TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/SOURCE-SPIN-ZERO-SECTIONS.txt
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000009 | C: eta^2=1-w^6, q_a=(-omega^a,0), omega^3=1, indices modulo3.
000010 | The already selected spin is S0=O(q0+q1+q2-K_C).
000011 | The actual sign line is L_a=O(q_a-q_(a+1)).
000012 | 
000013 | 1. Divisors and global section
000014 | Let I_+,I_- be the two points at infinity and represent the canonical divisor
000015 | by K_C=I_++I_-. Indeed dw/eta has a simple zero at each infinity and no
000016 | other zero or pole. Also
000017 |   div(w+omega^a)=2q_a-K_C.
000018 | For T_a=S0 tensor L_a use the literal divisor
000019 |   T_a=2q_a+q_(a+2)-K_C.
000020 | The meromorphic function
000021 |   f_a(w)=1/(w+omega^a)
000022 | has div f_a=K_C-2q_a. Therefore
000023 |   div f_a+T_a=q_(a+2).
000024 | It is a global holomorphic T_a section with its single simple zero there.
000025 | This gives a concrete representative of the previously proved
000026 |   H0(C,T_a)=C f_a, h0=h1=1.
000027 | It is not a globally defined ordinary scalar holomorphic function: its poles
000028 | are absorbed by the specified spin-line divisor.
000029 | 
000030 | 2. One compatible common spin quadratization
000031 | Put H_a=(w+omega^a)/(w+omega^(a+1)).
000032 | The already specified L_a^2 trivialization is represented by H_a, with
000033 | H0 H1 H2=1. A meromorphic differential representing S0^2 -> K is
000034 |   mu0=(w^3+1) dw/eta,
000035 |   div mu0=2(q0+q1+q2-K_C).
000036 | Their tensor product represents T_a^2 -> K by
000037 |   mu_a=mu0 H_a
000038 |       =(w+omega^a)^2(w+omega^(a+2)) dw/eta.
000039 | Thus, for psi_a represented by f_a,
000040 |   psi_a^2=f_a^2 mu_a=(w+omega^(a+2)) dw/eta.
000041 | Equality uses this explicit common algebraic normalization. Without that
000042 | normalization the right side is determined up to a nonzero scalar.
000043 | 
000044 | 3. Exact differential transport
000045 | For Omega_b=(w+omega^b) dw/eta, direct substitution gives
000046 |   c^*Omega_b=omega^2 Omega_(b-1),
000047 |   Q^*Omega_b=-i omega^b Omega_(-b),
000048 | where c(w,eta)=(omega w,eta) and Q(w,eta)=(1/w,-i eta/w^3).
000049 | The minus i includes the eta factor; dropping that factor changes the result.