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Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/MARKED-Q-DIVISOR-CENTRAL-TWIST.txt
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000001 | THE MARKED NEGATIVE DIVISOR RESTORES THE MISSING CENTRAL CHARACTER
000002 | Exact finite-point determinant test after theta-sign cancellation
000003 | 2026-10-26
000004 | 
000005 | VERDICT
000006 | 
000007 | PROVED AS A SOURCE-DETERMINED CENTRAL-CHARACTER LINE ON THE FULL THETA
000008 | FAMILY.  The ordinary theta determinant forgets the selected spin scalar
000009 | because its Euler
000010 | characteristic is zero.  The already marked negative branch divisor
000011 | 
000012 |     D_q=q_0+q_1+q_2
000013 | 
000014 | has odd length three.  Its finite evaluation determinant does not have that
000015 | cancellation.  If
000016 | 
000017 |     B_q=det H^0(D_q,S_0|D_q),
000018 | 
000019 | or equivalently one uses the permutation-sign-free norm line N_q, then the
000020 | source-selected hyperelliptic lift beta acts by -i on B_q and by +i on its
000021 | dual.  Therefore
000022 | 
000023 |     W_q=H^0(Jac(C),O(2 Theta)) tensor N_q^(-1)                  (1)
000024 | 
000025 | has
000026 | 
000027 |     beta|W_q=+i I_4,
000028 |     alpha^2|W_q=+i I_4,
000029 |     alpha^4|W_q=-I_4.                                          (2)
000030 | 
000031 | These are the missing native central relations, and the constant line twist
000032 | does not alter the cubic action.  No continuous
000033 | phase was fitted: D_q, S_0, alpha and its orientation were all selected
000034 | earlier by the original marked source.  The
000035 | remaining discrete choice is whether the physical boundary construction
000036 | uses the evaluation line or its dual.  The exact sequence below fixes the
000037 | dual when one imposes vanishing at D_q, but the original source has not yet
000038 | been shown to impose that boundary operation.
000039 | 
000040 | This does not yet make (1) the physical quantization space.  The full-family
000041 | statement requires the Poincare family to be normalized relative to S_0,
000042 | rather than additionally choosing pointwise rigidification at q_0.  Section
000043 | 4 proves that this
000044 | source-selected normalization makes the evaluation norm constant and is
000045 | c and Q covariant.  The original papers inspected here still do not insert
000046 | this marked-point determinant into the field or Calderon construction.
000047 | Equation (1) is the smallest exact family-level repair candidate; its use as
000048 | the source Hilbert or conformal-block module remains a new bridge statement
000049 | to prove.
000050 | 
000051 | 
000052 | 1. THE FINITE-POINT LINE IS A NATURAL BOUNDARY DETERMINANT
000053 | 
000054 | On
000055 | 
000056 |     C: eta^2=1-w^6,
000057 | 
000058 | write q_a=-omega^a, p_a=omega^a.  The source-selected spin line is
000059 | 
000060 |     S_0=O_C(D_q-K_C),       S_0^2=K_C.                          (3)
000061 | 
000062 | Because D_q is a reduced divisor of length three,
000063 | 
000064 |     H^0(D_q,S_0|D_q)=direct_sum_a S_0|q_a,
000065 |     B_q=det H^0(D_q,S_0|D_q).                                  (4)
000066 | 
000067 | The exact evaluation sequence
000068 | 
000069 |     0 -> S_0(-D_q) -> S_0 -> S_0|D_q -> 0                     (5)
000070 | 
000071 | makes B_q a canonical boundary term in determinant functoriality.  In fact,
000072 | by (3), S_0(-D_q)=O_C(-K_C).  Thus (4) is not an arbitrary one-dimensional
000073 | character appended to the theta module.
000074 | 
000075 | There are two closely related lines that must be distinguished:
000076 | 
000077 |     B_q = det H^0(D_q,S_0|D_q),
000078 | 
000079 |     N_q = B_q tensor det H^0(D_q,O_Dq)^(-1)
000080 |         = tensor_a S_0|q_a.                                    (6)
000081 | 
000082 | N_q is the ordinary norm or unordered tensor line.  Under a permutation of
000083 | the points, B_q contains the determinant sign; N_q cancels that sign against
000084 | the same permutation on O_Dq.  Both have the same beta action because beta
000085 | fixes every branch point.  Their alpha actions differ by one minus sign
000086 | because Q swaps q_1 and q_2.  Their alpha squares are identical.  Formula
000087 | (1) uses N_q only to avoid making an irrelevant ordering choice.
000088 | 
000089 | 
000090 | 2. THE q ORBIT CARRIES THE REQUIRED +i CHARACTER
000091 | 
000092 | The original quarter map is
000093 | 
000094 |     Q:(w,eta) |-> (1/w,-i eta/w^3),
000095 |     h=Q^2:(w,eta) |-> (w,-eta).                                (7)
000096 | 
000097 | At q_0=-1, eta is a local coordinate and Q sends eta to +i eta.
000098 | For the selected spin lift alpha this gives
000099 | 
000100 |     alpha^2=beta,       beta|S_0,q_0=+i.                        (8)
000101 | 
000102 | The cubic map c:(w,eta)|->(omega w,eta) cycles the three q points and
000103 | commutes with h.  Let gamma be an order-three lift of c to S_0.  The
000104 | commutator of gamma and beta is a scalar zeta.  From gamma^3=I and
000105 | beta^2=-I one obtains
000106 | 
000107 |     zeta^3=1,       zeta^2=1,
000108 | 
000109 | so zeta=1.  Hence beta has the same +i fiber action at every q_a.  Taking
000110 | the determinant or norm over the three points gives
000111 | 
000112 |     beta|B_q=beta|N_q=i^3=-i,
000113 |     beta|N_q^(-1)=+i.                                          (9)
000114 | 
000115 | The normalized c lift acts trivially on N_q: it cyclically permutes three
000116 | fibers, the cycle is even, and the product of its fiber maps is gamma^3=1.
000117 | Rescaling gamma by a cube root of unity also changes the product by its
000118 | third power and therefore changes nothing.  The twist preserves the cubic
000119 | operator rather than fitting a new one.
000120 | 
000121 | The ordinary theta space
000122 | 
000123 |     H_Theta=H^0(Jac(C),O(2 Theta))
000124 | 
000125 | has beta=+I under its canonical even linearization.  Combining this with
000126 | (9) proves the first relation in (2).  For alpha, Q fixes q_0 and exchanges
000127 | q_1,q_2.  Whether one uses B_q or N_q, functoriality gives
000128 | 
000129 |     alpha^2|N_q=beta|N_q=-i,
000130 |     alpha^2|N_q^(-1)=+i.                                      (10)
000131 | 
000132 | Replacing alpha by -alpha multiplies its action on the three-fiber line by
000133 | -1, so alpha itself is still determined only up to the already known global
000134 | sign.  Its square in (10) is sign independent.  Since alpha^2 acts as +I on
000135 | H_Theta, equations (1) and (10) prove the other two relations in (2).
000136 | 
000137 | 
000138 | 3. THE s-EXCHANGED POSITIVE LINE HAS THE CONJUGATE CHARACTER
000139 | 
000140 | The same calculation is not symmetric after retaining the marked Q
000141 | orientation.  At p_0=+1, formula (7) sends eta to -i eta.  Cubic covariance
000142 | then gives
000143 | 
000144 |     beta|S_0,p_a=-i       for all a,
000145 |     beta|N_p=(-i)^3=+i,
000146 |     beta|N_p^(-1)=-i,                                         (11)
000147 | 
000148 | where D_p=p_0+p_1+p_2 and N_p is defined as in (6).
000149 | 
000150 | The source involution s:(w,eta)|->(-w,eta) exchanges D_q and D_p and obeys
000151 | 
000152 |     s Q s=Q^(-1).                                               (12)
000153 | 
000154 | It therefore exchanges the two twisted spaces
000155 | 
000156 |     W_q=H_Theta tensor N_q^(-1),
000157 |     W_p=H_Theta tensor N_p^(-1),                               (13)
000158 | 
000159 | on which beta has the conjugate central characters +i and -i.  A
000160 | complex-linear operator cannot conjugate the scalar +i I on one fixed
000161 | four-dimensional space to -i I on that same space.  An antiunitary
000162 | identification or a changed complex structure would be extra data and is not
000163 | introduced here.
000164 | 
000165 | This does not damage the q candidate.  The accepted (4,3,2) triangle and
000166 | v117 coefficient source already select the negative q packet; s is not a
000167 | symmetry of that isolated A0* object.  If one demands the earlier six-line
000168 | s covariance, the correct object is the paired system W_q direct_sum W_p,
000169 | with s exchanging its two conjugate central-character blocks.  That paired
000170 | eight-dimensional system is not being identified with the native four-
000171 | dimensional module.
000172 | 
000173 | 
000174 | 4. THE SOURCE-SELECTED SPIN NORMALIZATION MAKES THE FAMILY LINE CONSTANT
000175 | 
000176 | The preceding central-character calculation must be checked over the whole
000177 | Jacobian A.  A Poincare bundle is defined only up to tensoring by a line
000178 | pulled back from A.  Normalizing it at q_0 produces a misleading flat factor.
000179 | Indeed, for a q_0-normalized Poincare bundle P one obtains
000180 |