TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/pfaff_source_review/NORMAL-KINETIC-PAIRING.txt Original file SHA-256: 28061f015df5c5bb2a07a12ffd1d45f84cab49fb84939606201a1efe77b49f62 Inclusive original lines: 37-94 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. 000037 | 2. Exact bilinear skew identity and its boundary term 000038 | Define on this same field-test space 000039 | beta_L(x,y)=integral_0^ell _new dr. 000040 | It is complex-bilinear. The C0 pairing b(x,y)= is symmetric, while 000041 | the B term in beta_L is skew since B*=B and C0 B C0=-B. Integration by 000042 | parts therefore gives exactly 000043 | beta_L(x,y)+beta_L(y,x) 000044 | =-. 000045 | With the EXISTING face order(0,ell) and C_faces=(C0,-C0), this is 000046 | -. 000047 | Reversing the overall Green orientation reverses both boundary signs. 000048 | The crucial relative minus sign between the two faces is unchanged. 000049 | On compactly supported interior fields beta_L is thus skew, for both scalar 000050 | quarter values and without the old condition a^J=-a. 000051 | 000052 | For a closed realization the actual trace domain must make this bilinear 000053 | boundary pairing vanish. An operator version requires the transported 000054 | adjoint-domain relation C0 Dom(L)=Dom(L*) and C0 L C0=-L* there. These 000055 | global boundary conditions were not automatically proved by the interior 000056 | formula. Invertibility/Fredholm data and any zero modes must also be retained 000057 | before taking a determinant or inverse. In particular, specifying the 000058 | Cauchy-solution graph is not by itself an invertible closed-boundary problem. 000059 | 000060 | 3. Charged cross pairing, retaining the same quarter parameter 000061 | Use the already proved whole unitary chart 000062 | T(v,bar(w))=Ev+C0Ew, 000063 | T^-1 B T=h directsum(-bar(h)), h=A gamma_F-i0 gamma_F Phi. 000064 | Since T is constant, 000065 | T^-1 L T=(partial_r+h) directsum(partial_r-bar(h)). 000066 | For x=T(v,bar(w)), y=T(v',bar(w')) this gives the exact expression 000067 | beta_L(x,y) 000068 | =integral[+<(partial_r-h)w',v>] dr. 000069 | After the displayed integration by parts, 000070 | beta_L(x,y) 000071 | =integral[-] dr+[]_0^ell, 000072 | K=partial_r+h. 000073 | Thus on the interior core this is the ordinary charged cross pairing. 000074 | There is no extra physical fermion copy: the second slot is the conjugate 000075 | test space already present in the complete selfdual chart. 000076 | 000077 | In a finite regulator with its matching boundary pairing, its skew matrix 000078 | in particle/conjugate order is 000079 | M=[[0,-K^T],[K,0]]. 000080 | Consequently Pf(M)=+/-det(K), with the sign fixed by the retained Berezin 000081 | orientation. The Grassmann half-quadratic is the charged barpsi K psi action. 000082 | The Pfaffian is not a freely inferred square root of the determinant on a 000083 | different field space. In particular a factor1/2 in a full doubled trace 000084 | cannot be carried into a comparison with the original spectral trace 000085 | without identifying that trace's actual space and real structure. 000086 | 000087 | 4. Same incoming/outgoing full graph, not just face occupations 000088 | D x=0 and L x=0 have exactly the same Cauchy solutions because -N is 000089 | invertible. For the existing normal-independent collar, put R=exp(-ell h). 000090 | The charged trace relation is v1=R v0. Its orthogonal graph projector is 000091 | G(h)=[[a,c],[c,1-a]], 000092 | a=(1+exp(-2ell h))^-1, c=(2cosh(ell h))^-1. 000093 | The full oriented chart is the existing 000094 | T_faces(v0,v1,bar(w0),bar(w1))