TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-collar-multifield-20261016/BOUNDARY-OPERATION-PROOF.txt
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Inclusive original lines: 40-66
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000040 | 2. First check: a raw boundary mask does not descend to the kernel quotient
000041 | 
000042 | Let P0=diag(I,0) select the incoming boundary. This label is geometric.
000043 | The naturally available positive comparison is K_G(x,y)=<x,G y>.
000044 | Its null space is Ker G. A linear map F on boundary data can act on its
000045 | quotient only if F Ker G is contained in Ker G. A represented adjoint pair
000046 | also needs the corresponding adjoint compatibility; a raw Hilbert adjoint
000047 | cannot simply be assumed to be the K_G adjoint.
000048 | 
000049 | At a scalar spectral mode T=t>0, let
000050 |  v=(1,t)/sqrt(1+t^2), w=(-t,1)/sqrt(1+t^2).
000051 | Then G=v v*, G w=0, whereas
000052 | 
000053 |  K_G(P0 w,P0 w)=t^2/(1+t^2)^2 > 0.
000054 | 
000055 | Thus P0 maps the ZERO kernel class to a NONZERO class. It is not an operation
000056 | on this minimal response-state quotient. This is an exact null-stability
000057 | failure in the actual collar graph, not a comparison of arbitrary toy models.
000058 | It forbids the inference 'a labeled boundary mask is automatically a physical
000059 | operator on the state reconstructed from this same K_G'. It does not forbid
000060 | physical boundary interventions governed by a DIFFERENT full history kernel.
000061 | 
000062 | Adding an invisible ancillary space cannot repair this exact intertwining
000063 | while preserving K_G: any realization W with W*W=G satisfies W w=0; an
000064 | operator O with O W=W P0 would imply W P0 w=0, contradicting the value above.
000065 | More memory cannot turn the same inconsistent quotient action into a map.
000066 |