TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/collar-current-source-bridge-20261021/DIRAC-LIFT-ADMISSIBILITY.md
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000001 | # Full Dirac lift of the relative endpoint family
000002 | 
000003 | Research continuation of `UR.COLLAR.FACE_CURRENT.01`, 21 October 2026.
000004 | Verdict: **conditional structural obstruction to an automatic P1 identification**.
000005 | The scalar current result in PROOF.md is retained. It must not be promoted to
000006 | the current of the entire selfadjoint Dirac module without the following test.
000007 | No general mathematical novelty or completed TFPT solution is claimed.
000008 | 
000009 | ## 1. The original module, not an added mirror model
000010 | 
000011 | Use the already present product Dirac realization, from
000012 | `seam-full-collar-reconstruction-20261015/PROOF.txt:21-61`:
000013 | 
000014 | ```
000015 | A = -i partial_theta + alpha,  alpha = 1/4 or 3/4,
000016 | B = diag(A,-A),   Jn = [[0,-I],[I,0]],   Gamma = diag(I,-I),
000017 | D = Jn (partial_r+B) = [[0,-partial_r+A],[partial_r+A,0]].
000018 | ```
000019 | 
000020 | These are two graded components of one operator, with the original periodic
000021 | frame and the opposite-quarter relabelling retained. They are not two boundary
000022 | faces, two independent vacua, or a new set of four-dimensional particles.
000023 | The scalar transport `L=partial_r+A` is exactly its chiral block `D_plus`.
000024 | But `D_plus` maps the positive graded module to the negative one; it is not
000025 | the entire selfadjoint D on one fixed module.
000026 | 
000027 | For `T=exp(-ell B)`, keep BOTH physical boundary faces and both components:
000028 | 
000029 | ```
000030 | G = [[a,c],[c,1-a]],
000031 | a=(1+exp(-2ell B))^-1, c=(2cosh(ell B))^-1.
000032 | ```
000033 | 
000034 | On the second face the occupation is
000035 | `n(B)=diag(n(A),1-n(A))`, where `n(A)=(1+exp(2ell A))^-1`.
000036 | This is the same original graph constructor, not a fitted covariance.
000037 | Identifying it with a pure charged-CAR state remains an explicit statistical
000038 | premise, as in PROOF.md. This identification is not derived from P1.
000039 | 
000040 | ## 2. The decisive cancellation is analytic, not a cutoff effect
000041 | 
000042 | For a complete orthogonal Dirac Cauchy projection C assume its actual normal
000043 | boundary unitary N satisfies `N C N*=1-C`. For the above finite collar it is
000044 | `N=diag(-Jn,Jn)`, and the identity follows from the actual graph formula.
000045 | It is the Lagrangian Cauchy property, not the geometric deck reflection and
000046 | not a definition of the original admissibility projector.
000047 | 
000048 | For bounded implementable generators X,Y, with `[C,X]` and `[C,Y]` Hilbert
000049 | Schmidt, use the charged-CAR cocycle
000050 | 
000051 | `c_C(X,Y)=Tr(C X (1-C) Y C - C Y (1-C) X C)`.
000052 | 
000053 | All displayed products are trace class. Cyclicity of products of Hilbert
000054 | Schmidt off-diagonal blocks gives `c_(1-C)(X,Y)=-c_C(X,Y)`.
000055 | If X and Y commute with N, unitary covariance also gives
000056 | `c_(1-C)(X,Y)=c_C(X,Y)`. Consequently **`c_C(X,Y)=0`**.
000057 | This is a sufficient condition; it does not classify all zero cocycles.
000058 | 
000059 | Smooth scalar multiplication on one boundary face, acting on the COMPLETE
000060 | Dirac fiber, commutes with N. On the circle its off-diagonal commutators are
000061 | Hilbert Schmidt. Therefore its full-module cocycle vanishes even on ONE face.
000062 | This cancellation is different from the cancellation BETWEEN the two faces
000063 | of a SINGLE scalar block established in PROOF.md.
000064 | 
000065 | In the product example, each component can be checked separately:
000066 | 
000067 | ```
000068 | c_plus(f,g) = sum_(k>0) k (conj(f_k) g_k - f_k conj(g_k)),
000069 | c_minus(f,g) = -c_plus(f,g),  c_vector(f,g)=0.
000070 | ```
000071 | 
000072 | The identity uses `n(-A)=1-n(A)` and the infinite telescoping argument already
000073 | proved for the scalar source. At `f=sin(4theta), g=1-cos(4theta)` these give
000074 | `-2i,+2i,0`. The commuting-profile Weyl rectangle has phase `exp(2ist)` on
000075 | the first chiral component, `exp(-2ist)` on the other, and 1 on the full
000076 | vector module. This does NOT exclude nonzero ordinary correlations, boundary
000077 | responses, mixed vector/axial currents, or global holonomies of other paths.
000078 | No finite matrix is asserted to represent the infinite CCR.
000079 | 
000080 | This lemma is restricted to the stated charged-CAR, Lagrangian, implementable
000081 | class. In higher boundary dimensions smooth multipliers need not have Hilbert
000082 | Schmidt commutators; the unrenormalized trace formula cannot simply be reused.