TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-full-collar-reconstruction-20261015/PROOF.txt
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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.