TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/SIGN-LINE-COVER-DICTIONARY.txt Original file SHA-256: 555f75a2d3c67347fa5f8e3c4f7f46e79d94cb4172176c84010224585dc842e7 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 | SIGN-LINE COVER DICTIONARY 000002 | Exact v117 coefficient lift on the genus-two source curve 000003 | 2026-10-26 000004 | 000005 | VERDICT 000006 | 000007 | PROVED. The sign lines missing from the September 28 source are the three 000008 | nonzero two-torsion differences inside the negative Weierstrass orbit, not the 000009 | antipodal differences: 000010 | 000011 | C: eta^2=1-w^6, 000012 | p_a=(omega^a,0), q_a=(-omega^a,0), omega=exp(2 pi i/3), 000013 | 000014 | L_a^q = O_C(q_a-q_(a+1)), a mod 3. (1) 000015 | 000016 | Their Kummer equations are 000017 | 000018 | h_a^2 = (w+omega^a)/(w+omega^(a+1)), 000019 | h_0 h_1 h_2 = 1. (2) 000020 | 000021 | The connected V4 cover Y_q -> C defined by (2) is unramified and has genus 000022 | five. Its descent over t=z^4 has the exact marked triangle passport 000023 | 000024 | t=0 : 4, t=1 : 3, t=infinity : 2. (3) 000025 | 000026 | This is the same oriented (4,3,2) S4 triangle cover proved for the v117 ODE. 000027 | Consequently the pulled-back coefficient local system splits exactly as 000028 | 000029 | pi^* E_v117 ~= L_0^q direct-sum L_1^q direct-sum L_2^q (4) 000030 | 000031 | with the canonical unitary flat structures of these two-torsion lines. 000032 | 000033 | The earlier antipodal proposal O(p_a-q_a) is explicitly excluded. It gives 000034 | passport (4,3,4) and genus three at the degree-24 triangle level. The 000035 | positive-orbit packet O(p_a-p_(a+1)) gives the reversed marking (2,3,4). 000036 | Neither can be the original v117 source. 000037 | 000038 | Equation (4) is an exact coefficient-local-system identification. It does 000039 | not yet identify charged CAR fields, their state, local support, or response 000040 | functionals. 000041 | 000042 | 000043 | 1. SOURCE STATUS 000044 | 000045 | The September 28 source, section 004, lines 203--327, already proves: 000046 | 000047 | * the exact reduction of the v117 Fuchs system to the finite triangle 000048 | problem with local orders (4,3,2); 000049 | * absence of the resonant logarithm at infinity; 000050 | * the faithful determinant-one three-dimensional S4 representation; 000051 | * the marked positive t=0 monodromy U^(-1); 000052 | * A4, of order twelve, for the four actual z-plane puncture monodromies; 000053 | * the genus-two curve C above; 000054 | * that the remaining signs must be a line local system on C. 000055 | 000056 | The source stops before writing that line system. Its six-point operator is 000057 | explicitly only a point intertwiner. It is not a field intertwiner or a seam 000058 | projector derivation. A targeted Atlas and old-source search found no older 000059 | formula equivalent to (1)--(2). 000060 | 000061 | 000062 | 2. THE THREE TWO-TORSION LINES 000063 | 000064 | At q_a, eta is a local parameter and w+omega^a has order two. Therefore 000065 | 000066 | div((w+omega^a)/(w+omega^(a+1))) 000067 | = 2 q_a - 2 q_(a+1). (5) 000068 | 000069 | There is no contribution at either infinity point. Hence every L_a^q is an 000070 | unramified order-two line in Pic^0(C). It is nontrivial: if q_a-q_(a+1) were 000071 | principal, its function would give a degree-one map from the genus-two curve 000072 | to P1. The product in (2) telescopes, so 000073 | 000074 | L_0^q tensor L_1^q tensor L_2^q = O_C. (6) 000075 | 000076 | The three classes are distinct nonzero elements of a rank-two subgroup of 000077 | Jac(C)[2]. Their connected Kummer cover has deck group V4. Since it is 000078 | unramified over C, 000079 | 000080 | g(Y_q)=1+4(g(C)-1)=5. (7) 000081 | 000082 | The rotation c:(w,eta)->(omega w,eta) cyclically permutes them: 000083 | 000084 | c^* L_a^q = L_(a-1)^q. (8) 000085 | 000086 | Thus the coefficient characters already carry the required V4 semidirect C3 000087 | structure. 000088 | 000089 | 000090 | 3. THE DECISIVE PASSPORT TEST 000091 | 000092 | First forget eta and form the degree-four cover 000093 | 000094 | Z_q = C(w,h_0,h_1), 000095 | h_a^2=(w+omega^a)/(w+omega^(a+1)). (9) 000096 | 000097 | It is a connected V4 cover of the w-sphere. It branches only at the three 000098 | negative points q_0,q_1,q_2, with inertia two. Riemann--Hurwitz gives 000099 | 000100 | 2g(Z_q)-2 = 4(-2) + 3(4/2)(2-1) = -2, 000101 | g(Z_q)=0. (10) 000102 | 000103 | The original relations give 000104 | 000105 | z^2 = (1+w^3)/(1-w^3), 000106 | t=z^4=((1+w^3)/(1-w^3))^2. (11) 000107 | 000108 | For the map w -> t: 000109 | 000110 | * q_a lies over t=0 with ramification two; 000111 | * w=0 and w=infinity lie over t=1 with ramification three; 000112 | * p_a lies over t=infinity with ramification two. 000113 | 000114 | The cover Z_q -> P1_t has degree 24. At q_a the V4 square root in (9) 000115 | combines with the order-two ramification of w -> t, giving order four. The 000116 | other two fibers are unramified in Z_q -> P1_w. Therefore its exact passport 000117 | is 000118 | 000119 | (e_0,e_1,e_infinity) = (4,3,2). (12) 000120 | 000121 | It is connected and transitive of degree 24. Its monodromy is a quotient of 000122 | Delta(4,3,2)=S4, whose order is 24. A transitive group on 24 sheets has order 000123 | at least 24, so the quotient has order exactly 24 and the cover is regular. 000124 | This is the unique marked regular S4 triangle cover with branch values 000125 | 0,1,infinity. 000126 | 000127 | The exact v117 matrices in the old F basis satisfy 000128 | 000129 | ord(U^(-1))=4, 000130 | ord(M)=3, 000131 | ord((M U^(-1))^(-1))=2, (13) 000132 | 000133 | and all have determinant one. Thus (12) has the same target positions, 000134 | positive t=0 clock, and determinant-one three-dimensional representation as 000135 | the original v117 system. No target matrix was inserted into the construction 000136 | of Z_q. 000137 | 000138 | 000139 | 4. EXACT FIBER PRODUCT AND A4 KERNEL 000140 | 000141 | The identity 000142 | 000143 | sqrt(t)=(1+w^3)/(1-w^3) (14) 000144 | 000145 | shows that C(z^2) is already a subfield of C(Z_q). It is also the unique 000146 | quadratic subfield of C(z)/C(t). The intersection cannot be the full cyclic 000147 | quartic field C(z), because S4 has no C4 quotient. Hence 000148 | 000149 | C(Z_q) intersect C(z) = C(z^2). (15) 000150 | 000151 | It follows that the composite has degree 000152 | 000153 | [C(Z_q,z):C(z)] = 24/2 = 12. (16) 000154 | 000155 | The corresponding subgroup of S4 is the kernel of its only nontrivial cyclic 000156 | quotient, namely A4. On the other hand 000157 | 000158 | C(C)=C(w,z), eta=z(1-w^3), (17) 000159 | 000160 | so 000161 | 000162 | C(Z_q,z) = C(C) C(Z_q) = C(Y_q). (18) 000163 | 000164 | Thus Y_q is exactly the A4 kernel cover of the original marked triangle 000165 | source. Restricting the determinant-one S4 representation to its normal V4 000166 | splits it into the three nontrivial V4 characters. Equations (1)--(2) realize 000167 | those characters, proving (4) without a fitted C4 matrix or an abstract 000168 | same-dimension argument. 000169 | 000170 | 000171 | 5. WHY THE OTHER TWO C3 PACKETS FAIL 000172 | 000173 | The positive-orbit packet 000174 | 000175 | L_a^p=O_C(p_a-p_(a+1)) 000176 | 000177 | branches only over the p_a on the w-sphere. Its degree-24 cover therefore has 000178 | orders 000179 | 000180 | (2,3,4). (19)