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Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/boundary_origin_20261027/uniqueness/INDEX-ONE-VERSUS-Q-DIVISOR-REVIEW.txt
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000001 | INDEPENDENT REVIEW: INDEX ONE VERSUS THE THREE-POINT q DIVISOR
000002 | 2026-10-27
000003 | 
000004 | VERDICT
000005 | 
000006 | The rank obstruction is correct for a claimed invertible Fredholm
000007 | identification of the old one-mode seam defect with the full D_q evaluation
000008 | defect.  The full defect changes the index by -3, while the old directed
000009 | Fourier seam shift changes it by -1.  An invertible Fredholm intertwiner
000010 | cannot identify those two operators.
000011 | 
000012 | There is a precise one-dimensional escape: at a c-linearized fiber, the
000013 | three evaluation fibers form the regular C3 representation and its invariant
000014 | character line has dimension one.  Imposing only that character condition
000015 | changes the index by -1.  This removes the rank obstruction, but it does not
000016 | produce the already required inverse norm line.  Its directed determinant
000017 | has the opposite beta phase.  Thus the invariant-character route is allowed
000018 | as a new typed test, not yet an identification with N_q^(-1).
000019 | 
000020 | 
000021 | 1. THE FULL DOMAIN DEFECT HAS INDEX MINUS THREE
000022 | 
000023 | Let
000024 | 
000025 |   E=S_0 tensor L,
000026 |   D_q=q_0+q_1+q_2.
000027 | 
000028 | For genus two, deg(E)=1 and therefore
000029 | 
000030 |   chi(E)=deg(E)+1-g=0.
000031 | 
000032 | After imposing vanishing at all three q points,
000033 | 
000034 |   deg(E(-D_q))=-2,
000035 |   chi(E(-D_q))=-3.                                             (1)
000036 | 
000037 | The evaluation sequence
000038 | 
000039 |   0 -> E(-D_q) -> E -> E|D_q -> 0                             (2)
000040 | 
000041 | accordingly changes the Fredholm index by -3.  Equivalently,
000042 | 
000043 |   lambda(E(-D_q)) tensor lambda(E)^(-1)
000044 |     =det H^0(D_q,E|D_q)^(-1)                                  (3)
000045 | 
000046 | is the inverse determinant of a rank-three quotient.
000047 | 
000048 | If the recovered Fourier seam operation is a directed index-minus-one
000049 | Fredholm defect, no bounded invertible intertwiner can turn it into (2):
000050 | Fredholm index is invariant under such an intertwining.  This conclusion is
000051 | limited to an asserted identity of the two full operators.  It does not
000052 | exclude a projection, an equivariant summand, a noninvertible pushforward, or
000053 | additional kernel and cokernel data.
000054 | 
000055 | 
000056 | 2. THE CUBIC CHARACTER DECOMPOSITION CONTAINS AN INDEX-ONE SUMMAND
000057 | 
000058 | At a fiber carrying a c linearization, c cyclically permutes the three q
000059 | fibers.  Hence
000060 | 
000061 |   H^0(D_q,E|D_q)=F_1 direct_sum F_omega direct_sum F_omega2,   (4)
000062 | 
000063 | with each character line one-dimensional.  Composing evaluation with the
000064 | projection to F_1 and requiring only this invariant component to vanish gives
000065 | a codimension-one domain condition.  Its index change is -1.
000066 | 
000067 | This construction is not available fiberwise for generic L in Jac(C).
000068 | Generally c sends E_L to E_(cL), rather than acting on one fixed E_L.  A
000069 | single-fiber invariant line requires a c-fixed L together with a
000070 | linearization, or an explicitly constructed equivariant family over a c
000071 | orbit.  The source-selected L=O fiber satisfies this requirement, but it
000072 | does not justify the same reduction for the generic theta family.
000073 | 
000074 | The nontrivial character lines are exchanged by Q because Q conjugates c to
000075 | c^(-1).  The invariant line is the only one-dimensional character summand
000076 | stable under that relation.  Thus it is the correct smallest equivariant
000077 | rank test, but equivariance alone does not determine its determinant
000078 | orientation.
000079 | 
000080 | 
000081 | 3. THE BETA PHASE EXCLUDES THE NAIVE IDENTIFICATION
000082 | 
000083 | On the marked q fiber, beta acts by +i on every evaluation fiber.  It
000084 | therefore acts as follows:
000085 | 
000086 |   beta|F_1=+i,
000087 |   beta|F_1^(-1)=-i,                                           (5)
000088 | 
000089 | whereas on the full permutation-free norm,
000090 | 
000091 |   beta|N_q=i^3=-i,
000092 |   beta|N_q^(-1)=+i.                                           (6)
000093 | 
000094 | The directed index-minus-one relative determinant obtained by imposing the
000095 | invariant condition is F_1^(-1), by the same convention as (3).  Its beta
000096 | phase is -i, opposite to the +i phase of N_q^(-1).  In spin notation, its
000097 | alpha square is likewise -i rather than +i.
000098 | 
000099 | The undualized invariant quotient F_1 has the desired +i scalar, but it is
000100 | the opposite determinant orientation.  Identifying it with the old directed
000101 | defect would reverse the relative line or the index convention.  Such a
000102 | reversal must come from the original operator orientation; it cannot be made
000103 | solely to match the phase.
000104 | 
000105 | The full norm product succeeds because it multiplies all three character
000106 | lines before dualizing:
000107 | 
000108 |   det(F_1 direct_sum F_omega direct_sum F_omega2)^(-1)
000109 |     has phase (i^3)^(-1)=+i.                                  (7)
000110 | 
000111 | One character line and the product of all three are therefore not
000112 | interchangeable, even though both can be one-dimensional determinant lines.
000113 | 
000114 | 
000115 | 4. EXACT SCOPE
000116 | 
000117 | The test proves:
000118 | 
000119 |   * the old one-mode index-minus-one shift is not invertibly identical to the
000120 |     full three-point D_q domain restriction;
000121 | 
000122 |   * a c-invariant rank-one restriction is possible at a c-linearized fiber;
000123 | 
000124 |   * its naturally directed inverse determinant has the wrong beta and alpha
000125 |     square for N_q^(-1);
000126 | 
000127 |   * the full product of three character lines has the correct phase but the
000128 |     index-minus-three domain defect.
000129 | 
000130 | It does not prove that no original boundary construction can produce
000131 | N_q^(-1).  A valid construction may retain the full three-point determinant,
000132 | use a noninvertible index pushforward, or contain a separately justified
000133 | orientation or duality.  What is excluded is the simple claim that the old
000134 | single Fourier defect is already the full q evaluation defect, or that its
000135 | c-invariant part is automatically the inverse q norm.
000136 | 
000137 | The next required identity remains operator-typed: the original source must
000138 | specify which evaluation character or full determinant enters, with which
000139 | relative orientation, on a c and Q compatible domain.  No phase-only
000140 | identification supplies that information.
000141 | 
000142 | 
000143 | FILES CHECKED
000144 | 
000145 |   * research-20261027-boundary-origin/source/
000146 |     ORIGINAL-BOUNDARY-NORM-ORIGIN.txt
000147 | 
000148 |   * research-20261027-boundary-origin/originals/
000149 |     INVERSE-Q-POINT-NORM-ORIGIN.txt
000150 | 
000151 |   * research-20261026-real-boundary/
000152 |     MARKED-Q-DIVISOR-CENTRAL-TWIST.txt
000153 | 
000154 |   * research-20261026-real-boundary/
000155 |     SPIN-RIGIDIFIED-THETA-MODULE.txt