TFPT scientific source excerpt Original repository path: experiments/theory-contracts/e-series-compiler-closure-20261023/COVERS.txt Original file SHA-256: cdd04f5f38819b60ddd8488eb1c3c316c6849829387205e78ec5e4bf767f80bf Inclusive original lines: 1-185 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 | MATH.SEAM.ESERIES.COVERS.04 000002 | 23 October 2026 000003 | 000004 | Question and object class 000005 | 000006 | Fix the marked base 000007 | 000008 | B° = P1 minus {1,i,-1,-i,infinity}, f(x)=x^4-1, 000009 | 000010 | with the four finite meridians ordered positively and the base quarter 000011 | rotation q(x)=i x. Infinity is punctured because the existing degree-three 000012 | curve is ramified there. This puts all covers below in the same unbranched 000013 | marked-cover category. Their compactifications are used only for genus and 000014 | branch data. 000015 | 000016 | For n in {2,3,4,6,12}, let 000017 | 000018 | C_n : u_n^n=f(x), Delta_n(x,u_n)=(x,zeta_n u_n), 000019 | 000020 | where zeta_n=exp(2 pi i/n). A morphism is over the fixed x-coordinate, 000021 | preserves the ordered finite marks, and sends the chosen deck generator to 000022 | the chosen deck generator. Deck inversion is therefore not silently 000023 | quotiented out. This is the required marking for comparing the existing 000024 | omega sheet with the original compiler sigma. 000025 | 000026 | The supplied predecessors are: 000027 | 000028 | C_2 : y^2=x^4-1, the parity cover of the given all-J monodromy; 000029 | C_3 : w^3=x^4-1, the existing native mu3 curve; 000030 | C_4 : z^4=x^4-1, the kernel cover of the given C4 monodromy J. 000031 | 000032 | No new modulus, metric, coupling, state rule or physical identification is 000033 | introduced. 000034 | 000035 | 1. Minimal common covers 000036 | 000037 | The normalization of C_2 times_B C_3 is the connected curve 000038 | 000039 | C_6 : t^6=f(x), 000040 | C_6 -> C_2, y=t^3, 000041 | C_6 -> C_3, w=t^2. 000042 | 000043 | It is the minimal common marked cover. Indeed the two cyclic monodromy 000044 | characters are reduction of the same winding exponent modulo 2 and modulo 3. 000045 | Their kernel intersection is the kernel modulo lcm(2,3)=6. Equivalently, 000046 | Z/6 -> Z/2 times Z/3 is the Chinese-remainder isomorphism. Hence any connected 000047 | cover dominating both has degree divisible by 6. The displayed C_6 attains 000048 | that degree. 000049 | 000050 | C_6 does not dominate the full C_4 kernel cover. Adding that requirement 000051 | gives the connected normalization 000052 | 000053 | C_12 : s^12=f(x), 000054 | C_12 -> C_2, y=s^6, 000055 | C_12 -> C_3, w=s^4, 000056 | C_12 -> C_4, z=s^3, 000057 | C_12 -> C_6, t=s^2. 000058 | 000059 | This time the kernel intersection is the kernel modulo lcm(4,3)=12, so degree 000060 | 12 is minimal. The degree-six and degree-twelve curves are forced fibre 000061 | products of already supplied marked covers. They are not freely chosen 000062 | auxiliary models. 000063 | 000064 | 2. Peripheral monodromy, branching and genus 000065 | 000066 | With positive meridians (gamma_1,...,gamma_4,gamma_infinity), the C_n deck 000067 | exponents are 000068 | 000069 | rho_n = (1,1,1,1,-4) mod n. 000070 | 000071 | Their sum is zero modulo n, as required by the sphere relation. At each 000072 | finite mark there is one point with ramification index n. Over infinity 000073 | there are gcd(n,4) points, each of ramification index n/gcd(n,4). Thus 000074 | 000075 | n peripheral exponents infinity compact genus 000076 | 2 (1,1,1,1,0) 2 points, e=1 1 000077 | 3 (1,1,1,1,2) 1 point, e=3 3 000078 | 4 (1,1,1,1,0) 4 points, e=1 3 000079 | 6 (1,1,1,1,2) 2 points, e=3 7 000080 | 12 (1,1,1,1,8) 4 points, e=3 15 000081 | 000082 | Riemann-Hurwitz gives, uniformly, 000083 | 000084 | g(C_n)=(3n-gcd(n,4)-2)/2. 000085 | 000086 | Every displayed morphism reduces the peripheral exponent vector modulo its 000087 | target degree. In particular the nontrivial infinity monodromy of C_3 is 000088 | retained on C_6 and C_12; it is not erased by the fact that C_2 and C_4 are 000089 | unramified at infinity. 000090 | 000091 | 3. Plain lifts and the existing mu3 lift 000092 | 000093 | Because f(i x)=f(x), every algebraic lift of q to C_n is 000094 | 000095 | Q_{n,k}(x,u_n)=(i x,zeta_n^k u_n), k in Z/n. 000096 | 000097 | The plain lift is Q_n=Q_{n,0}. On C_2 the two lifts (i x,+/-y) both have 000098 | order 4. On C_4 all four lifts have order 4. This is the exact finite 000099 | ambiguity already isolated in the predecessor. The cover by itself does 000100 | not select a lift. 000101 | 000102 | The existing mu3 rotation is different from the plain lift: 000103 | 000104 | D_3=Delta_3=(x,omega w), 000105 | R_3=Q_3 D_3=(i x,omega w). 000106 | 000107 | It satisfies 000108 | 000109 | R_3^4=D_3, R_3^9=Q_3, ord(R_3)=12, 000110 | 000111 | whereas Q_3 has order 4. This is the algebraic marked-cover realization of 000112 | the existing Burau and covered-seam relation r^4=omega. The relation uses 000113 | the non-plain lift. Merely writing x -> i x selects Q_3 and loses the deck 000114 | factor. 000115 | 000116 | 4. Exact compiler dictionary 000117 | 000118 | The original lattice chart has commuting operators 000119 | 000120 | J(x0,x1,...)=(-x1,x0,...), ord(J)=4, 000121 | sigma(x)=x[4,5,0,1,2,3,6,7], ord(sigma)=3, 000122 | c=J sigma, ord(c)=12, 000123 | 000124 | and the actual checked identities 000125 | 000126 | c^4=sigma, c^9=J. 000127 | 000128 | The cover clock has the same marked factorization, not only the same order: 000129 | 000130 | J <-> Q_3 plain base-quarter lift, 000131 | sigma <-> D_3 oriented mu3 deck, 000132 | c=J sigma <-> R_3=Q_3 D_3. 000133 | 000134 | This preserves the generator, the distinguished C3 kernel, the C4 quotient, 000135 | the embedded plain C4 splitting, and both power identities. Hence 000136 | 000137 | -> 000138 | 000139 | is an isomorphism of marked cyclic clock groups. Equality of order is only 000140 | one consequence of this stronger dictionary. 000141 | 000142 | On the common covers, let Delta_n remain the chosen deck generator. The 000143 | compiler-marked lifts are 000144 | 000145 | R_6 = Q_6 Delta_6^4 = (i x,omega_bar t), 000146 | R_12 = Q_12 Delta_12^4 = (i x,omega s). 000147 | 000148 | Both have exact order 12, fourth power Delta_n^4 of order 3, and ninth power 000149 | the plain lift Q_n. Their descents are 000150 | 000151 | C_12 -> C_6 : R_12 -> R_6, 000152 | C_12 -> C_4 : R_12 -> Q_4, 000153 | C_12 -> C_3 : R_12 -> R_3, 000154 | C_12 -> C_2 : R_12 -> Q_2, 000155 | C_6 -> C_3 : R_6 -> R_3, 000156 | C_6 -> C_2 : R_6 -> Q_2. 000157 | 000158 | Here omega_bar on t is not a reversed mu3 marking: squaring t to w sends 000159 | omega_bar^2=omega, so the C_3 deck orientation is the fixed original one. 000160 | 000161 | 5. Uniqueness and the no-free-parameter conclusion 000162 | 000163 | A lift of R_3 to C_6 is Q_6 Delta_6^k with k=1 or 4. A lift to C_12 has 000164 | k=1,4,7 or 10. These are exactly the solutions of k=1 mod 3. Requiring the 000165 | existing compiler marking J=c^9 to be the plain base lift adds 000166 | 000167 | 9k=0 mod n. 000168 | 000169 | It selects k=4 uniquely for n=6 and n=12. Thus the complete data already in 000170 | the program 000171 | 000172 | finite peripheral weight +1, 000173 | oriented sigma deck, 000174 | plain compiler-J lift, 000175 | c=J sigma, 000176 | 000177 | force the common-cover clock. There is no continuous parameter and no new 000178 | discrete choice. If the J=c^9 marking is forgotten, the two or four lifts 000179 | above reappear. If the conjugate mu3 sheet is chosen, sigma is replaced by 000180 | sigma^{-1} and the marked compiler generator becomes J sigma^{-1}=c^5. That 000181 | is a different explicit marking, not an unrecorded freedom in this result. 000182 | 000183 | 6. Actual carrier characters 000184 | 000185 | The actual matrices give a sharper statement than equality of clock order.