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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000001 | ACTUAL SOURCE SPIN SIGN KERNEL: EXPLICIT ZERO SECTIONS
000002 | Independent analytic continuation, 2026-10-29. No new source or enumeration.
000003 | 
000004 | Reused source data:
000005 |   elliptic-moebius-source-bridge-20261026/SOURCE-SELECTED-THETA-SPIN.txt;
000006 |   SIGN-LINE-COVER-DICTIONARY.txt:11-31,62-84;
000007 |   CLOCK-Q-PACKET-COMPATIBILITY.txt:61-84,105-170;
000008 |   aps_source_review/SOURCE-SPIN-SIGN-KERNEL-REVIEW.txt:19-69.
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.
000050 | Hence c sends the section line a to a-1 and Q sends it to -a-1,
000051 | as the existing global divisor calculation already requires.
000052 | 
000053 | 4. Explicit phase boundary and a compatible representative
000054 | For any spin-compatible section transport in the normalization above, write
000055 |   c^*psi_a=u_a psi_(a-1), Q^*psi_a=v_a psi_(-a-1).
000056 | Their squares must satisfy
000057 |   u_a^2=omega^2, v_a^2=-i omega^(a+2).
000058 | These equations alone do not choose the signs or source lift.
000059 | 
000060 | For completeness there is a common representative using the old Kummer
000061 | choice epsilon=(1,-1,-1). The meromorphic Q maps on S0 and L_a are
000062 |   xi/w, and epsilon_a omega/H_(-a-1), respectively, xi^2=-i.
000063 | They follow from Q^*mu0=-i w^-2 mu0 and
000064 | Q^*H_a=omega^2/H_(-a-1). Applying them to f_a(1/w) gives
000065 |   Q^*psi_a=epsilon_a xi omega^(1-a) psi_(-a-1).
000066 | Consequently Q^2 has factors (i,-i,i) and Q^4=-I on all three section lines.
000067 | Changing the spin lift xi to -xi multiplies Q by -1, preserving those squares.
000068 | Compatible changes of Kummer frames conjugate the displayed basis phases.
000069 | An order-three c spin representative can similarly be chosen with
000070 |   c^*psi_a=omega psi_(a-1).
000071 | The exact originally adopted lifts must be compared before fixing these
000072 | representatives as the physical phases; pullback versus flat-solution
000073 | monodromy also retains the old inverse convention.
000074 | 
000075 | In the column convention Q^*psi_a=v_a psi_(-a-1), set
000076 |   xi=exp(-i pi/4), B=diag(1,1,-omega).
000077 | The resulting section matrix satisfies the exact single-generator identity
000078 |   Q_kernel=(-xi) B A_q B^-1,
000079 | and spec Q_kernel=(-xi){1,i,-i}.
000080 | This is a spin phase times an explicit conjugacy to the old coefficient A_q.
000081 | It is not equality to v117 or native kappa. The SAME basis change must also
000082 | transport c: its new matrix is B^-1 (omega C_cyclic) B, where C_cyclic sends
000083 | index a to a-1. No full c/Q intertwiner follows from the Q spectrum alone.
000084 | 
000085 | 5. What is now explicitly evaluable
000086 | The three zero sections, their zeros, squares and c/Q transport can now be
000087 | evaluated directly from the existing curve and sign packet.
000088 | The three squared sections lie in the two-dimensional H0(C,K):
000089 |   psi_0^2+omega psi_1^2+omega^2 psi_2^2=0
000090 | in the normalization of section2. This does not identify the three distinct
000091 | spin lines with one ordinary differential bundle.
000092 | No P1 operator insertion, CAR quantization, boundary domain, selected state,
000093 | OS positivity or physical time is inferred from these holomorphic sections.
000094 | The original closed-operator kernel/cokernel count and Seed scope are retained.
000095 |