TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/graded_boundary_functor/PROOF.txt Original file SHA-256: 708dad21942cc00dd56d3348eacee3f0d90ccfc53e3f4987e39ded00ddc70841 Inclusive original lines: 1-149 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 | GRADED BOUNDARY CLIFFORD TYPING AND THE EXISTING SCALAR SOURCE 000002 | 29 October 2026 000003 | 000004 | Result and exact scope 000005 | There is a constructive alternative to adding a charge conjugation Jgamma_n 000006 | on the original complex Hilbert space. Use the standard geometric Clifford 000007 | boundary construction with its changed complex structure. In the existing 000008 | free product collar this identifies the ENTIRE oriented normal/two-face 000009 | field space and complete geometric selfdual projector with the old scalar 000010 | charged two-face CAR data. Under the physical covariance premise in section3 000011 | it also transports the full source state. The map retains the circle 000012 | evolution and local smearing chart. It does not 000013 | postselect a right normal component or add another chiral determinant. 000014 | This is a source-typing construction inside the canonical Clifford functor. 000015 | It does not prove that the thin P1 tuple chooses this physical quantization, 000016 | the P2 coefficient multiplicity, the interacting bosonic state, or OS time. 000017 | It is not equivalent to the earlier full COMPLEX CAR quantization of both 000018 | normal components as independent charged species; its real structure differs. 000019 | 000020 | 1. Original collar data and the primary construction being applied 000021 | Let V=L2(S1,C^d), A=-i partial_theta+alpha on periodic H1, with the old complex 000022 | unit i0. Use the already studied free product data, with no new matrix fit: 000023 | 000024 | H0=V directsum V, Gamma=diag(1,-1), 000025 | N=gamma_n=[[0,-1],[1,0]], B=Gamma A, 000026 | D=N(partial_r+B). 000027 | 000028 | N^2=-I, N^*=-N, Gamma^2=I, and NGamma=-Gamma N. 000029 | The standard geometric construction changes the complex structure on the 000030 | underlying REAL Hilbert space and uses normal multiplication for conjugation: 000031 | 000032 | Ical=i0 Gamma, C0=i0 N. 000033 | 000034 | Here C0 is OLD-complex-linear. It is NEW-complex-antilinear, because 000035 | C0 Ical=-Ical C0. Confusing these two complex structures would invalidate 000036 | the construction. Use the Hermitian metric whose real part is Re h0 and 000037 | whose imaginary part is Re h0(Ical u,v). Then C0 is antiunitary, C0^2=I, 000038 | and B is selfadjoint and complex-linear for Ical, with C0 B C0=-B. 000039 | These are full H1 operator identities; the matrices are constant and do not 000040 | change the domain. No assumption on the square of the original charge J is 000041 | used to prove them. They do not identify C0 with that original J. 000042 | 000043 | Primary reference: Ludewig and Roos, The Chiral Anomaly of the Free Fermion 000044 | in Functorial Field Theory, Ann. Henri Poincare21 (2020),1191-1233, 000045 | DOI10.1007/s00023-020-00893-6, section3.3, equations36-38 and Theorem3.12. 000046 | Their J is the re-complexifying grading operator, not the TFPT charge J. 000047 | The reference constructs the anomaly twist and coherent Clifford bimodule 000048 | gluing; it is not cited as a proof of a complete interacting physical functor. 000049 | https://link.springer.com/article/10.1007/s00023-020-00893-6 000050 | 000051 | 2. The real field space and the charged chart are explicit 000052 | The fixed real form is 000053 | 000054 | W_R=Fix(C0)={(v,i0 v)/sqrt(2):v in V}. 000055 | 000056 | The displayed map is a real isometry from the realification of V. Thus its 000057 | real Clifford algebra has the usual charged CAR(V) complexification. More 000058 | explicitly, regarding the left normal copy as a conjugate complex space, 000059 | 000060 | T(v,bar(w))=(v,i0 w), T:V directsum bar(V) -> (H0,Ical) 000061 | 000062 | is unitary and new-complex-linear, and transports the canonical antiunitary 000063 | flip (v,bar(w))->(w,bar(v)) to C0. It transports B to 000064 | 000065 | A directsum (-bar(A)). 000066 | 000067 | Both normal entries are retained as field and adjoint test slots. They are 000068 | not two independent charged fermion copies in this functor. Conversely, 000069 | quantizing CAR(H0) using the OLD complex structure is a different statistical 000070 | assignment. The old full-complex-vector cancellation theorem retains exactly 000071 | that premise and is not contradicted by the new typing. 000072 | This map does not identify a finite spinor STATE module Lambda_even C5 with 000073 | elementary field labels. Its odd Clifford maps still require the full parity 000074 | module and change parity. No physical species count is inferred from it. 000075 | 000076 | 3. The entire two-face covariance, including its crossing blocks 000077 | Let ell>0. On the two labeled faces the already given graph is 000078 | 000079 | G(B)=[[a(B),c(B)],[c(B),I-a(B)]], 000080 | a(x)=(1+exp(-2ell x))^-1, c(x)=(2cosh(ell x))^-1. 000081 | 000082 | Opposite outward normals give the real structure 000083 | C_faces=C0 directsum(-C0), not C0 directsum C0. Therefore 000084 | C_faces G(B) C_faces=I-G(B), on both complete boundary Hilbert spaces. 000085 | The charged chart, with particles first and holes second, is 000086 | 000087 | T_faces(v0,v1,bar(w0),bar(w1))=(v0,i0 w0,v1,-i0 w1). 000088 | 000089 | It is constant, preserves supports and H1 domains, and transports G(B) to 000090 | 000091 | G(A) directsum (I-bar(G(A))). (1) 000092 | 000093 | Indeed on right normal slots the graph is G(A). On left normal slots it is 000094 | G(-A), while the opposite factors +i0,-i0 turn its off-diagonal +c(A) into 000095 | -c(A); this is exactly I-bar(G(A)) on the conjugate slots. Equation(1) is 000096 | the selfdual presentation of the OLD full scalar charged CAR covariance, 000097 | not just equality of individual face occupations. The second labeled face 000098 | retains n(A)=(1+exp(2ell A))^-1; it has not been changed into a vacuum. 000099 | If the Clifford graph is used as the actual quantum covariance, its pure 000100 | projector state is uniquely determined by the old positive-projector theorem. 000101 | Equation(1) then preserves every CAR word and every defined composite on the 000102 | same supports, without another quasifree-state or independent-face premise. 000103 | This last physical use of the functor is an explicit premise, not a theorem 000104 | deduced merely by naming the classical Calderon projector. 000105 | 000106 | 4. Circle time, gauge action and geometric reflection 000107 | The new-complex one-particle unitary generator obeys the exact identity 000108 | 000109 | Ical B = i0 A I_normal. (2) 000110 | 000111 | Thus exp(t Ical B) is precisely the old scalar circle evolution exp(i0 t A) 000112 | on both normal slots. It commutes with C0. The canonical CAR map therefore 000113 | preserves the original scalar circle action; it does not use exp(i0 t B) 000114 | as if it were the same group. That latter group generally fails to preserve 000115 | this real form. Gauge maps exp(i0 Q f(theta)) with internal Hermitian Q and 000116 | real scalar f commute with C0 and Ical and have their old charged action on 000117 | the particle chart. They need not leave the fixed background covariance 000118 | invariant; gauge covariance transports the background as usual. 000119 | For the explicitly given source state the GNS implementation of the preserved 000120 | circle group, fixed on its cyclic vector, transports with the CAR map. This 000121 | does not equate tangential circle time with a positive radial OS Hamiltonian. 000122 | 000123 | In the free collar, put S=NGamma and R=face_swap tensor S. One has R^2=I, 000124 | R Ical R=-Ical, R C_faces R=C_faces, and R G(B) R=G(B). 000125 | It is the covariant geometric reflected field map in this collar chart. 000126 | Its compatibility with the particular original domains/reference and its 000127 | identification with physical OS reflection still need that actual map. 000128 | No reflection-positive interacting reconstruction is claimed from a symbol 000129 | calculation. The unbounded exp(-ell B) is not called a Hilbert contraction. 000130 | 000131 | 5. What this changes in the origin search 000132 | Within this standard graded Clifford boundary construction, an additional 000133 | freely chosen charge conjugation or separate right-fermion source is not 000134 | needed to obtain the old scalar CAR source: the above chart constructs it 000135 | from Gamma,N and the complete graph. This is a more concrete sufficient route 000136 | than arbitrary field/state matching, using the original product operator. 000137 | The remaining question for this route is whether the original TFPT source 000138 | implements this functor with its full charge, marks and interactions. The 000139 | complex retyping is not silently declared equal to the old independent full 000140 | complex CAR assignment, and the finite Yukawa extension is not assumed from 000141 | the free calculation. Neither original 3+2 selection nor E8 charged completion 000142 | nor the physical whole-theory endpoint follows from equations(1)-(2) alone. 000143 | 000144 | Verification 000145 | The analytic domain/spectral-calculus argument above is supplemented by the 000146 | exact normal and two-face matrices in check_typing.py. No finite Fourier 000147 | window or fitted state is used to establish(1)-(2). It is a concrete 000148 | application of standard fermionic boundary quantization, not a newly proved 000149 | general quantization theorem.