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Original repository path: experiments/theory-contracts/seam-joint-conjugation-20261015/PROOF.txt
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000039 | 2. THE SCALAR-PAIRING TYPE CORRECTION
000040 | 
000041 | Let E=C^5 with the stated Hermitian metric and oriented orthonormal wedge basis.
000042 | Write S=Lambda E=S+ direct-sum S-, epsilon_i for creation and iota_i for its
000043 | adjoint. The original v109 identity correctly concerns COMPLEX-BILINEAR forms:
000044 | S+ tensor S- contains one singlet, while S+ tensor S+ contains none.
000045 | 
000046 | The actual Hermitian form is of type conjugate(S+) tensor S+. It is invariant
000047 | on S+ itself. Thus epsilon=I is already a sheet-even involution with the
000048 | invariant scalar form <f,epsilon g>=<f,g>. This contradicts the claimed general
000049 | Hermitian implication 'scalar iff sheet-odd'. It is not asserted to be the
000050 | physical Calderon operator. The contradiction concerns the advertised finite
000051 | selection criterion, not a full P1 countermodel or all RP realizations.
000052 | 
000053 | More generally, the complex-linear Spin(10) commutant on S+ direct-sum S- is
000054 | C I_+ direct-sum C I_- (inequivalent irreducibles). An invariant Hermitian form
000055 | <f,epsilon g> requires epsilon in that commutant. A nonzero linear sheet-odd
000056 | epsilon cannot satisfy this untwisted covariance. A geometrical or outer-
000057 | automorphism-twisted reflection has different hypotheses and is not excluded.
000058 | 
000059 | v110 :85-101 substitutes 1+1==2 and 0+0==0 for a kernel selection proof.
000060 | Its explicit block matrices and weight counts do not test that inference.
000061 | Architecture :1548-1560 labels it exact; that interpretation cannot be reused
000062 | as a theorem about the Hermitian/RP kernel. The Atlas already records R3 as
000063 | assumed, but its overview still repeats the stronger ledger claim.
000064 | 
000065 | The correct charge-conjugation operator is the PREVIOUSLY constructed
000066 | J_C=B K, B=product_(i=1..5)(epsilon_i+iota_i), K coefficient conjugation.
000067 | It is in fact Spin(10)-equivariant, extending the previous SU5 statement.
000068 | For Hermitian h, rho(h)=dGamma(h)-tr(h)I/2 satisfies J_C rho(h) J_C^-1=-rho(h).
000069 | For i<j let A=epsilon_i epsilon_j, A*=iota_j iota_i. Then
000070 | J_C A J_C^-1=-A*, J_C A* J_C^-1=-A, and both Hermitian generators
000071 | A+A* and i(A-A*) change sign. Antiunitarity therefore makes J_C commute with
000072 | all compact Spin(10) group elements. These 25+20 directions span its Lie algebra.
000073 | 
000074 | The reversion signs are already in B. For I subset {1,...,5},
000075 | J_C u_I=(-1)^(sum_(i in I)(5-i)) u_(I complement).
000076 | The correct bilinear scalar is beta(s,t)=<J_C s,t>
000077 | =[rev(s) wedge t]_top, rev|Lambda^k=(-1)^(k(k-1)/2).
000078 | This pairs opposite chiralities and does not identify J_C with a linear deck.