TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/SIX-LINE-THETA-COMPLETION.txt Original file SHA-256: 99676497d1de1769807c1e34208f45e12e95b6e9460618b1d84aa3e51e64675f 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 | SIX LINE COMPLETION AND THE THETA BOUNDARY 000002 | Exact closure of the q coefficient packet under the old inversion s 000003 | 2026-10-26 000004 | 000005 | VERDICT 000006 | 000007 | PROVED at the flat coefficient and cover level. 000008 | 000009 | Let 000010 | 000011 | C: eta^2=1-w^6, 000012 | p_a=(omega^a,0), q_a=(-omega^a,0), a mod 3, 000013 | 000014 | and let the two C3 packets be 000015 | 000016 | L_a^q=O_C(q_a-q_(a+1)), 000017 | L_a^p=O_C(p_a-p_(a+1)). (1) 000018 | 000019 | The negative packet alone is the exact v117 coefficient system selected by 000020 | the marked (4,3,2) passport. It is preserved by the old quarter lift Q, but 000021 | it is not preserved by 000022 | 000023 | s:(w,eta) |-> (-w,eta). 000024 | 000025 | Instead, s exchanges the q and p packets. Therefore the minimal closure of 000026 | the accepted v117 packet under the old six point symmetry is exactly 000027 | 000028 | E_6 = direct-sum_a (L_a^q direct-sum L_a^p). (2) 000029 | 000030 | The six classes in (2) generate all of Jac(C)[2]. Their joint connected 000031 | Kummer cover is the maximal elementary two cover of C, of degree 16 and 000032 | genus 17. The old operator T=sc lifts to this cover and, after a flat frame 000033 | choice, acts as a six cycle. Its two three dimensional Fourier blocks are 000034 | P_c and -P_c and hence, through the already printed F, are M and -M. Thus 000035 | the old point identity M direct-sum (-M) has an exact coefficient-cover 000036 | realization after the forced q+p completion. 000037 | 000038 | This statement concerns the C6 action generated by c and s. The separately 000039 | marked quarter map Q and its positive-loop orientation remain the ones proved 000040 | in CLOCK-Q-PACKET-COMPATIBILITY. The calculation below does not identify T 000041 | with Q, combine them into a new clock, or prove a joint field dictionary. 000042 | 000043 | This does not yet identify the six coefficient lines with charged fields. 000044 | It also does not derive the native four dimensional G31 representation. 000045 | The completed cover supplies the abelian F2^4 translation layer. G31 also 000046 | contains a nonabelian central theta extension and the full Sp4(2)=S6 action. 000047 | Neither is forced by c and s alone. 000048 | 000049 | 000050 | 1. WHAT THE OLD SOURCE ALREADY SAYS 000051 | 000052 | The September 28 source proves the v117 triangle system, constructs C, and 000053 | defines 000054 | 000055 | c:(w,eta) |-> (omega w,eta), 000056 | s:(w,eta) |-> (-w,eta), 000057 | T=sc. (3) 000058 | 000059 | On the six Weierstrass points, T is a six cycle and the printed matrix 000060 | J_pts satisfies 000061 | 000062 | J_pts^dagger T J_pts = M direct-sum (-M). (4) 000063 | 000064 | The same source explicitly fences (4) as a point intertwiner. The accepted 000065 | SIGN-LINE-COVER-DICTIONARY later identifies the actual v117 sign lines with 000066 | the q packet in (1). The present test asks whether (3) also acts on those 000067 | lines. That is a stronger question than the point permutation in (4). 000068 | 000069 | 000070 | 2. THE SMALLEST OBSTRUCTION AND ITS MINIMAL REPAIR 000071 | 000072 | Use the normalized Kummer equations 000073 | 000074 | (h_a^q)^2=(w+omega^a)/(w+omega^(a+1)), 000075 | (h_a^p)^2=(w-omega^a)/(w-omega^(a+1)), 000076 | product_a h_a^q = product_a h_a^p = 1. (5) 000077 | 000078 | Direct substitution gives 000079 | 000080 | s^*((h_a^q)^2)=(h_a^p)^2, 000081 | s^*((h_a^p)^2)=(h_a^q)^2. (6) 000082 | 000083 | Equivalently, 000084 | 000085 | s^*L_a^q=L_a^p, s^*L_a^p=L_a^q. (7) 000086 | 000087 | This is the exact obstruction for the isolated q packet: no s equivariant 000088 | structure can exist on it because its set of isomorphism classes is carried 000089 | to a disjoint set. Adding the p orbit is both sufficient and minimal. 000090 | It does not replace the q packet in the v117 triangle problem. The p packet 000091 | still has the reversed marked passport (2,3,4); it enters only as the image 000092 | required by s. 000093 | 000094 | Equations (5) also construct a lift rather than merely count dimensions. 000095 | One may choose 000096 | 000097 | s^*h_a^q=h_a^p, s^*h_a^p=h_a^q, (8) 000098 | c^*h_a^q=h_(a-1)^q, c^*h_a^p=h_(a-1)^p. (9) 000099 | 000100 | These substitutions preserve every defining equation and give s^2=c^3=1 000101 | and sc=cs. Hence T=sc has order six on the joint cover. Other lifts differ 000102 | from (8) by a deck transformation. This deck torsor is the exact residual 000103 | sign ambiguity; the six point permutation by itself does not choose a 000104 | preferred lift. 000105 | 000106 | 000107 | 3. THE TWO PLANES FILL Jac(C)[2] 000108 | 000109 | For a genus two hyperelliptic curve, a nonzero two torsion class is represented 000110 | by an unordered pair of the six Weierstrass points, with even subsets taken 000111 | modulo complement. The three nonzero elements of the q plane are 000112 | 000113 | {q_0,q_1}, {q_1,q_2}, {q_2,q_0}, 000114 | 000115 | and the three nonzero elements of the p plane are the analogous p pairs. 000116 | No q pair equals a p pair, even after taking the complement. Thus 000117 | 000118 | V_q intersect V_p = 0, 000119 | dim_F2 V_q = dim_F2 V_p = 2, 000120 | Jac(C)[2] = V_q direct-sum V_p. (10) 000121 | 000122 | The Weil pairing is nondegenerate on each plane: two distinct nonzero pairs 000123 | inside the same three point orbit meet in one point. The two planes are 000124 | orthogonal because a q pair and a p pair are disjoint. Therefore (10) is an 000125 | orthogonal sum of two symplectic planes. 000126 | 000127 | The six lines in (2) are only six of the fifteen nonzero classes. Their 000128 | tensor products generate the remaining nine mixed classes. They must not be 000129 | identified with the six Weierstrass points themselves. The two six element 000130 | sets carry matching T cycles, but they have different mathematical types. 000131 | 000132 | Let Y_16 be the normalization obtained by adjoining two independent q roots 000133 | and two independent p roots from (5). By (10), 000134 | 000135 | Gal(Y_16/C) = (Z/2)^4, 000136 | [Y_16:C]=16. (11) 000137 | 000138 | The cover is unramified, so Riemann-Hurwitz gives 000139 | 000140 | g(Y_16)=1+16(g(C)-1)=17. (12) 000141 | 000142 | Because the four characters span Jac(C)[2], Y_16 is the maximal connected 000143 | elementary abelian two cover of C. This is a canonical cover. A chosen lift 000144 | of s or T remains a torsor under its deck group, as stated after (9). 000145 | 000146 | 000147 | 4. RECOVERY OF THE OLD M direct-sum (-M) BLOCKS 000148 | 000149 | Order coefficient frames as 000150 | 000151 | (e_0^q,e_1^q,e_2^q,e_0^p,e_1^p,e_2^p). 000152 | 000153 | Using covariant transport, c acts by P_c on each triple and s exchanges the 000154 | triples. Therefore 000155 | 000156 | T = [[0,P_c],[P_c,0]]. (13) 000157 | 000158 | The fixed block Fourier transform 000159 | 000160 | H_6 = (1/sqrt(2)) [[I,I],[I,-I]] 000161 | 000162 | gives 000163 | 000164 | H_6^dagger T H_6 = P_c direct-sum (-P_c). (14) 000165 | 000166 | The old source already proves F^dagger M F=P_c. Applying F on both blocks 000167 | turns (14) into (4). For pullback rather than covariant transport, all cycles 000168 | are inverted; this is the same orientation reversal already present between 000169 | the geometric clock and positive-loop monodromy. 000170 | 000171 | Thus (4) is no longer only a coincidence of six point spectra. It is the 000172 | block decomposition of the explicit coefficient-cover lift (8)-(9). What 000173 | remains absent is an operator map from these flat coefficient frames to the 000174 | charged CAR fields and their state. 000175 | 000176 | 000177 | 5. WHY THIS IS NOT YET G31 000178 | 000179 | The exact native G31 calculation in E8.ST31.01 gives a normal subgroup N64 000180 | and quotient data