TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/CLOCK-Q-PACKET-COMPATIBILITY.txt
Original file SHA-256: fe37f20bbb2d959dfe678ca1bf435a69067cfcb859a3d36769294bbe9bf61c32
Inclusive original lines: 22-54
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000022 | 
000023 |     f(z)=(z^2-1)/(z^2+1),
000024 |     w^3=f(z),
000025 |     eta=z(1-w^3),
000026 |     C: eta^2=1-w^6.                                                (1)
000027 | 
000028 | A direct calculation gives
000029 | 
000030 |     f(i z)=1/f(z).                                                  (2)
000031 | 
000032 | Therefore a lift of the quarter map z->i z has w' equal to a cube root of
000033 | w^(-3).  The marking used in the September source is
000034 | 
000035 |     Q:(z,w,eta) |-> (i z, 1/w, -i eta/w^3).                       (3)
000036 | 
000037 | The eta factor is not optional.  It follows from eta=z(1-w^3):
000038 | 
000039 |     eta' = i z(1-w^(-3)) = -i eta/w^3.                             (4)
000040 | 
000041 | For q_a=-omega^a one has
000042 | 
000043 |     1/q_a = -omega^(-a) = q_(-a).                                 (5)
000044 | 
000045 | Thus Q preserves the negative orbit {q_0,q_1,q_2}.  It also preserves the
000046 | positive orbit separately.  The apparent q to p exchange came from dropping
000047 | the minus sign in the reciprocal of q_a.
000048 | 
000049 | Applying (3) twice gives
000050 | 
000051 |     Q^2:(z,w,eta) |-> (-z,w,-eta),                                 (6)
000052 | 
000053 | which is the hyperelliptic involution of C over the w-sphere.  In particular
000054 | Q has order four on C.