TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/pfaff_source_review/SELFDUAL-KINETIC-PAIRING.txt Original file SHA-256: 5543c7cf0360985377e388cbd80c8e2eb3faa554f5c6e7e1f50bb1fae88bb065 Inclusive original lines: 1-117 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 | THE SAME COLLAR HAS A CANONICAL SELFDUAL KINETIC PAIRING 000002 | 29 October 2026. Constructive continuation of the fixed-quarter candidate review. 000003 | 000004 | 1. Exact question and retained data 000005 | The old pairing _old fails the scalar-quarter skew test. 000006 | This does not imply that the GIVEN canonical Clifford source lacks a kinetic 000007 | pairing. Test that question on the same full field space, with no new copy, 000008 | no new charge J and no fitted kernel. Retain the original N, Gamma, B, 000009 | D=N(partial_r+B), Ical=i_old Gamma and C0=i_old N from 000010 | ../graded_boundary_functor/{PROOF,FINITE-COUPLED}.txt. 000011 | The fixed smooth even connection and bounded smooth odd Phi may be retained. 000012 | B is selfadjoint, new-complex-linear, and C0 B C0=-B. C0 is new antiunitary. 000013 | The whole bulk D is new-antilinear. Assertions about differential pairings 000014 | first use smooth fields compactly supported in the open product collar. 000015 | 000016 | 2. Remove the original normal symbol, rather than replace the field 000017 | Because N^2=-I, the already present operator determines 000018 | 000019 | L=-N D=partial_r+B. 000020 | 000021 | L is new-complex-linear, and its homogeneous solutions are exactly those 000022 | of D. In the SAME T chart, with Lplus=partial_r+h, one obtains 000023 | 000024 | T^-1 L T=Lplus directsum (-bar(Lplus^*)), 000025 | Lplus^*=-partial_r+h, 000026 | 000027 | where * is the formal interior adjoint, not an invented closed boundary 000028 | domain. Equivalently the conjugate block is partial_r-bar(h). 000029 | In contrast C0D gives diag(-i_old Lplus,bar[-i_old(partial_r-h)]). 000030 | Multiplication by Ical is not multiplication by i_old on both T slots. 000031 | Confusing those two phases gives the wrong pairing sign. Two entire old 000032 | Dirac copies would double the full field space and are unnecessary. 000033 | 000034 | 3. A genuine skew form in the already declared NEW complex type 000035 | On the interior core, L^*=-partial_r+B and 000036 | 000037 | C0 L C0=partial_r-B=-L^*. 000038 | 000039 | Thus beta_L(x,y)=_new is complex bilinear and antisymmetric. 000040 | This uses C0 B C0=-B, which holds at the scalar quarter as well. It does 000041 | not import the failed premise a^J=-a for the DIFFERENT old Jloc pairing. 000042 | For fields with endpoints the exact remaining boundary expression is 000043 | 000044 | beta_L(x,y)+beta_L(y,x)=[_new]_(r=0)^(r=ell). 000045 | 000046 | A closed kinetic domain must therefore kill this expression; open boundary 000047 | fields retain it and need the original oriented endpoint prescription. 000048 | The interior result alone is not a global invertible fermion determinant. 000049 | On the same product core, L^*L=D^2. Thus this kinetic bridge retains the 000050 | full squared Dirac spectrum; it does not evade the original log-profile 000051 | comparison or identify an untransported eta/determinant phase. 000052 | 000053 | 4. Same field count and the charged action 000054 | Let chi denote the independent conjugate test slot in T, not another 000055 | charged species. In the charged coordinates the formal skew matrix is 000056 | 000057 | A_L=[[0,-Lplus^t],[Lplus,0]]. 000058 | 000059 | Here t is the bilinear transpose including integration by parts. For 000060 | Grassmann variables ordered as (psi,chi), the half quadratic form becomes 000061 | chi^t Lplus psi, with its endpoint expression retained as above. On an 000062 | invertible finite compatible regulator, 000063 | 000064 | Pf(A_L)=(-1)^(n(n+1)/2) det(Lplus). 000065 | 000066 | The sign is the fixed ordered orientation, not a freely selected physical 000067 | phase. There is no new square-root reduction of the charged determinant. 000068 | After the original domain, orientation and source coupling are supplied, 000069 | the existing exact Berezin/Schur formula may therefore be reused with this 000070 | Lplus. The calculation does not determine those original physical inputs. 000071 | ROOT-SELFDUAL-KINETIC-CHECK.json independently checks the transpose relation 000072 | and Pfaffian formula for a genuinely complex Hermitian h with both scalar 000073 | quarters. Its finite derivative is an algebra control, not a boundary proof. 000074 | 000075 | 5. Entire two-face graph and source response 000076 | For normal-independent B, Lx=0 gives x(ell)=exp(-ell B)x(0). 000077 | Its closed Cauchy graph is EXACTLY the existing graph for D, including 000078 | all crossing blocks. Its orthogonal projector is the already established 000079 | 000080 | G(B)=[[a(B),c(B)],[c(B),1-a(B)]], 000081 | a(b)=(1+exp(-2ell b))^-1, c(b)=1/(2cosh(ell b)). 000082 | 000083 | With C_faces=C0 directsum(-C0) and the existing T_faces chart this becomes 000084 | G(h) directsum(1-bar(G(h))). The negative sign on the outgoing conjugate 000085 | slot is essential. No equality of two independent scalar quarter domains 000086 | is used. The given positive pure-projector CAR theorem then fixes all 000087 | correlations if this is the ACTUAL Clifford quantum source. The given 000088 | circle action and its GNS transport are those already proved for B/h. 000089 | This is not an equation G=L^-1. The latter inverse only exists after an 000090 | actual global domain is supplied; its source contractions and unnormalized 000091 | amplitude have their previously audited additional data. 000092 | 000093 | 6. What is now gained, and what is not identified 000094 | The full same-D source has an explicit kinetic bridge to its charged 000095 | first-order block, as well as the existing complete two-face graph. It is 000096 | unnecessary to guess a new paired Dirac field or freely add a conjugation 000097 | inside this already GIVEN Clifford construction. This removes a type and 000098 | kinetic ambiguity for that route. It is not a new chosen auxiliary model. 000099 | The new beta_L is NOT the unchanged old beta_Jloc,D. It changes neither 000100 | the old exclusion nor the real Dirac spectral profile by a naming choice. 000101 | Selecting this quantum functor as the physical original P1 implementation, 000102 | identifying its actual domain/reference/determinant line and bosonic weight, 000103 | and deriving physical OS time are still the precise original comparisons. 000104 | The complete operational Seed already includes state/time; no extra free 000105 | measure is demanded for it. No E8/SM or full physical promotion follows here. 000106 | 000107 | Sources checked 000108 | - graded_boundary_functor/PROOF.txt:20-140, especially the entire T_faces map. 000109 | - graded_boundary_functor/FINITE-COUPLED.txt:14-100, actual h and total grade. 000110 | - graded_boundary_functor/charge_transport/DERIVED-PAIRING.txt:17-125. 000111 | - seam-collar-multifield-20261016/PROOF.txt:271-326, existing exact integration. 000112 | - Ludewig/Roos, Ann. Henri Poincare21(2020)1191-1233, 000113 | https://doi.org/10.1007/s00023-020-00893-6, section3.3 equations36-38. 000114 | Their exact re-complexification/normal real structure is used as already 000115 | stated in the existing proof. Section4.1 distinguishes anomaly twist from 000116 | a physical free-fermion natural transformation. This note's local L/skew 000117 | calculation is explicit above and is not attributed as a new theorem there.