TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/marked-seam-monodromy-20261023/THEORY.txt
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000117 | 3. Quarter-clock lift group and a genuine central class
000118 | 
000119 | The chosen existing J lift j has j^4=1: its fourth power is a character,
000120 | so checking it on the eight integral generators proves the claim on L.
000121 | The generator phases and their products are retained in the certificate.
000122 | Consequently the FULL preimage of <J> in the sign-lift group is
000123 |   Gamma=A semidirect <j>, |Gamma|=256*4=1024.            (3.1)
000124 | This extension splits; it is not central.
000125 | 
000126 | Here is an intrinsic normal form. The pair(L,J) is a torsion-free module
000127 | of rank4 over Z[i], a PID, and hence is free. Reduction modulo2 makes
000128 | J=1+N with N^2=0, rank N=4. The same holds on the dual A. Choose Jordan
000129 | coordinates A=F2^m direct_sum F2^m, m=4, so
000130 |   J.(u,v)=(u,v+u).
000131 | Then Gamma consists of (u,v,k), k in Z4, with multiplication
000132 |   (u,v,k)(a,b,l)=(u+a,v+b+(k mod2)a,k+l mod4).           (3.2)
000133 | This is an explicit presentation, so its associativity also follows
000134 | directly from the action law. Inverses are
000135 |   (u,v,k)^-1=(u,v+(k mod2)u,-k).
000136 | Its center is Z={(0,v,2e)}, order2^(m+1)=32; its derived group is
000137 | {(0,v,0)}, order2^m=16. These follow by taking commutators with j and
000138 | with the u-coordinate basis elements. In particular j and a nonzero u
000139 | do not commute: the extension over A is not central despite splitting.
000140 | 
000141 | THEOREM A (central class and every subgroup restriction).
000142 | Let V=Gamma/Z, written (u,e) in F2^m direct_sum F2. With center coordinates
000143 | (v,t) for (0,v,2t), the section s(u,e)=(u,0,e) has full central cocycle
000144 |   kappa((u,e),(a,f))=(e a,e f).                         (3.3)
000145 | Its quadratic square invariant and alternating commutator are
000146 |   Q(u,e)=(e u,e),
000147 |   B((u,e),(a,f))=(e a+f u,0).
000148 | The class [kappa] in H2(V,Z), with trivial V action, restricts to zero on
000149 | a subgroup W<=V if and only if W is contained in U={(u,0)}.
000150 | 
000151 | PROOF. Multiply the sections using(3.2): the v term is ea; the carry of
000152 | e+f in Z4 is2ef. This proves(3.3); bilinearity proves the cocycle law.
000153 | On U the displayed section is a homomorphism. If W contains (u,1),
000154 | every preimage of this order2 element has square (0,u,2), nonidentity.
000155 | Multiplication by a central element cannot change that square since Z
000156 | has exponent2. Therefore even that cyclic subgroup has no section;
000157 | neither does W. This proves necessity, sufficiency and nontriviality.
000158 | 
000159 | COROLLARY (which scalar phases are removable).
000160 | For a central character chi_(ell,t)(v,e)=(-1)^(ell.v+t e), the scalar
000161 | multiplier is (-1)^(e ell.a+t e f). As a U1-valued cohomology class its
000162 | t term is the coboundary of beta(u,e)=i^(t e). Its ell term is nontrivial
000163 | iff ell!=0: its commutator with (a,0) and (0,1) is(-1)^(ell.a), which
000164 | no scalar coboundary changes. Different ell give distinct commutators.
000165 | Thus the32central characters give16distinct marked U1 classes,15nonzero,
000166 | each occurring twice. A nontrivial Z2 extension of a cyclic subgroup
000167 | must not therefore automatically be called a nontrivial U1 multiplier.
000168 |