TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/e-series-compiler-closure-20261023/COVERS.txt
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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 |