TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/ELLIPTIC-COVER-COMPARISON-REVIEW.txt Original file SHA-256: ff40821957cab57540041a9f4261a95f4bd70bd8971b38a0b8a612e448ac95cc Inclusive original lines: 71-201 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. 000071 | u^2=x^4=w^3+1=v^3+1. 000072 | 000073 | The involution iota:(x,w)->(-x,w) therefore gives the degree-two quotient 000074 | 000075 | C3 -> E_omega, (x,w) |-> (v=w,u=x^2), 000076 | E_omega: u^2=v^3+1. (1.1) 000077 | 000078 | C3 has genus 3 and E_omega has genus 1. The fixed points of iota are the 000079 | three points x=0, w^3=-1 and the unique point over infinity. Riemann-Hurwitz 000080 | checks the four branch points: 000081 | 000082 | 2g(C3)-2 = 4 = 2(2g(E_omega)-2)+4. 000083 | 000084 | The double cover E_omega->P1_v is branched at 000085 | 000086 | {-1,-omega,-omega^2,infinity}. 000087 | 000088 | For that ordering its cross-ratio is 000089 | 000090 | rho_bar=(1-i sqrt(3))/2, 000091 | rho_bar^2-rho_bar+1=0. 000092 | 000093 | The unordered orbit is {rho,rho_bar}; therefore j(E_omega)=0. This is an 000094 | actual quotient of the existing C3. No equianharmonic curve has been added. 000095 | 000096 | 2. Existing j=1728 image 000097 | ------------------------- 000098 | 000099 | The existing C2 is already elliptic. Its degree-two map to P1_x is branched 000100 | at 000101 | 000102 | {1,i,-1,-i}. 000103 | 000104 | The ordered cross-ratio (1,i;-1,-i) is 2. The Legendre formula 000105 | 000106 | j(lambda)=256(1-lambda+lambda^2)^3 / 000107 | (lambda^2(1-lambda)^2) 000108 | 000109 | then gives j(C2)=1728. This is the harmonic four-point orbit already checked 000110 | in v492. 000111 | 000112 | 3. Exact common image from the existing C12 000113 | --------------------------------------------- 000114 | 000115 | The two maps from C12 are 000116 | 000117 | pi_i(x,s) =(x,y=s^6) in C2, 000118 | pi_omega(x,s) =(v=s^4,u=x^2) in E_omega. (3.1) 000119 | 000120 | Their degrees are 6 and 8. The second is the existing degree-four map 000121 | C12->C3 followed by (1.1). 000122 | 000123 | Inside C(C12)=C(x,s), the two pulled-back image fields generate 000124 | 000125 | C(x,s^6) C(s^4,x^2) 000126 | = C(x,s^6,s^4) 000127 | = C(x,s^2), (3.2) 000128 | 000129 | because s^2=s^6/s^4. With t=s^2 this is the existing C6 field 000130 | 000131 | C(C6)=C(x,t), t^6=x^4-1. 000132 | 000133 | Equivalently, the paired map (pi_i,pi_omega) has kernel 000134 | 000135 | ={1,s->-s}. 000136 | 000137 | The normalization of its image is C6. The paired image forgets only the sign 000138 | of s. It does not retain the whole C4 kernel cover: z=s^3 is not contained in 000139 | C(x,s^2). Thus the common elliptic image is concrete, but it is not the full 000140 | C12 common kernel cover. 000141 | 000142 | 4. Deck and clock actions 000143 | -------------------------- 000144 | 000145 | Let 000146 | 000147 | Delta=Delta12: (x,s)->(x,zeta12 s), 000148 | Q=Q12: (x,s)->(i x,s), 000149 | R1=Q Delta, 000150 | R4=Q Delta^4. 000151 | 000152 | R1 is the lift of the original C3 mixed operation. R4 is the unique 000153 | compiler-marked lift called R12 in the pinned cover dictionary. Their actions 000154 | on the two elliptic images are 000155 | 000156 | C2: (x,y) E_omega: (v,u) 000157 | Delta (x,-y) (omega v,u) 000158 | Q (i x,y) (v,-u) 000159 | R1 (i x,-y) (omega v,-u) 000160 | R4 (i x, y) (omega v,-u). 000161 | 000162 | Consequently Delta has order 2 on C2, order 3 on E_omega and order 6 on the 000163 | pair. Both mixed lifts have order 4 and 6 on the respective images, hence 000164 | exact order 12 on the pair. On the normalized C6 image they act as 000165 | 000166 | Delta(x,t)=(x,zeta6 t), 000167 | R1(x,t) =(i x,zeta6 t), 000168 | R4(x,t) =(i x,zeta6^4 t). (4.1) 000169 | 000170 | The pure deck order drops from 12 to 6 while the mixed clock action remains 000171 | faithful of order 12. Clock order 12 therefore cannot be identified with 000172 | cover degree 12. 000173 | 000174 | The compiler marking is unchanged. R1 is not the compiler-marked generator: 000175 | R1^9 is not Q. R4 obeys the existing marks R4^9=Q and R4^4=Delta^4. 000176 | 000177 | 5. Relation to the family-D3 tetrahedron 000178 | ----------------------------------------- 000179 | 000180 | CUTS.txt supplies an unordered projective set of four axes 000181 | 000182 | (1,1,1), (1,1,-1), (1,-1,1), (1,-1,-1). 000183 | 000184 | It explicitly states that the fixed D5+D3 split determines this unordered set 000185 | but does not determine a vertex. Choose the sign representatives 000186 | 000187 | (1,1,1), (1,-1,-1), (-1,1,-1), (-1,-1,1). 000188 | 000189 | They are the vertices of a regular oriented tetrahedron. Normalize them and 000190 | use the Bloch stereographic coordinate associated with the existing projector 000191 | P(n), 000192 | 000193 | z(n)=(n1+i n2)/(1-n3). 000194 | 000195 | For the displayed ordering their cross-ratio is 000196 | 000197 | rho=(1+i sqrt(3))/2, 000198 | rho^2-rho+1=0. 000199 | 000200 | Reversing the orientation conjugates rho. Hence the branch values of (1.1) 000201 | and the family-D3 tetrahedron have the same unordered projective four-point