TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/SOURCE-SELECTED-THETA-SPIN-INDEPENDENT-REVIEW.txt
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000001 | INDEPENDENT REVIEW: SOURCE SELECTED THETA SPIN
000002 | 2026-10-26
000003 | 
000004 | Reviewed sources
000005 | 
000006 |   SOURCE-SELECTED-THETA-SPIN.txt
000007 |   SHA-256 cf4478191ccd75d62a2f049a54193541009bed24877df9cbe0ed9f942d07d717
000008 | 
000009 |   HYPERELLIPTIC-SPIN-QUARTER.txt
000010 |   SHA-256 68163e1664b76d9cbd2b28ffe8314acb9fefec78a14a4ce9b557d4e8240138eb
000011 | 
000012 | VERDICT
000013 | 
000014 | ACCEPTED.  The new result closes the theta-characteristic and half-turn-lift
000015 | choices from the already marked q|p curve data.  It does not close the module
000016 | map to the native Gaussian E8 clock.
000017 | 
000018 | 1. THETA CHARACTERISTIC AND PARITY
000019 | 
000020 | For every Weierstrass point r on C, 2r is a canonical divisor.  Hence
000021 | 
000022 |   S_0=O_C(q_0+q_1+q_2-K_C)
000023 | 
000024 | has degree one and 2S_0 is isomorphic to K_C.  Moreover
000025 | 
000026 |   div((w^3+1)/eta)=q_0+q_1+q_2-p_0-p_1-p_2,
000027 | 
000028 | so exchanging the q and p sides leaves S_0 unchanged.
000029 | 
000030 | The even-parity argument is valid.  An effective degree-one theta
000031 | characteristic on a genus-two hyperelliptic curve is O_C(r) for a Weierstrass
000032 | point r.  If S_0 were one of these six classes, use the q expression when r is
000033 | a q point and the equivalent p expression when r is a p point.  Cancelling r
000034 | and replacing K_C by twice another point in the same triple would make the
000035 | difference of two distinct points principal.  That would define a degree-one
000036 | map to P1, impossible for a genus-two curve.  Thus h^0(S_0)=0 and S_0 is even.
000037 | 
000038 | 2. UNIQUENESS FROM THE CUBIC SOURCE SYMMETRY
000039 | 
000040 | The automorphism c cycles each of the q and p triples, so it fixes S_0.  On
000041 | 
000042 |   Jac(C)[2]=V_q direct-sum V_p
000043 | 
000044 | its restriction to either two-dimensional F2 plane has order three and no
000045 | nonzero fixed vector.  Therefore Jac(C)[2]^c=0.  Since theta characteristics
000046 | form an affine Jac(C)[2] torsor, S_0 is the unique c-invariant theta
000047 | characteristic.  This uses the already selected source action rather than an
000048 | additional theta marking.
000049 | 
000050 | 3. Q LIFT AND PULLBACK CONVENTION
000051 | 
000052 | The marked map Q preserves S_0 and fixes q_0.  At q_0, eta is a local
000053 | coordinate and
000054 | 
000055 |   Q^*eta=i eta,       Q^*deta=i deta.
000056 | 
000057 | For a square-root-compatible map alpha:Q^*S_0->S_0, its fiber scalar lambda at
000058 | q_0 therefore obeys lambda^2=i.  The fourth pullback iterate has value
000059 | lambda^4=-1 at the fixed point and is a global scalar, proving
000060 | 
000061 |   alpha^4=-I,         alpha^8=I.
000062 | 
000063 | The composition
000064 | 
000065 |   beta=alpha composed with Q^*alpha:(Q^2)^*S_0->S_0
000066 | 
000067 | is type-correct.  Replacing alpha by -alpha leaves beta unchanged.  In the
000068 | stated pullback convention its fiber value at q_0 is lambda^2=+i, and
000069 | beta^2=alpha^4=-I.  Thus Q selects the sign-independent order-four projective
000070 | spin lift of h=Q^2.  This is not an honest C2-linearization, whose cocycle
000071 | square would be +I.
000072 | 
000073 | 4. SCOPE
000074 | 
000075 | The proof now selects the even spin line S_0, the Q lift up to an overall sign
000076 | that cancels in beta, and the induced local +i phase of beta.  It does not construct an
000077 | equivariant identification of the resulting theta or spin module with the
000078 | native four-dimensional Gaussian E8 module or its central generator J.  The
000079 | separate obstruction to the literal identification with the Gaussian code
000080 | quotient V=L/(1+i)L therefore remains in force.  That quotient labels Pauli
000081 | contexts, not Pauli translations.  It is not an obstruction to the correctly
000082 | typed theta target Jac(C)[2] -> N64/mu4.  For that target the source cubic and
000083 | the frozen native sigma both have zero-dimensional fixed space; the marked
000084 | symplectic and Heisenberg intertwiner remains to be constructed.