TFPT scientific source excerpt
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000106 | ## 3. Pure mapping class quotient of the original unmarked stratum
000107 | 
000108 | This section adds a different equivalence to the original four meridian model.
000109 | It does not quotient fixed reflection markings by incompatible base changes.
000110 | The standard pure mapping class generators are the two adjacent Dehn twists,
000111 | represented by squared Hurwitz moves b0^2,b1^2. The two twist generation of
000112 | PMod(S0,4) is recalled in Section2 of Ferihe Atalan, MPIM24-26:
000113 | https://archive.mpim-bonn.mpg.de/5138/1/mpim-preprint_2024-26.pdf
000114 | 
000115 | Use b_i(x,y)=(x*y*x^-1,x). With N=I+A, one multiplication gives
000116 | 
000117 | ```
000118 | b_i(ci,cj) = (N*ci+A*cj, ci), j=i+1;
000119 | b_i^2: ci -> ci+N*(ci+cj), cj -> cj+N*(ci+cj).
000120 | ```
000121 | 
000122 | N^2=0 proves these squared moves are involutions and commute on the character
000123 | tuples. The meridian product is preserved. On the product1 space b2^2=b0^2
000124 | modulo uniform character gauge: N*(c0+c1)=N*(c2+c3), so their difference is
000125 | the same N*(c0+c1) in all four positions. They preserve each peripheral
000126 | Gamma conjugacy class. Hence the full pure mapping class image on the original
000127 | Gamma moduli is C2 x C2, generated by P0=b0^2 and P1=b1^2.
000128 | 
000129 | The product1 character space has dimension24. Uniform character gauge has
000130 | dimension4, leaving a vector space Q of dimension20. A,P0,P1 induce commuting
000131 | involutions on Q. For masks0..7, with bits representing A,P0,P1, the respective
000132 | fixed point counts are
000133 | 
000134 | ```
000135 | [1048576,4096,65536,4096,65536,4096,4096,65536].
000136 | ```
000137 | 
000138 | Burnside gives 157696 orbits. Independently every one of the1048576 points
000139 | of Q was enumerated. Taking the least encoded integer among its eight images
000140 | gives precisely157696 different normal forms. Their orbit census, measured in
000141 | the previously counted Gamma classes, is
000142 | 
000143 | ```
000144 | size1:   4096 orbits;
000145 | size2:  46080 orbits;
000146 | size4: 107520 orbits.
000147 | ```
000148 | 
000149 | The counts sum to157696; the size weighted sum is526336. This is the complete
000150 | pure mapping class quotient in the original ordered quarter clock stratum.
000151 | It includes no puncture permutations or arbitrary W(E8) transformations.