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000001 | INDEPENDENT REVIEW: THE CORRECT NATIVE THETA TARGET
000002 | 2026-10-26
000003 | 
000004 | VERDICT
000005 | 
000006 | TYPE CORRECTION PROVED.  The natural finite target for Jac(C)[2] in the native
000007 | Clifford system is the Pauli translation quotient N64/mu4, not the Gaussian
000008 | code quotient V=L/(1+i)L.
000009 | 
000010 | 1. TWO DIFFERENT FOUR-BIT MODULES
000011 | 
000012 | Verification v783 proves both statements needed to separate the types:
000013 | 
000014 |   * N64/mu4 is the sixteen-element Pauli translation space on which
000015 |     G31/N64=Sp(4,2) acts by conjugation;
000016 |   * V=L/(1+i)L labels the sixteen Gaussian code classes.  Its fifteen
000017 |     nonzero classes correspond to the fifteen maximal commuting Pauli
000018 |     contexts, not to the fifteen nonzero Pauli translations.
000019 | 
000020 | The same verification computes dim Hom_G31(V,N64/mu4)=0.  Their relation is
000021 | the point-line outer duality of W(3,2), not a linear identification.  The
000022 | previous fixed-space mismatch with V is therefore a valid code-context result
000023 | but not a theta-group obstruction.
000024 | 
000025 | 2. ACTION OF THE FROZEN NATIVE CUBIC ON THE PAULI QUOTIENT
000026 | 
000027 | The frozen sigma is an order-three Clifford unitary with complex trace one.
000028 | Its conjugation action on End(C^4) consequently has trace
000029 | 
000030 |   tr(Ad_sigma)=tr(sigma) conjugate(tr(sigma))=1.
000031 | 
000032 | In the Pauli basis this adjoint action is a signed permutation.  A projective
000033 | Pauli fixed by an order-three element cannot acquire sign -1: cubing such an
000034 | action would still give -1, contradicting sigma^3=I.  Every fixed Pauli thus
000035 | contributes +1 to the adjoint trace.  Since the identity is already fixed and
000036 | the trace is one, no nonidentity Pauli is fixed.  Equivalently,
000037 | 
000038 |   dim_F2 (N64/mu4)^sigma=0,
000039 |   chi_sigma(x)=(x^2+x+1)^2.
000040 | 
000041 | This exactly matches the source action c_src on
000042 | Jac(C)[2]=V_q direct-sum V_p.  The two order-three actions belong to the same
000043 | fixed-point-free symplectic conjugacy class.
000044 | 
000045 | 3. CONSEQUENCE FOR THE SOURCE-SELECTED SPIN RESULT
000046 | 
000047 | The selected even theta characteristic S_0 and the projective order-four lift
000048 | beta remain valid.  The code-quotient mismatch no longer blocks a theta-to-
000049 | Pauli comparison.  It also does not construct that comparison.  The remaining
000050 | task is an explicitly marked symplectic isomorphism
000051 | 
000052 |   Jac(C)[2] -> N64/mu4
000053 | 
000054 | together with its compatible lift to the two central theta or Heisenberg
000055 | extensions and an equivariant identification of the resulting four-
000056 | dimensional modules.  Equality of the order-three conjugacy class is a
000057 | necessary compatibility test, not that missing marked construction.