TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-full-collar-reconstruction-20261015/PROOF.txt
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000015 | Lines 65-80 already supply a boundary tuple including D_+ and B_Sigma and an exact
000016 | product collar D_+=gamma_n(partial_n+B_Sigma). Lines 130-136 and 154-166 assert that
000017 | the geometric deck reflection R_tau is F=2P_C-I. Lines 480-520 supply an operational
000018 | seed containing the algebra, time, reflection, state, winding and collar operator.
000019 | The present results do not remove those supplied inputs by renaming them outputs.
000020 | 
000021 | 2. The geometric reflection is not the Calderon polarization
000022 | 
000023 | Let C be the ORTHOGONAL projection onto the Cauchy data of a symmetric Dirac operator,
000024 | and let J_n be its normal Clifford action, J_n*=-J_n and J_n^2=-I. Under the usual
000025 | Dirac hypotheses the Cauchy space is Lagrangian for <J_n u,v>. Consequently
000026 | J_n C J_n*=I-C. For F=2C-I this gives J_n F=-F J_n.
000027 | The algebraic operator R_alg=J_n F is a selfadjoint involution, satisfies
000028 | R_alg F R_alg=-F and R_alg F=J_n. This is a reconstructed, generally nonlocal
000029 | operator; it is not automatically the local geometric lift of a deck reflection.
000030 | If R and F are anticommuting involutions then (RF)^2=-I, whereas R=F would give
000031 | (RF)^2=I. The algebraic quarter turn is not a derivation of charged holonomy i.
000032 | 
000033 | An explicit Dirac calculation: on a circle let A_alpha=-i partial_theta+alpha,
000034 | alpha=1/4, in the periodic frame. On TWO GRADED COMPONENTS OF THE SAME DOMAIN put
000035 | B=diag(A_alpha,-A_alpha), J_n=[[0,-I],[I,0]], and R=[[0,I],[I,0]]. Then
000036 | J_n^2=-I, R^2=I, R J_n R=-J_n, R B R=-B. The actual differential operator
000037 | D=J_n(partial_r+B) commutes with the geometric reflection psi(r)->R psi(-r).
000038 | On the half-infinite cylinder C_half=1_(0,infinity)(B), and
000039 | R C_half R=I-C_half, not C_half. Thus its Cauchy range is not the +1 eigenspace of R.
000040 | 
000041 | There is also a compact version; the conclusion is not an artifact of an infinite
000042 | collar. Double [0,ell]xS^1 to the r-circle of length 2ell with constant D as above.
000043 | The involution r->-r fixes r=0 and r=ell, and has the local lift R. The Cauchy
000044 | projection of the original finite cylinder is G_ell in section 3. On its two
000045 | boundary components the deck lift is diag(R,R). On any nonzero B mode it conjugates
000046 | G_ell=[[A,Cross],[Cross,I-A]] to [[I-A,Cross],[Cross,A]], which differs from G_ell.
000047 | In particular it cannot equal 2G_ell-I, an operator that commutes with G_ell.
000048 | Do not confuse this deck operation, which fixes each boundary, with reflection
000049 | WITHIN the original cylinder: (u,v)->(Rv,Ru); the latter preserves graph(exp(-ell B)).
000050 | 
000051 | More generally, a lift over the fixed boundary is pointwise in the bundle, while
000052 | the principal symbol of 2C-I depends on tangential cotangent direction for an
000053 | ordinary Dirac boundary. These cannot be identified just from reflection regularity.
000054 | If R_tau is instead DEFINED as 2C-I, the algebraic statement is true by definition,
000055 | but its identification with a geometric reflection has become an extra hypothesis.
000056 | This corrects the stated geometric inference; it is not a no-go for all TFPT seams.
000057 | 
000058 | For a generic compact interior, C need not equal the positive spectral projector
000059 | of B. The half-cylinder equality used above is exact in its specified realization.
000060 | The two-face finite-cylinder calculation below uses the actual solutions, not that
000061 | generic replacement. Orthogonalizing a Calderon projection is also explicit here.
000062 | 
000063 | 3. The complete two-face datum retains the signed operator
000064 | 
000065 | Assume a given selfadjoint B on H, a known normal length ell>0, identified/labeled
000066 | boundary fibers, and an exact normal-independent product equation
000067 | (partial_r+B)psi=0 on the retained interval [0,ell]. Write T=exp(-ell B).
000068 | By the spectral theorem T is positive, injective, densely defined and closed,
000069 | possibly unbounded. The L2-closed endpoint Cauchy space is
000070 | L_ell={(u,Tu): u in D(T)}, where
000071 | D(T)={u: integral exp(-2ell lambda) dmu_B,u(lambda)<infinity}.
000072 | Finite spectral-cutoff solutions form a core for this closed graph. No claim is
000073 | made that every such endpoint pair has an H1 solution up to both boundary faces.
000074 | 
000075 | The orthogonal graph projection is
000076 | G_ell = [[A,Cross],[Cross,I-A]],
000077 | A=(I+T^2)^(-1)=(I+exp(-2ell B))^(-1),
000078 | Cross=T(I+T^2)^(-1)=(2cosh(ell B))^(-1).
000079 | This follows by projecting each spectral two-vector onto span(1,t), or from the
000080 | closed-operator graph projection theorem. All four displayed blocks are bounded.
000081 | Both A and I-A are injective, with dense range; 0 and 1 can be spectral limit
000082 | points but not eigenvalues. The original signed B is recovered uniquely by
000083 | B=(2ell)^(-1) log(A/(I-A)),
000084 | D(B)={u: integral_(0,1) |log(a/(1-a))|^2 dmu_A,u(a)<infinity}.
000085 | This formula keeps signs and magnitudes and transforms covariantly with the fibers.
000086 | 
000087 | Likewise T is the spectral quotient Cross/A on D(T). As a raw composition,
000088 | Cross A^(-1) has domain Ran(A)=D(T^2); its closure is T. Omitting this closure
000089 | would give the wrong domain in infinite dimension. Example: B e_n=-n e_n,
000090 | u_n=exp(-3ell n/2) is in D(T) but not D(T^2).
000091 | If ell is not independently known, the reconstruction determines only ell B.
000092 | If the face identification is missing, a positive transfer cannot simply be
000093 | declared in an arbitrary frame. Recovering a spectral list is not recovering
000094 | the matrix/operator action on the given local fields.
000095 | 
000096 | 4. A common state and time from that same datum, with explicit field premise
000097 | 
000098 | Now additionally assume that the actual field algebra is charged CAR(H direct_sum H)
000099 | and that its state is the gauge-invariant quasifree state with covariance G_ell.
000100 | These are field/statistical premises, not consequences of P1 alone. A projection
000101 | covariance defines a pure quasifree state. Restriction to the second labeled face
000102 | has occupation n_ell=I-A=(I+exp(2ell B))^(-1), with 0<n_ell<I spectrally.
000103 | For the free CAR dynamics a(f)->a(exp(itB)f), this is the quasifree KMS state
000104 | at inverse temperature beta=2ell (using the convention where occupation is
000105 | (I+exp(beta B))^(-1)). Its one-particle modular logarithm is
000106 | log((I-n_ell)/n_ell)=2ell B.
000107 | Thus the covariance and time generator are read from the SAME two-face datum;
000108 | one does not fit a second Hamiltonian to a prescribed occupation list. In infinite
000109 | dimension this is a state on the CAR algebra, not automatically a trace-class
000110 | Gibbs density matrix in an arbitrarily chosen Fock representation.
000111 | 
000112 | This is the known faithful-covariance inverse from the earlier Atlas contract
000113 | seam-state-dynamics-injectivity-20260918, now linked to the exact finite-collar
000114 | boundary equation. Neither the logarithm nor quasifree purification is new general
000115 | mathematics. The constructive contribution here is the corrected TFPT data path.
000116 | As ell->infinity, n_ell tends strongly to 1_(-infinity,0)(B)+1/2*1_{0}(B).
000117 | The sign-only limit loses the energy magnitudes that the finite two-face data keep.
000118 | On that limit one must not continue using a faithful-state logarithm.
000119 | 
000120 | Normal distance and physical time are distinct. Analytic continuation of the
000121 | positive transfer constructs a unitary one-particle group exp(itB). Its physical
000122 | identification, time unit and event interpretation require the original field
000123 | realization and scale. Neither a finite matrix test nor KMS uniqueness proves them.