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Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/HYPERELLIPTIC-SPIN-QUARTER.txt
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000001 | THE HYPERELLIPTIC HALF TURN HAS AN ORDER FOUR SPIN LIFT
000002 | An exact typed survivor after the literal Q to J quotient obstruction
000003 | 2026-10-26
000004 | 
000005 | VERDICT
000006 | 
000007 | PROVED.  Let C be the genus two curve
000008 | 
000009 |     C: eta^2=1-w^6
000010 | 
000011 | and let h be its hyperelliptic involution.  Every one of the sixteen theta
000012 | characteristics S on C is h invariant.  For each specified S, either of the
000013 | two lifts of h that is compatible with S^2=K_C has square -I.  Thus every
000014 | theta characteristic carries an order four spin lift of the geometric order
000015 | two half turn.
000016 | 
000017 | This proves the exact geometric mechanism by which the half turn h=Q^2 can
000018 | produce a central quarter phase.  It does not select one of the sixteen S,
000019 | choose between the two signs of its lift, or identify the resulting spin
000020 | module with the native Gaussian E8 lattice module.  In particular it does not
000021 | undo the separate proof that the literal action Q on Jac(C)[2] cannot map to
000022 | the central Gaussian J on L/(1+i)L.
000023 | 
000024 | 
000025 | 1. ACTION OF THE HYPERELLIPTIC INVOLUTION ON Pic^1
000026 | 
000027 | Choose a Weierstrass point W.  It is fixed by h and the hyperelliptic fiber at
000028 | W is 2W, so
000029 | 
000030 |     K_C isomorphic O_C(2W).                                      (1)
000031 | 
000032 | Every degree one line bundle has a unique expression
000033 | 
000034 |     L=O_C(W) tensor M,       M in Jac(C).                         (2)
000035 | 
000036 | The hyperelliptic involution induces [-1] on Jac(C).  Hence
000037 | 
000038 |     h^*L = O_C(W) tensor M^(-1)
000039 |           = K_C tensor L^(-1).                                  (3)
000040 | 
000041 | Equivalently, for every degree one divisor class D,
000042 | 
000043 |     hD is linearly equivalent to K_C-D.                          (4)
000044 | 
000045 | This argument takes place on all of Pic^1.  It does not assume that D is an
000046 | effective point divisor.
000047 | 
000048 | 
000049 | 2. ALL SIXTEEN THETA CHARACTERISTICS ARE INVARIANT
000050 | 
000051 | Let S be any theta characteristic:
000052 | 
000053 |     S^2 isomorphic K_C.                                          (5)
000054 | 
000055 | Equations (3) and (5) give
000056 | 
000057 |     h^*S isomorphic K_C tensor S^(-1) isomorphic S.               (6)
000058 | 
000059 | There are 2^(2g)=16 such line bundles at genus two.  Therefore h fixes every
000060 | theta characteristic as an isomorphism class.  This is stronger than the
000061 | modulo two statement that h acts trivially on Jac(C)[2]: it identifies each
000062 | member of the affine theta characteristic torsor with its pullback.
000063 | 
000064 | 
000065 | 3. THE COMPATIBLE LIFT SQUARES TO -I
000066 | 
000067 | Fix an isomorphism
000068 | 
000069 |     alpha:S tensor S -> K_C.                                     (7)
000070 | 
000071 | Because of (6), choose an isomorphism beta:h^*S->S.  The two maps from
000072 | (h^*S)^2 to K_C obtained from alpha, beta, and the differential action of h
000073 | on K_C differ by a nonzero complex scalar.  Rescaling beta by a square root
000074 | makes the diagram compatible with (7).  Exactly two compatible choices
000075 | remain, beta and -beta.
000076 | 
000077 | Now evaluate at any Weierstrass fixed point.  Take a local coordinate u with
000078 | 
000079 |     h(u)=-u.                                                      (8)
000080 | 
000081 | Then h^*(du)=-du.  If beta acts on the one dimensional spin fiber at that
000082 | point by the scalar lambda, compatibility with alpha gives
000083 | 
000084 |     lambda^2=-1,
000085 |     lambda=+i or -i.                                             (9)
000086 | 
000087 | The square beta composed with h^*beta is a holomorphic automorphism of S and
000088 | therefore a constant scalar on the connected compact curve.  Its value at the
000089 | fixed point is lambda^2=-1.  Consequently
000090 | 
000091 |     beta composed with h^*beta = -I_S,                            (10)
000092 | 
000093 | so each compatible lift has order four.  Changing beta to -beta changes its
000094 | quarter phase sign but leaves (10) unchanged.
000095 | 
000096 | 
000097 | 4. PRECISE CONSEQUENCE FOR THE CLOCK DICTIONARY
000098 | 
000099 | For the original marked curve,
000100 | 
000101 |     Q^2=h.
000102 | 
000103 | The correct typed central phase candidate is therefore the spin lift beta of
000104 | the half turn h, not the action of Q on the two torsion label space.  The two
000105 | facts coexist exactly:
000106 | 
000107 |   * h acts as [-1] on integral H_1(C,Z) and as the identity on Jac(C)[2];
000108 |   * its compatible action on every chosen square root of K_C squares to -I.
000109 | 
000110 | Thus an internal quarter phase can survive after the underlying two torsion
000111 | label has forgotten the sign.  This is the same spin cocycle mechanism as the
000112 | older order four lift of an order two base operation, now proved directly on
000113 | the actual genus two curve C.
000114 | 
000115 | Three source selections remain necessary before this lift can be called the
000116 | native Gaussian J:
000117 | 
000118 |   1. the original construction must select a particular theta characteristic
000119 |      S among the sixteen;
000120 |   2. it must select one of the two compatible signs beta and -beta;
000121 |   3. it must construct an equivariant identification of the resulting spin or
000122 |      theta module with the native Gaussian E8 module, including its central
000123 |      mu4 phase.
000124 | 
000125 | The curve, its q and p coefficient packets, and their two torsion action do
000126 | not yet supply those three markings.  No native J identity is claimed here.
000127 | 
000128 | 
000129 | SOURCE ANCHORS
000130 | 
000131 |   * research-20261026-real-boundary/CLOCK-Q-PACKET-COMPATIBILITY.txt,
000132 |     lines 22--54: the actual Q and h=Q^2 on C.
000133 |   * research-20261026-real-boundary/Q-NATIVE-CLOCK-TEST.txt:
000134 |     the separately proved literal quotient obstruction and its exact scope.
000135 |   * Standard genus two hyperelliptic facts used explicitly above:
000136 |     K_C is a hyperelliptic fiber, h acts by [-1] on Jac(C), and theta
000137 |     characteristics form a torsor of size 2^(2g) over Jac(C)[2].