TFPT scientific source excerpt Original repository path: experiments/theory-contracts/seam-full-collar-reconstruction-20261015/PROOF.txt Original file SHA-256: 9f4effed34316798c6a35e1ac0ec2832cd3720180f476cb8747bb32359b56e6a Inclusive original lines: 21-61 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000021 | 2. The geometric reflection is not the Calderon polarization 000022 | 000023 | Let C be the ORTHOGONAL projection onto the Cauchy data of a symmetric Dirac operator, 000024 | and let J_n be its normal Clifford action, J_n*=-J_n and J_n^2=-I. Under the usual 000025 | Dirac hypotheses the Cauchy space is Lagrangian for . Consequently 000026 | J_n C J_n*=I-C. For F=2C-I this gives J_n F=-F J_n. 000027 | The algebraic operator R_alg=J_n F is a selfadjoint involution, satisfies 000028 | R_alg F R_alg=-F and R_alg F=J_n. This is a reconstructed, generally nonlocal 000029 | operator; it is not automatically the local geometric lift of a deck reflection. 000030 | If R and F are anticommuting involutions then (RF)^2=-I, whereas R=F would give 000031 | (RF)^2=I. The algebraic quarter turn is not a derivation of charged holonomy i. 000032 | 000033 | An explicit Dirac calculation: on a circle let A_alpha=-i partial_theta+alpha, 000034 | alpha=1/4, in the periodic frame. On TWO GRADED COMPONENTS OF THE SAME DOMAIN put 000035 | B=diag(A_alpha,-A_alpha), J_n=[[0,-I],[I,0]], and R=[[0,I],[I,0]]. Then 000036 | J_n^2=-I, R^2=I, R J_n R=-J_n, R B R=-B. The actual differential operator 000037 | D=J_n(partial_r+B) commutes with the geometric reflection psi(r)->R psi(-r). 000038 | On the half-infinite cylinder C_half=1_(0,infinity)(B), and 000039 | R C_half R=I-C_half, not C_half. Thus its Cauchy range is not the +1 eigenspace of R. 000040 | 000041 | There is also a compact version; the conclusion is not an artifact of an infinite 000042 | collar. Double [0,ell]xS^1 to the r-circle of length 2ell with constant D as above. 000043 | The involution r->-r fixes r=0 and r=ell, and has the local lift R. The Cauchy 000044 | projection of the original finite cylinder is G_ell in section 3. On its two 000045 | boundary components the deck lift is diag(R,R). On any nonzero B mode it conjugates 000046 | G_ell=[[A,Cross],[Cross,I-A]] to [[I-A,Cross],[Cross,A]], which differs from G_ell. 000047 | In particular it cannot equal 2G_ell-I, an operator that commutes with G_ell. 000048 | Do not confuse this deck operation, which fixes each boundary, with reflection 000049 | WITHIN the original cylinder: (u,v)->(Rv,Ru); the latter preserves graph(exp(-ell B)). 000050 | 000051 | More generally, a lift over the fixed boundary is pointwise in the bundle, while 000052 | the principal symbol of 2C-I depends on tangential cotangent direction for an 000053 | ordinary Dirac boundary. These cannot be identified just from reflection regularity. 000054 | If R_tau is instead DEFINED as 2C-I, the algebraic statement is true by definition, 000055 | but its identification with a geometric reflection has become an extra hypothesis. 000056 | This corrects the stated geometric inference; it is not a no-go for all TFPT seams. 000057 | 000058 | For a generic compact interior, C need not equal the positive spectral projector 000059 | of B. The half-cylinder equality used above is exact in its specified realization. 000060 | The two-face finite-cylinder calculation below uses the actual solutions, not that 000061 | generic replacement. Orthogonalizing a Calderon projection is also explicit here.