TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/HYPERELLIPTIC-SPIN-QUARTER-INDEPENDENT-REVIEW.txt
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000001 | INDEPENDENT REVIEW: HYPERELLIPTIC SPIN QUARTER
000002 | 2026-10-26
000003 | 
000004 | Reviewed source
000005 | 
000006 |   HYPERELLIPTIC-SPIN-QUARTER.txt
000007 |   SHA-256 68163e1664b76d9cbd2b28ffe8314acb9fefec78a14a4ce9b557d4e8240138eb
000008 | 
000009 | VERDICT
000010 | 
000011 | ACCEPTED WITH ONE TERMINOLOGICAL BOUNDARY.
000012 | 
000013 | For a fixed Weierstrass point W, every L in Pic^1(C) has the unique form
000014 | O(W) tensor M with M in Jac(C).  Since the hyperelliptic involution h fixes W,
000015 | acts as inversion on Jac(C), and K_C is O(2W), the asserted identity
000016 | 
000017 |   h^*L isomorphic K_C tensor L^(-1)
000018 | 
000019 | holds on all of Pic^1(C).  Applying it to S^2 isomorphic K_C proves that all
000020 | sixteen theta characteristics are h invariant as isomorphism classes.
000021 | 
000022 | After fixing alpha:S tensor S -> K_C, any isomorphism beta:h^*S->S can be
000023 | rescaled to intertwine alpha with the differential action of h on K_C.  The
000024 | normalization leaves exactly beta and -beta.  At a Weierstrass fixed point,
000025 | choose u with h(u)=-u.  The canonical action on du is then multiplication by
000026 | -1, so compatibility forces the fiber scalar lambda of beta to satisfy
000027 | lambda^2=-1.  Since beta composed with h^*beta is a global scalar automorphism
000028 | of S, evaluation at that fixed point proves
000029 | 
000030 |   beta composed with h^*beta = -I_S.
000031 | 
000032 | Thus beta is an order-four projective spin lift of the order-two geometric
000033 | involution.  It is not an honest C2-linearization, whose cocycle condition
000034 | would require square +I.  The main report is correct when it calls beta a
000035 | compatible spin lift.
000036 | 
000037 | The stated scope is also correct.  The argument selects neither a theta
000038 | characteristic S nor one of the signs beta and -beta.  It supplies no
000039 | equivariant identification with the native Gaussian E8 module and does not
000040 | remove the separately proved obstruction to identifying the literal Q action
000041 | on Jac(C)[2] with the native central J.