TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/CANONICAL-BRANCHED-SPIN-PULLBACK.txt
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000023 | 2. Explicit square-compatible map, not only equality in Picard
000024 | Retain the old common meromorphic spin frame Phi0, with
000025 |   Phi0^2=mu0=(w^3+1)dw/eta, div(Phi0)=Dq-K.
000026 | Direct differentiation of the actual z gives
000027 |   dz=3w^2/[eta(1-w^3)] dw,
000028 |   pi*(dz/z^2)=3w^2/[(w^3+1)eta] dw.
000029 | For f=w/(w^3+1),
000030 |   div(f)=Rhalf-Dq-(Dq-K),
000031 |   pi*(dz/z^2)=3 f^2 mu0.
000032 | Therefore the meromorphic canonical pullback frame is EXACTLY
000033 |   e_can=sqrt(3) f Phi0, e_can^2=pi*(dz/z^2).
000034 | Its divisor is Rhalf-Dq, as required. Multiplying by z gives the equally
000035 | useful frame pulled from sqrt(dz):
000036 |   sqrt(dz)=z e_can=(sqrt(3)w/eta)Phi0,               (1)
000037 |   mu0/dz=eta^2/(3w^2), div(sqrt(dz))=Rhalf-Dp.
000038 | The positive real sqrt(3) is the displayed derivative normalization;
000039 | the remaining common spin sign is the ordinary sign of a spin frame.
000040 | No fitted quarter or relative source scalar occurs in this identity.
000041 | 
000042 | 3. The original c,Q and Real lifts, with the actual spin phase
000043 | Use the already fixed order-three spin representative and marked quarter:
000044 |   c(w,eta)=(omega w,eta), c*Phi0=omega^2 Phi0,
000045 |   Q(z,w,eta)=(iz,1/w,-i eta/w^3), Q*Phi0=(xi/w)Phi0,
000046 |   xi=exp(-i pi/4).
000047 | Then c*f=omega f and Q*f=w f. Consequently
000048 |   c*e_can=e_can, Q*e_can=xi e_can,
000049 |   c*sqrt(dz)=sqrt(dz), Q*sqrt(dz)=i xi sqrt(dz)=bar(xi)sqrt(dz).