, or the native G31 carrier. 000022 | 000023 | 000024 | 1. THE ACTUAL Q ACTION ON THE GENUS TWO TWO TORSION 000025 | 000026 | Use the already selected curve and negative sign packet 000027 | 000028 | C: eta^2=1-w^6, 000029 | q_a=-omega^a, 000030 | L_a^q=O_C(q_a-q_(a+1)), a in Z/3, 000031 | 000032 | with L_0^q+L_1^q+L_2^q=0 in Jac(C)[2]. The marked geometric lift is 000033 | 000034 | Q:(z,w,eta) |-> (i z,1/w,-i eta/w^3). 000035 | 000036 | The divisor calculation already proved 000037 | 000038 | Q^* L_a^q = L_(-a-1)^q. (1) 000039 | 000040 | Thus Q exchanges L_0^q and L_2^q and fixes L_1^q. In the basis 000041 | 000042 | e_0=L_0^q, e_1=L_1^q, L_2^q=e_0+e_1, 000043 | 000044 | the columns of the reduced action are Qe_0=e_0+e_1 and Qe_1=e_1: 000045 | 000046 | [1 0] 000047 | A_Q = [1 1] over F2. (2) 000048 | 000049 | Hence 000050 | 000051 | A_Q != I, A_Q^2=I, rank(A_Q-I)=1. (3) 000052 | 000053 | The identical calculation applies to the positive packet L_a^p. The accepted 000054 | six line result gives 000055 | 000056 | Jac(C)[2]=V_q direct_sum V_p, 000057 | Q_bar=diag(A_Q,A_Q). (4) 000058 | 000059 | Therefore Q_bar is a nonidentity involution with a two dimensional fixed 000060 | space. This does not conflict with the fact that Q has order four on C: 000061 | 000062 | Q^2:(z,w,eta) |-> (-z,w,-eta) 000063 | 000064 | is the hyperelliptic involution. It induces [-1] on Jac(C), which becomes the 000065 | identity on Jac(C)[2]. 000066 | 000067 | 000068 | 2. THE NATIVE GAUSSIAN J ACTION 000069 | 000070 | The frozen native chart of E8.CLIFFORD2Q.01 uses the Gaussian lattice L and 000071 | 000072 | Jx=i x, 000073 | V_E8=L/(1+i)L. (5) 000074 | 000075 | For every x in L, 000076 | 000077 | ix-x=(i-1)x=i(1+i)x in (1+i)L. (6) 000078 | 000079 | Thus 000080 | 000081 | J_bar=I on V_E8. (7) 000082 | 000083 | This is also the exact content of the Lean certificate 000084 | mu4_acts_trivially_on_labels and of the native G31 calculation: the central 000085 | mu4 lies in the order 64 kernel of G31 -> Sp(4,2). 000086 | 000087 | 000088 | 3. THE DECISIVE INTERTWINER TEST 000089 | 000090 | Assume that a proposed literal source dictionary supplied an invertible map 000091 | 000092 | phi:Jac(C)[2] -> V_E8 000093 | 000094 | and required phi Q_bar=J_bar phi. By (7), 000095 | 000096 | phi Q_bar=phi. 000097 | 000098 | Multiplication by phi^(-1) gives Q_bar=I, contradicting (2) and (4). Hence 000099 | 000100 | #{invertible phi with phi Q_bar=J_bar phi}=0. (8) 000101 | 000102 | The failure is conjugacy invariant. A change among the 720 symplectic frames 000103 | cannot turn the nonidentity element Q_bar into the identity. 000104 | 000105 | 000106 | 4. EXACT SOURCE SCOPE 000107 | 000108 | The September source proves the following different statement. Its geometric 000109 | quarter lift Q has order four on C, and its selected lift on the three negative 000110 | coefficient lines is conjugate to the v117 clock U, with positive source 000111 | monodromy U^(-1). That rank three statement remains intact. The inspected 000112 | source passage contains no identification of Q with the scalar Gaussian J on 000113 | L/(1+i)L. 000114 | 000115 | This obstruction excludes only the simultaneous choices 000116 | 000117 | source object = Jac(C)[2], 000118 | native object = L/(1+i)L, 000119 | marked generator Q |-> central J. 000120 | 000121 | It does not exclude a source map sending Q to a different native element whose 000122 | Sp(4,2) image is conjugate to diag(A_Q,A_Q). It also does not exclude a lift 000123 | to the thicker module L/2L, where the ramified jet relation J=1+epsilon retains 000124 | nilpotent information that is erased modulo (1+i). Neither alternative is 000125 | proved here. 000126 | 000127 | The earlier v784 comparison belongs to a different curve and abelian surface. 000128 | It is unnecessary for (8) and is not imported as an exclusion for C. 000129 | 000130 | 000131 | 5. THE ONE TYPE CORRECT SURVIVING CENTRAL PHASE CANDIDATE 000132 | 000133 | The square h=Q^2 is a better typed place to look for a central quarter phase. 000134 | On integral first homology and on holomorphic one forms, the hyperelliptic 000135 | involution acts by -I; explicitly, 000136 | 000137 | h^*(dw/eta)=-dw/eta. (9) 000138 | 000139 | It acts trivially only after reduction to two torsion. If an h invariant 000140 | theta characteristic kappa with kappa^2=K_C is selected together with a spin 000141 | linearization, a local half form s satisfying s^2=dw/eta can transform as 000142 | 000143 | h_tilde(s)=+i s or -i s. (10) 000144 | 000145 | Such a spin lift has order four although h has order two on C. This is the 000146 | same type of central spin cocycle mechanism already present in the older A4 000147 | spin lift result: an order two base action can lift with square -I and produce 000148 | the unordered quarter pair. 000149 | 000150 | Equation (10) is only a surviving typed candidate for the native central J. 000151 | The present q and p coefficient packets do not select a theta characteristic, 000152 | its linearization, or a map of its central phase to the native mu4 in G31. 000153 | Those are the three required source premises. Without them, identifying J 000154 | with the spin lift of Q^2 would be a new choice rather than a reconstruction. 000155 | 000156 | 000157 | SOURCE ANCHORS 000158 | 000159 | [OMITTED: non-scientific conversation or execution metadata] 000160 | [OMITTED: non-scientific conversation or execution metadata] 000161 | [OMITTED: non-scientific conversation or execution metadata] 000162 | [OMITTED: non-scientific conversation or execution metadata] 000163 | [OMITTED: non-scientific conversation or execution metadata] 000164 | [OMITTED: non-scientific conversation or execution metadata] 000165 | [OMITTED: non-scientific conversation or execution metadata] 000166 | [OMITTED: non-scientific conversation or execution metadata] 000167 | [OMITTED: non-scientific conversation or execution metadata] 000168 | [OMITTED: non-scientific conversation or execution metadata] 000169 | [OMITTED: non-scientific conversation or execution metadata] 000170 | [OMITTED: non-scientific conversation or execution metadata] 000171 | [OMITTED: non-scientific conversation or execution metadata] 000172 | [OMITTED: non-scientific conversation or execution metadata] 000173 | [OMITTED: non-scientific conversation or execution metadata] 000174 | [OMITTED: non-scientific conversation or execution metadata] 000175 | [OMITTED: non-scientific conversation or execution metadata] 000176 | [OMITTED: non-scientific conversation or execution metadata]