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Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/CUBIC-NATIVE-MARKING-INDEPENDENT-REVIEW.txt
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000001 | INDEPENDENT REVIEW: CUBIC NATIVE MARKING TEST
000002 | 2026-10-26
000003 | 
000004 | Reviewed source
000005 | 
000006 |   CUBIC-NATIVE-MARKING-TEST.txt
000007 |   SHA-256 361a7c61680ae2450a3b62a34a71cfe7525d65433f123f0ffea3668d9f44de48
000008 | 
000009 | VERDICT
000010 | 
000011 | ACCEPTED.  The corrected theorem uses the type-correct native theta target
000012 | N64/mu4.  The earlier fixed-space obstruction remains valid only for the
000013 | distinct Gaussian code and Pauli-context module V=L/(1+i)L.
000014 | 
000015 | 1. TARGET TYPE AND ORDER-THREE ACTION
000016 | 
000017 | Verification v783 distinguishes two four-bit Sp(4,2) modules.  N64/mu4 is the
000018 | Pauli translation space.  The fifteen nonzero classes of V=L/(1+i)L label the
000019 | fifteen maximal commuting Pauli contexts.  Their relation is the outer
000020 | point-line duality of W(3,2), and v783 finds no invertible linear module
000021 | intertwiner.  A genus-two theta translation space must therefore first be
000022 | compared with N64/mu4.
000023 | 
000024 | On Jac(C)[2]=V_q direct-sum V_p, the source cubic has matrix
000025 | diag(C_0,C_0), with C_0=[[1,1],[1,0]].  Its characteristic polynomial is
000026 | (x^2+x+1)^2 and its fixed space is zero.
000027 | 
000028 | In Pauli bit order (x_1,z_1,x_2,z_2), the four conjugation formulas in the
000029 | report give the columns of
000030 | 
000031 |   Sigma=[[0,0,1,0],
000032 |          [0,1,0,1],
000033 |          [1,0,1,0],
000034 |          [0,1,0,0]].
000035 | 
000036 | Direct F2 multiplication confirms Sigma^3=I, ker(Sigma-I)=0 and
000037 | chi_Sigma=(x^2+x+1)^2.  This also agrees with the character check:
000038 | tr_C(sigma)=1 implies tr_End(Ad_sigma)=1, and an order-three Clifford element
000039 | cannot fix a projective Pauli with negative sign.  Thus only the identity
000040 | Pauli is fixed.
000041 | 
000042 | 2. EXPLICIT INTERTWINER
000043 | 
000044 | For the columns stated in the report,
000045 | 
000046 |   Phi=[[0,1,0,1],
000047 |        [0,1,1,1],
000048 |        [1,0,1,0],
000049 |        [1,1,1,0]],
000050 | 
000051 | exact multiplication gives
000052 | 
000053 |   Phi^T Omega Phi=diag(Omega_2,Omega_2),
000054 |   Sigma Phi=Phi diag(C_0,C_0).
000055 | 
000056 | Hence Phi is invertible, symplectic and intertwines the actual marked source
000057 | cubic with conjugation by the frozen native sigma on Pauli translations.
000058 | 
000059 | 3. INVARIANT PLANES AND RESIDUAL COUNT
000060 | 
000061 | An exhaustive enumeration of the 35 two-dimensional subspaces of F2^4 gives
000062 | exactly five Sigma-invariant planes.  Three are isotropic and exactly two are
000063 | symplectic.  The two symplectic planes are precisely
000064 | 
000065 |   W_q={0,IY,YZ,YX},
000066 |   W_p={0,ZY,XY,YI}.
000067 | 
000068 | Their mutual pairing is zero, so they are orthogonal complements.  Therefore
000069 | Sigma canonically selects the unordered pair {W_q,W_p}, and Phi maps the
000070 | source unordered pair {V_q,V_p} to it.
000071 | 
000072 | On either symplectic plane the centralizer of the order-three action has order
000073 | three.  The planes may also be exchanged, giving 3*3*2=18 symplectic
000074 | intertwiners.  Equivalently, an independent enumeration of Sp(4,2) gives
000075 | centralizer order 18.  The source cubic alone does not choose one of them.
000076 | 
000077 | 4. PRECISE SCOPE
000078 | 
000079 | The code-label calculation with fixed-space dimension two is still correct,
000080 | but it concerns the context module and is not a theta-to-Pauli obstruction.
000081 | The present result proves the first generator and unordered plane-pair match
000082 | on the correct translation modules.  It does not yet show that one of the
000083 | eighteen maps also intertwines s, Q, the selected projective spin lift beta,
000084 | the central theta extension, and the four-dimensional Schrodinger module.
000085 | 
000086 | The comparison with v796 is correctly retained as an explanation of the
000087 | outer context-versus-translation class swap.  The present Phi is instead a
000088 | direct internal symplectic map for this curve and does not import v784.