TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/pfaff_source_review/CLIFFORD-KINETIC-INDEPENDENT.txt Original file SHA-256: 59d098be379a8a21003e1d703862abceb137308566f68c66bb97ffb0b4f0ed1a Inclusive original lines: 1-100 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 CLIFFORD COLLAR HAS A COMPLEX-LINEAR SKEW KINETIC FORM 000002 | Independent analytic followthrough, 29 October 2026 000003 | 000004 | VERDICT: the proposed root signs and finite Pfaffian identity are correct. 000005 | This supplies a concrete kinetic pairing within the GIVEN re-complexified 000006 | Clifford source. It does not equate that pairing with the old Jloc/D form 000007 | or select its original physical integration, determinant reference or phase. 000008 | 000009 | 1. Same operators, new explicitly typed kinetic expression 000010 | 000011 | Use exactly FINITE-COUPLED.txt and charge_transport/DERIVED-PAIRING.txt: 000012 | N^2=-I, D=N(partial_r+B), Ical=i_old Gamma_tot, C0=i_old N, 000013 | C0^2=I, C0 Ical=-Ical C0, C0 B C0=-B. 000014 | B is selfadjoint and complex-linear in the new structure. C0 is antiunitary 000015 | there; N and D are new-antilinear. On the interior smooth common core put 000016 | 000017 | L=-N D=partial_r+B. 000018 | 000019 | Thus L is new-complex-linear. Multiplication by the fixed invertible N 000020 | does not change its solutions or their Cauchy boundary data. With the NEW 000021 | Hermitian metric and formal partial_r^*=-partial_r, 000022 | 000023 | C0 L C0=partial_r-B=-L^*. 000024 | 000025 | Consequently beta_new(x,y)=_new is complex-bilinear and skew. 000026 | Here the scalar product is linear in its second argument. This is a 000027 | formal/core identity; integration by parts on a finite collar retains 000028 | boundary terms unless the chosen source domain cancels them. For a closed 000029 | operator version one needs an Ical-stable domain and C0 Dom(L)=Dom(L^*). 000030 | Those are concrete compatibility conditions, not consequences of kernel 000031 | equality. A finite regulator must preserve the same adjoint conditions. 000032 | 000033 | For the given normal-independent product collar, direct multiplication 000034 | also gives L^*L=D^2 on the common interior core. 000035 | This is an operator identity, not a choice of logarithmic trace profile. 000036 | No normal-dependent graph formula is inferred here. 000037 | 000038 | 2. Full charged chart and the exact skew matrix 000039 | 000040 | In the existing unitary chart T(v,bar(w))=Ev+C0 Ew, 000041 | B=diag(h,-bar(h)), C0(v,bar(w))=(w,bar(v)). 000042 | Set L+=partial_r+h. Since L+^*=-partial_r+h on the same core, 000043 | 000044 | T^-1 L T=diag(L+,-bar(L+^*)). 000045 | 000046 | For x=T(v,bar(w)), y=T(v',bar(w')), the exact pairing is 000047 | 000048 | beta_new(x,y)=-. 000049 | 000050 | In ordered coordinates(v,bar(w)), its complex-bilinear matrix is 000051 | 000052 | A_new=[[0,-L+^t],[L+,0]]. 000053 | 000054 | The transpose includes the actual normal derivative and its adjoint domain; 000055 | it is not merely the transpose of an internal h matrix. For an N-dimensional 000056 | invertible finite regulator and ordered basis(v1,...,vN,bar(w1),...,bar(wN)), 000057 | 000058 | Pf(A_new)=(-1)^(N(N+1)/2) det(L+). 000059 | 000060 | For fixed orientation its sign cancels in a relative ratio. Thus the FULL 000061 | skew form exactly gives the charged complex determinant, rather than an 000062 | additional square root of that determinant. The original coefficient and 000063 | antiparticle labels remain present; no new physical half-species or Majorana 000064 | reduction has been inferred from the conjugate test slots. 000065 | 000066 | 3. What this closes and what it leaves to the actual source prescription 000067 | 000068 | This removes the statement that the given Clifford source has no concrete 000069 | compatible complex-linear Euclidean kinetic form. It has L with the fixed 000070 | geometric C0 pairing above. The previous old-complex pairing 000071 | beta_D=_old 000072 | uses a different conjugation, metric type and operator. It is not rewritten 000073 | as beta_new by this calculation. In particular the present skewness does 000074 | not require the old Jloc charge-odd-connection hypothesis; its hypotheses 000075 | are instead precisely the already supplied geometric Clifford identities. 000076 | 000077 | The same solution graph is still an orthogonal Cauchy-data projector G. 000078 | It is not an inverse L Green operator on an unchanged invertible field 000079 | space. An actual source boundary condition and source coupling must specify 000080 | the Green contraction. A regulator/integration with that data then supplies 000081 | the determinant, as in the existing solved Schur source construction. 000082 | For example an actual trace closure is additional boundary data; it cannot 000083 | be inferred from the open-collar homogeneous solution graph alone. 000084 | 000085 | Even when the normalized graph state is fixed, multiplying its generating 000086 | functional by a source-independent nonzero scalar changes no normalized 000087 | CAR correlations. The original unnormalized determinant/reference and its 000088 | orientation or renormalization are therefore not selected by G. If these 000089 | are supplied by the actual existing source prescription, they are reused; 000090 | they need not be invented again. They simply do not follow from the kernel 000091 | identity L=-ND or the algebraic Pfaffian factorization above. 000092 | 000093 | Finally this is not a construction of a determinant-line anomaly functor 000094 | on arbitrary bordisms, nor a comparison with the primary paper's physical 000095 | J/Pfaffian line. It does not fix the primitive profile or choose the relative 000096 | proper-time measure discussed in PFAFF-INDEPENDENT-REVIEW.txt. Its positive 000097 | outcome is the exact common kinetic operator and charged determinant INSIDE 000098 | the specified Clifford source, with all conjugate slots retained. 000099 | 000100 | No new source class, Atlas mutation, test run or pinned-source change.