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: 1-143 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. 000001 | CONSTRUCTIVE REPAIR: THE EXISTING NORMAL KINETIC OPERATOR 000002 | 29 October 2026. Independent analytical review; no new field, J choice, 000003 | source measure, test suite, original edit or Atlas mutation. 000004 | 000005 | Result 000006 | L=-N D=partial_r+B gives a correct new-complex skew kinetic pairing with 000007 | the EXISTING C0. It works at scalar quarter holonomy too. It retains the 000008 | same whole Cauchy graph and gives the canonical charged cross action. 000009 | This repairs a mathematical Pfaffian PRESENTATION inside the given Clifford 000010 | quantization. It does not prove that the original P1 source selects that 000011 | quantization or identify its unnormalized determinant with Z_adm. 000012 | 000013 | 1. Fixed data and exact type 000014 | Use the entire existing coupled chart, not a right-channel reduction: 000015 | D=N(partial_r+B), N^2=-I, 000016 | B=Z A-S Phi, A=-i0 partial_theta+alpha, 000017 | Gamma=Z tensor gamma_F, Ical=i0 Gamma, C0=i0 N, 000018 | C0^2=I, C0 Ical=-Ical C0, C0 B C0=-B. 000019 | The background Phi is the given bounded smooth selfadjoint odd coefficient, 000020 | normal-independent here. The actual scalar alpha=1/4 or3/4 is unchanged. 000021 | Inner products below are NEW-complex Hermitian products, linear in the 000022 | second argument. C0 is antiunitary for this structure, not the old Jloc. 000023 | 000024 | N and D are new-complex-antilinear, whereas 000025 | L=-N D=partial_r+B 000026 | is new-complex-linear. On the common interior smooth core, 000027 | L*= -partial_r+B, 000028 | C0 L C0=partial_r-B=-L*. 000029 | The multiplication by -N is fixed by the existing normal symbol; it is not 000030 | a fitted identification, an extra conjugation or Ical C0D. 000031 | 000032 | Sources: graded_boundary_functor/{PROOF.txt,FINITE-COUPLED.txt}; 000033 | charge_transport/DERIVED-PAIRING.txt:17-30,83-95. The earlier failed scalar 000034 | quarter pairing _old is a DIFFERENT form. Its counterexample 000035 | remains correct and does not apply to the new pairing below. 000036 | 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)) 000095 | =(Ev0+C0Ew0, Ev1-C0Ew1). 000096 | The conjugate solutions give w1=-R^-1 w0, so their graph is precisely the 000097 | orthogonal complement of the conjugate charged graph. Therefore 000098 | T_faces^-1 G(B) T_faces=G(h) directsum(I-bar(G(h))). 000099 | Both crossing blocks c and both labeled face covariances are retained. 000100 | The alpha dependence is present in h and R. Opposite outward-normal signs 000101 | do not require choosing a second independent quarter source. 000102 | 000103 | For unbounded h the exponentials are used on their actual graph domains 000104 | and the bounded projector is defined by spectral calculus, as in the old 000105 | proof. This is not an assertion of a well-posed arbitrary L2 Cauchy ODE. 000106 | The existing normal-dependent extension replaces R^-1 by(R*)^-1 and uses 000107 | the actual elliptic Cauchy space or a finite regulator; it need not be 000108 | re-derived here. See normal_dependent_transport/PROOF.txt:sections2-3. 000109 | 000110 | 5. What is repaired, what is reused, and the exact remaining identification 000111 | Given that the full graph is the quantum CAR covariance, the existing 000112 | positive-projector theorem fixes every CAR word. The new L pairing is a 000113 | correct kinetic presentation of that same conditional charged/Clifford source. 000114 | It repairs the local skew typing of the candidate at fixed scalar alpha. 000115 | It neither revives _old as skew nor identifies C0 with physical J_F. 000116 | The whole existing HS reflection and same-parameter circle action remain 000117 | the results of GRADED-HS-REFLECTION-AND-TIME.txt, with beta=2ell. 000118 | 000119 | The Atlas already contains the relevant stronger sewing statement: 000120 | finding:collar-car-projection-multifield-ray, 000121 | finding:original-operational-seed-and-given-green-pairing. 000122 | Their actual source seam-collar-multifield-20261016/PROOF.txt:34-128,449-480 000123 | was reread. It fixes the oriented multi-field operator RAY as direct sum 000124 | of exterior powers of R and explicitly does not recover its overall complex 000125 | scalar, vacuum determinant or unnormalized partition function. This is 000126 | reused, not announced as a new completion of all source normalization. 000127 | 000128 | A chosen fixed-domain Gaussian integral can supply a determinant scalar; 000129 | showing it equals the ORIGINAL relative source requires its same reference, 000130 | zero-mode convention, determinant-line orientation/eta phase and actual 000131 | bosonic background weight. The normalized graph and its Wick/Pfaffian 000132 | moments alone do not prove that equality. For a complete given Seed these 000133 | objects are to be compared with its existing state and source, not selected 000134 | again. The original physical use of the Clifford functor remains exactly 000135 | the explicit premise of graded_boundary_functor/PROOF.txt:99-104. 000136 | 000137 | Finally r evolution of this first-order kinetic problem, its charged circle 000138 | automorphism, the HS Liouvillian and a physical OS transfer are distinct 000139 | previously typed actions. This corrected pairing supplies no new physical 000140 | time identification. The constructive next comparison is the actual 000141 | fixed-domain charged K functional with the original source insertions and 000142 | reference, with the full T_faces graph retained. No new source platform is 000143 | needed to perform that comparison.