TFPT scientific source excerpt Original repository path: experiments/theory-contracts/marked-seam-monodromy-20261023/THEORY.txt Original file SHA-256: 83f6f464aff89acb5cc4194244a5636d4a4ff62600cdf5ee3c33ded4410f963a Inclusive original lines: 169-251 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. 000169 | 4. Complete four-puncture classification for the quarter-clock stratum 000170 | 000171 | Use Y=CP1, Sigma=(1,i,-1,-i), X=Y minus Sigma. Ordered positive meridians 000172 | gamma0,...,gamma3 have product1 and give pi1(X)=F3. Fix 000173 | rho(gamma_i)=J for all i. 000174 | This is valid because J^4=1. We classify ALL lifts rhohat:F3->Gamma of 000175 | this fixed rho, modulo simultaneous conjugation by the full Gamma. 000176 | No puncture permutation or extra spatial-reflection equivariance is imposed. 000177 | 000178 | THEOREM B (classification, obstruction, orbit and stabilizer). 000179 | The conjugacy classes over J are C_u={(u,v,1):v in F2^m}, indexed by u. 000180 | For four prescribed peripheral classes C_(u0),...,C_(u3): 000181 | (i) a lifted representation exists iff u0+u1+u2+u3=0; 000182 | (ii) if all four u_i agree, there are2^(2m) isomorphism classes; 000183 | (iii) otherwise there are2^(2m-1) classes. 000184 | The corresponding ordered-tuple orbit sizes are2^m and2^(m+1); 000185 | the stabilizer orders are2^(m+2) and2^(m+1), respectively. 000186 | As groups these stabilizers are C4 times(C2)^m in the diagonal case and 000187 | the center Z(Gamma)=(C2)^(m+1) in the nondiagonal case. 000188 | 000189 | PROOF WITH NORMAL FORMS. Write M_i=(u_i,v_i,1). The product relation is 000190 | sum u_i=0, sum v_i=u1+u3. (4.1) 000191 | Conjugation by (w,z,k) fixes all u_i and changes 000192 | v_i -> v_i+w+(k mod2)u_i. (4.2) 000193 | Hence C_u is exactly one class. For any zero-sum profile choose v0=0 000194 | using w, put x=v1,y=v2, and v3=x+y+u1+u3. Thus all solutions have been 000195 | constructed. The only remaining action on (x,y) is translation by 000196 | d=(u1+u0,u2+u0). (4.3) 000197 | If d=0, (4.1) makes the whole profile diagonal; all2^(2m) pairs remain. 000198 | If d!=0, the translation pairs them freely. Formula(4.2) also gives the 000199 | orbit sizes; divide |Gamma|=2^(2m+2) by these to obtain stabilizers. 000200 | More explicitly a stabilizer element(w,z,k) must have 000201 | w=(k mod2)u_i for every i. For a nondiagonal profile this forces w=0 000202 | and even k, giving precisely Z. For u_i=t it gives arbitrary z and k, 000203 | generated by(t,0,1) of order4 and the independent central v basis. 000204 | Every isomorphism in the marked problem is precisely a simultaneous 000205 | conjugation. The normal forms therefore establish completeness. 000206 | 000207 | For m=4 there are4096admissible ordered profiles,16diagonal and4080other. 000208 | Their complete coarse moduli has 000209 | 16*256+4080*128=526336points. 000210 | There are4096orbits of size16 with stabilizer64, and522240orbits of size32 000211 | with stabilizer32. There are2^24=16777216based tuples in total; the 000212 | orbit-size mass agrees exactly. All counts concern the specified finite 000213 | group and peripheral projection; they are not a G31 census. 000214 | 000215 | EXAMPLES AND COUNTEREXAMPLES. The profile(1,0,0,0) has no solution by(4.1). 000216 | The profile(0,0,0,0) has256classes, not one. Choosing(x,y)=(0,0),(1,0), 000217 | (1,2) in that profile gives monodromy images of orders4,8,16, so equal 000218 | peripheral conjugacy classes need not determine a marked representation. 000219 | For profile(0,1,0,1), (x,y) and(x+1,y) are equivalent; the moduli has128 000220 | points. Bit labels1,2 denote independent vectors in F2^4. 000221 | 000222 | THEOREM C (complete monodromy-image order invariant). 000223 | For the normal form above set a=u1+u0,b=u2+u0, and 000224 | r=dim span{(a,x),(b,y),(0,a),(0,b)} in F2^(2m). 000225 | Then |im rhohat|=4*2^r, with r<=4. For m=4 the class counts are 000226 | image order4:16;8:720;16:14880;32:80640;64:430080. 000227 | In particular none of these representations surjects onto Gamma. 000228 | 000229 | PROOF. The first three meridians generate the image. Using M0 as the 000230 | order4 lift, their ratios M1 M0^-1 and M2 M0^-1 are the two A elements 000231 | (a,x),(b,y). Their conjugates under M0 add (0,a),(0,b); further 000232 | conjugation adds nothing because N^2=0. These span exactly the kernel 000233 | of the image projection to C4. Its intersection with is trivial, 000234 | so the order formula follows. 000235 | For the census, rank(a,b) is0,1or2. Rank0 gives16profiles; the pair(x,y) 000236 | has ranks0,1,2 in counts1,45,210 per profile. Rank1 gives720profiles. 000237 | After replacing the two kernel generators by an invertible pair, one 000238 | has u parts(a,0), a!=0. There are16quotient pairs with y in and112 000239 | with y outside, giving image16or32. Rank2 gives3360profiles,128classes 000240 | each, all with r=4. The stated totals follow and sum to526336. 000241 | 000242 | The coarse finite moduli is discrete, so each point is a connected 000243 | component in this finite topology. The groupoid additionally retains the 000244 | computed isotropy groups. No algebraic scheme or analytic singularity 000245 | structure is being inferred merely from a finite orbit set. 000246 | 000247 | Every M_i has exact order4: its square is (0,u_i,2). Thus these 000248 | representations also factor through the four-cone orbifold presentation 000249 | . They do not extend as ordinary 000250 | local systems across a filled puncture, where its meridian must become1. 000251 |