TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/SOURCE-SELECTED-THETA-SPIN.txt Original file SHA-256: cf4478191ccd75d62a2f049a54193541009bed24877df9cbe0ed9f942d07d717 Inclusive original lines: 1-174 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 ORIGINAL q|p PARTITION SELECTS THE THETA SPIN QUARTER 000002 | Exact source selection on the genus two curve 000003 | 2026-10-26 000004 | 000005 | VERDICT 000006 | 000007 | PROVED. The already reconstructed partition of the six Weierstrass points 000008 | into the negative orbit q={q_0,q_1,q_2} and the positive orbit 000009 | p={p_0,p_1,p_2} selects a single even theta characteristic 000010 | 000011 | S_0=O_C(q_0+q_1+q_2-K_C). (1) 000012 | 000013 | The same class is obtained from the p orbit. It is the unique theta 000014 | characteristic fixed by the original cubic source symmetry c. Hence the 000015 | previously open choice among sixteen spin lines is closed by existing source 000016 | data. 000017 | 000018 | Moreover, the marked orientation of the actual geometric quarter map Q fixes 000019 | a specific order four projective spin lift beta of h=Q^2. A compatible lift 000020 | alpha of Q itself has order eight and satisfies alpha^4=-I; its square beta is 000021 | independent of the remaining choice alpha versus -alpha and obeys beta^2=-I. 000022 | 000023 | This proves a source selected half turn to internal quarter phase on the 000024 | actual genus two curve. It does not identify its theta module with the native 000025 | Gaussian E8 module or prove that beta is the native J. 000026 | 000027 | 000028 | 1. THE q|p PARTITION DEFINES ONE THETA CHARACTERISTIC 000029 | 000030 | On 000031 | 000032 | C: eta^2=1-w^6 000033 | 000034 | write 000035 | 000036 | p_a=omega^a, q_a=-omega^a, a in Z/3. 000037 | 000038 | Every branch point r satisfies 2r linearly equivalent to K_C. Therefore the 000039 | degree one line bundle (1) satisfies 000040 | 000041 | 2S_0 000042 | =O_C(2q_0+2q_1+2q_2-2K_C) 000043 | =O_C(3K_C-2K_C) 000044 | =K_C. (2) 000045 | 000046 | It is thus a theta characteristic. The meromorphic function 000047 | 000048 | (w^3+1)/eta 000049 | 000050 | has divisor 000051 | 000052 | q_0+q_1+q_2-p_0-p_1-p_2. (3) 000053 | 000054 | Consequently 000055 | 000056 | O_C(q_0+q_1+q_2-K_C) 000057 | isomorphic O_C(p_0+p_1+p_2-K_C). (4) 000058 | 000059 | Thus interchanging the two sides of the partition does not change S_0. 000060 | 000061 | The theta characteristic is even. Indeed, an effective degree one theta 000062 | characteristic is O_C(r) for a Weierstrass point r and is one of the six odd 000063 | classes. Formula (1), or equivalently (4), is not any of these classes: after 000064 | subtracting a point on the chosen side and using K_C equivalent to twice one 000065 | of the other two points, equality would make the difference of two distinct 000066 | points principal. That would give a degree one map C->P1, impossible for a 000067 | genus two curve. Hence h^0(S_0)=0 and S_0 is even. 000068 | 000069 | 000070 | 2. THE ORIGINAL CUBIC SYMMETRY MAKES THE CHOICE UNIQUE 000071 | 000072 | The source automorphism 000073 | 000074 | c:(w,eta) |-> (omega w,eta) 000075 | 000076 | cycles each triple, so c^*S_0 isomorphic S_0. Theta characteristics form an 000077 | affine torsor over Jac(C)[2]. Relative to S_0, every other theta 000078 | characteristic is S_0 tensor L for a unique L in Jac(C)[2]. 000079 | 000080 | The accepted six line calculation gives 000081 | 000082 | Jac(C)[2]=V_q direct_sum V_p. (5) 000083 | 000084 | On each two dimensional F2 plane, c is an element of order three with no 000085 | nonzero fixed vector. Hence 000086 | 000087 | Jac(C)[2]^c=0. (6) 000088 | 000089 | If S_0 tensor L were also c invariant, (6) would force L=0. Therefore 000090 | 000091 | S_0 is the unique c invariant theta characteristic. (7) 000092 | 000093 | This uniqueness uses the actual c action and the already proved q|p packet. 000094 | It is not an added theta marking. 000095 | 000096 | 000097 | 3. THE MARKED Q ORIENTATION SELECTS THE LIFT OF h=Q^2 000098 | 000099 | The original quarter map is 000100 | 000101 | Q:(z,w,eta) |-> (i z,1/w,-i eta/w^3). (8) 000102 | 000103 | It preserves the q triple and therefore Q^*S_0 isomorphic S_0. Fix the 000104 | square root isomorphism S_0^2 isomorphic K_C and choose a compatible 000105 | projective spin lift 000106 | 000107 | alpha:Q^*S_0 -> S_0. (9) 000108 | 000109 | There are exactly two normalized choices, alpha and -alpha. 000110 | 000111 | The Weierstrass point q_0=-1 is fixed by Q. The coordinate eta is a local 000112 | parameter there. Formula (8) gives, at q_0 where w^3=-1, 000113 | 000114 | eta |-> i eta, 000115 | d eta |-> i d eta. (10) 000116 | 000117 | If alpha acts on the spin fiber at q_0 by lambda, compatibility with 000118 | S_0^2=K_C gives 000119 | 000120 | lambda^2=i. (11) 000121 | 000122 | The fourth iterate of alpha is a global scalar automorphism of S_0, and at 000123 | q_0 it equals 000124 | 000125 | lambda^4=-1. 000126 | 000127 | Therefore 000128 | 000129 | alpha^4=-I, alpha^8=I. (12) 000130 | 000131 | Thus alpha is an order eight spin lift of the order four map Q. Define 000132 | 000133 | beta=alpha composed with Q^*alpha, (13) 000134 | 000135 | which is the induced lift of h=Q^2. Equations (12) and (13) give 000136 | 000137 | beta^2=-I. (14) 000138 | 000139 | Replacing alpha by -alpha leaves beta unchanged because the two signs cancel 000140 | in (13). The actual orientation eta |-> i eta therefore selects the +i local 000141 | phase of beta at q_0 and removes the sign ambiguity that remained when h was 000142 | considered alone. 000143 | 000144 | Beta is a projective spin lift of order four. It is not an honest C2 000145 | linearization, whose cocycle square would be +I. 000146 | 000147 | 000148 | 4. WHAT IS NOW SELECTED, AND WHAT IS NOT 000149 | 000150 | The original source data now determine, without an added fit: 000151 | 000152 | * the even spin line S_0, uniquely by c invariance; 000153 | * the order eight Q lift up to its harmless overall sign; 000154 | * the sign independent order four lift beta of h=Q^2; 000155 | * the local central phase beta=+i at the marked fixed point q_0. 000156 | 000157 | One type change remains. No current source constructs an equivariant 000158 | identification between the theta or spin module generated from (S_0,beta) and 000159 | the native Gaussian E8 module L, its four dimensional theta representation, 000160 | or its central mu4 generator J. The literal quotient identification Q |-> J 000161 | on Jac(C)[2] and L/(1+i)L remains excluded by the separate two by two test. 000162 | 000163 | 000164 | SOURCE ANCHORS 000165 | 000166 | * research-20261026-real-boundary/SIX-LINE-THETA-COMPLETION.txt: 000167 | the exact q and p planes, their direct sum Jac(C)[2], and the c action. 000168 | * research-20261026-real-boundary/CLOCK-Q-PACKET-COMPATIBILITY.txt, 000169 | lines 22--73: the actual marked Q and its action on the q packet. 000170 | * research-20261026-real-boundary/HYPERELLIPTIC-SPIN-QUARTER.txt: 000171 | every theta characteristic is h invariant and every compatible h lift 000172 | squares to -I. 000173 | * research-20261026-real-boundary/Q-NATIVE-CLOCK-TEST.txt: 000174 | the still valid scoped obstruction to the literal quotient map Q |-> J.