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Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/THETA-FUNCTOR-SPIN-SIGN-INDEPENDENT-REVIEW.txt
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000001 | INDEPENDENT REVIEW: THETA FUNCTOR SPIN SIGN TEST
000002 | 2026-10-26
000003 | 
000004 | Reviewed source
000005 | 
000006 |   THETA-FUNCTOR-SPIN-SIGN-TEST.txt
000007 |   SHA-256 a5128ab5c2b1602bd64a7b6610ba9282707d6ec65c839051ecb42ccc50bfb910
000008 | 
000009 | VERDICT
000010 | 
000011 | ACCEPTED FOR THE ORDINARY DETERMINANT-OF-COHOMOLOGY AND STANDARD KUMMER
000012 | FUNCTORS, WITH THE REPORT'S EXPLICIT EXCLUSION OF PFAFFIAN AND METAPLECTIC
000013 | REFINEMENTS.
000014 | 
000015 | 1. DETERMINANT WEIGHT AND COCYCLE
000016 | 
000017 | For E_L=S_0 tensor L on the genus-two curve, deg(E_L)=g-1=1 and therefore
000018 | chi(E_L)=0 for every L in Jac(C).  A scalar lambda on E_L acts on
000019 | 
000020 |   det R Gamma(E_L)=det H^0(E_L) tensor det H^1(E_L)^(-1)
000021 | 
000022 | with weight lambda^(h^0-h^1)=lambda^chi=1.  This remains true on the theta
000023 | divisor because Riemann-Roch gives h^0=h^1 fiberwise; it is not a generic-fiber
000024 | argument.
000025 | 
000026 | The normalized Poincare bundle supplies the required family statement.  The
000027 | hyperelliptic involution fixes q_0 and induces inversion on Jac(C).  An
000028 | isomorphism between the corresponding Poincare pullbacks is unique after its
000029 | value on {q_0} times Jac(C) is fixed.  Its square is therefore the identity by
000030 | the same rigidification.  Consequently functorial composition gives
000031 | 
000032 |   beta_D^2=det R Gamma(beta^2)=det R Gamma(-I)=+I,
000033 |   alpha_D^4=det R Gamma(alpha^4)=+I.
000034 | 
000035 | There is no untracked scalar in these equations.  Removing the rigidification
000036 | would reintroduce precisely an additional phase choice.
000037 | 
000038 | 2. STANDARD SECOND-ORDER THETA SECTIONS
000039 | 
000040 | The determinant line is O(-Theta) in the convention used in the report, with
000041 | the dual convention changing no zero Euler exponent.  Since S_0 is even and
000042 | non-effective, H^0(S_0)=H^1(S_0)=0.  The determinant fiber at the origin is
000043 | therefore canonically C and its induced involution is +1.  This pins the
000044 | standard symmetric linearization rather than its negative.
000045 | 
000046 | For O(2Theta), the four standard second-order theta functions are individually
000047 | even under z->-z.  Hence the normalized inversion lift acts as +I on all of
000048 | H^0(Jac(C),O(2Theta)).  This agrees with the determinant calculation and with
000049 | the factorization of the Kummer map through A/{+1,-1}.
000050 | 
000051 | Multiplying this lift by i would produce square -I and scalar iI_4, but that is
000052 | an extra mu4 central-character choice.  It is not recovered from the ordinary
000053 | determinant functor, S_0, or the local beta phase.
000054 | 
000055 | 3. CONSEQUENCE AND SCOPE
000056 | 
000057 | The source-selected spin relation beta^2=-I and its +i fiber value do not by
000058 | themselves imply beta->J=iI_4 on the ordinary four-dimensional second-order
000059 | theta module.  Thus the finite Pauli and Clifford candidate remains compatible
000060 | but conditional on an additional phase-sensitive extension dictionary.
000061 | 
000062 | This is not a no-go for a Pfaffian, metaplectic, or otherwise spin-refined
000063 | construction.  Such a refinement can retain square-root phase data, but none
000064 | is supplied by the ordinary determinant and Kummer functors checked here.
000065 | The next positive bridge would have to construct that refinement from the
000066 | original source and compare its central character and full Heisenberg action
000067 | with native N64 and J.