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Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/THETA-FUNCTOR-SPIN-SIGN-TEST.txt
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000001 | THE THETA FUNCTOR FORGETS THE SELECTED SPIN CENTRAL SIGN
000002 | Smallest decisive test for the geometric theta to native Pauli bridge
000003 | 2026-10-26
000004 | 
000005 | VERDICT
000006 | 
000007 | PROVED FOR THE STANDARD DETERMINANT-OF-COHOMOLOGY THETA FUNCTOR.  The
000008 | source-selected spin line S_0 and its projective hyperelliptic lift beta obey
000009 | 
000010 |     beta^2=-I on S_0.
000011 | 
000012 | Nevertheless, the induced lift on the theta determinant line squares to
000013 | +I.  The reason is exact and does not depend on a basis: the relevant Euler
000014 | characteristic is zero.  Passing to O(2 Theta) cannot recover the lost sign.
000015 | With the standard symmetric linearization, inversion acts as +I on all four
000016 | second-order theta functions.
000017 | 
000018 | Thus the existing geometric theta functor does not derive
000019 | 
000020 |     beta  |->  J=i I_4.
000021 | 
000022 | The already verified Pauli quotient and finite lift remain compatible.  To
000023 | obtain the native central phase one still needs a marked mu_4 theta-group or
000024 | metaplectic linearization whose central character is specified separately.
000025 | Projectively this distinction disappears; linearly it is exactly the
000026 | missing datum.
000027 | 
000028 | 
000029 | 1. THE DETERMINANT LINE AND ITS SCALAR WEIGHT
000030 | 
000031 | Let A=Jac(C), and let S_0 be the unique source-selected even theta
000032 | characteristic from SOURCE-SELECTED-THETA-SPIN.  Over L in A consider
000033 | 
000034 |     E_L=S_0 tensor L,
000035 | 
000036 | and the determinant-of-cohomology line, in one standard convention,
000037 | 
000038 |     D_L=det R Gamma(C,E_L)
000039 |        =det H^0(C,E_L) tensor det H^1(C,E_L)^(-1).              (1)
000040 | 
000041 | Changing to the dual convention reverses every exponent below and changes
000042 | nothing because that exponent is zero.  Since deg(S_0)=g-1=1 and g=2,
000043 | 
000044 |     chi(E_L)=deg(E_L)+1-g=0.                                  (2)
000045 | 
000046 | A scalar lambda I on E_L acts functorially on (1) by
000047 | 
000048 |     lambda^(h^0(E_L)-h^1(E_L))=lambda^chi(E_L)=1.              (3)
000049 | 
000050 | This also holds on the theta divisor, where both cohomology groups can be
000051 | nonzero: Riemann-Roch still gives h^0=h^1.  Formula (3) is therefore a
000052 | fiberwise identity on the determinant line, not a generic-point argument.
000053 | 
000054 | Use the Poincare bundle on C x A, normalized along {q_0} x A at the marked
000055 | fixed point q_0, to form this family over A.  The normalization does not hide
000056 | another central character.  An isomorphism between the relevant normalized
000057 | Poincare pullbacks is unique after its value on the rigidifying fiber is
000058 | fixed.  Dropping that rigidification would allow one to multiply the lift by
000059 | an arbitrary scalar, but doing so is precisely the additional phase choice
000060 | at issue, not part of the canonical functor.
000061 | 
000062 | The determinant line is O(-Theta) in convention (1), or O(Theta) after
000063 | dualizing.  This sign convention has no bearing on (3).
000064 | 
000065 | 
000066 | 2. APPLYING THE TEST TO THE SELECTED SPIN LIFT
000067 | 
000068 | Let h=Q^2 be the hyperelliptic involution.  It induces inversion [-1] on A.
000069 | The already proved source-selected lift
000070 | 
000071 |     beta:h^*S_0 -> S_0
000072 | 
000073 | satisfies beta^2=-I.  Functoriality of determinant of cohomology produces a
000074 | lift beta_D of [-1] to D.  Its square is the determinant action of beta^2:
000075 | 
000076 |     beta_D^2
000077 |       =det R Gamma(beta^2)
000078 |       =det R Gamma(-I)
000079 |       =(-1)^chi(E_L)
000080 |       =+I.                                                       (4)
000081 | 
000082 | Equivalently, if alpha_D is the determinant lift induced from the selected
000083 | order-eight Q lift alpha, then
000084 | 
000085 |     alpha_D^4=det R Gamma(alpha^4)=det R Gamma(-I)=+I.           (5)
000086 | 
000087 | This cannot equal the already constructed native linear candidate, whose
000088 | fourth power is -I.
000089 | 
000090 | So the order-four spin lift becomes an involution on the theta determinant
000091 | line.  The selected local +i phase at q_0 is real information on S_0, but
000092 | the ordinary determinant functor cancels it between H^0 and H^1.
000093 | 
000094 | This is the decisive functorial obstruction.  It is stronger than observing
000095 | that projectivization forgets scalars: the scalar has already disappeared
000096 | on the determinant line before taking projective sections.
000097 | 
000098 | 
000099 | 3. O(2 THETA) DOES NOT RESTORE THE PHASE
000100 | 
000101 | Squaring or dualizing beta_D still gives a lift whose square is +I.  In the
000102 | standard analytic theta structure, a basis of H^0(A,O(2 Theta)) consists of
000103 | the four second-order theta functions
000104 | 
000105 |     Theta[epsilon](tau,z),       epsilon in (Z/2)^2.
000106 | 
000107 | Each is even in z.  Hence the standard symmetric lift of [-1] acts by +I on
000108 | the whole four-dimensional space, and the associated Kummer map factors
000109 | through A/{+1,-1}.  This agrees with (4) and (5).
000110 | 
000111 | One may multiply this linear lift by i and thereby make it square to -I and
000112 | act as i I_4.  That multiplication is a new choice of mu_4 central
000113 | character.  It is not selected by determinant of cohomology, by S_0 alone,
000114 | or by beta after (3).  In projective space +I and iI have the same image, so
000115 | the Kummer representation cannot decide this choice.
000116 | 
000117 | Mumford's theta group supplies exactly the correct general home for such a
000118 | linear phase: its central G_m acts with its chosen central character in the
000119 | Schrodinger representation.  The existence of that theta group does not
000120 | identify its mu_4 with the native G31 center.  Such an identification is
000121 | additional marked theta-structure or metaplectic data.
000122 | 
000123 | 
000124 | 4. WHAT THE OLDER NATIVE SOURCES DO AND DO NOT SUPPLY
000125 | 
000126 | The targeted source audit gives no hidden sign-preserving functor.
000127 | 
000128 |   * verification/v783_two_qubit_clifford.py, especially P4.1--P4.4,
000129 |     constructs the native extension
000130 | 
000131 |         mu_4 . Pauli_16 . Sp(4,2)
000132 | 
000133 |     and separates the Pauli translation module from the duality-twisted
000134 |     Gaussian code-label module.  It fixes the native central mu_4, but it
000135 |     contains no determinant-of-cohomology construction from (C,S_0,beta).
000136 | 
000137 |   * verification/v784_curve_code_2torsion.py concerns the separate genus
000138 |     three curve y^3=x^4-1 and a finite A[2] refinement for a type (1,3)
000139 |     polarization.  Its own statement is explicitly a two-torsion comparison.
000140 |     It neither constructs H^0(Jac(C),O(2 Theta)) for the present genus two
000141 |     curve nor transports the spin scalar beta^2=-I.
000142 | 
000143 |   * SIX-LINE-THETA-COMPLETION already records that the general theta-group
000144 |     construction exists only after the corresponding theta data are chosen.
000145 |     Equation (4) now identifies the exact missing datum: it is the linear
000146 |     central phase, not the Pauli quotient.
000147 | 
000148 | 
000149 | 5. EXACT STATUS OF THE BRIDGE
000150 | 
000151 | The current source chain therefore reaches the following sharp boundary.
000152 | 
000153 | PROVED FROM THE ORIGINAL CURVE SOURCE:
000154 | 
000155 |   * S_0 is the unique c-invariant even theta characteristic;
000156 |   * Q selects beta with beta^2=-I;
000157 |   * Jac(C)[2] with c matches the correctly typed native Pauli translation
000158 |     module at the finite symplectic level;
000159 |   * compatible native matrix lifts with the desired finite relations exist.
000160 | 
000161 | PROVED FAILURE OF THE CANONICAL NEXT FUNCTOR:
000162 | 
000163 |   * determinant of cohomology sends beta^2=-I to +I because chi=0;
000164 |   * the canonical |2 Theta| representation is even and hence cannot recover
000165 |     the lost mu_4 phase.
000166 | 
000167 | STILL REQUIRED FOR A LINEAR NATIVE IDENTIFICATION:
000168 | 
000169 |   * a source-selected theta structure, metaplectic refinement, or other
000170 |     phase-sensitive construction that maps the geometric central lift to
000171 |     the native J=iI_4 and intertwines the four-dimensional modules.
000172 | 
000173 | A Pfaffian or spin-refined determinant could in principle retain square-root
000174 | phase data, but no such source construction has been identified in the
000175 | original material inspected here.  Introducing one would be a new premise
000176 | and would need its own functorial comparison.  The present result does not
000177 | exclude such a refinement; it proves that the ordinary theta determinant and
000178 | Kummer module do not perform the required central-phase transport.
000179 | 
000180 |