TFPT scientific source excerpt
Original repository path: documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/sections/001-section-primitive-core-and-carrier-interface-handoff.tex
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000122 |     \item the doubled operator and its even and odd parts are
000123 |     \[
000124 |     D=D_+\cup_{\Sigma}D_+^{\mathrm{op}},
000125 |     \qquad
000126 |     D_{\mathrm{ref}}:=\frac12\bigl(D+\taudbl^{\ast}D\taudbl^{\ast}\bigr),
000127 |     \qquad
000128 |     D_{\mathrm{rel}}:=\frac12\bigl(D-\taudbl^{\ast}D\taudbl^{\ast}\bigr);
000129 |     \]
000130 |     \item on boundary Cauchy data the induced reflection $R_{\taudbl}$ satisfies
000131 |     \[
000132 |     \operatorname{Ran}(P_C)=\ker(R_{\taudbl}-1),
000133 |     \qquad
000134 |     \ker(P_C)=\ker(R_{\taudbl}+1),
000135 |     \qquad
000136 |     \iC=R_{\taudbl}.
000137 |     \]
000138 | \end{enumerate}
000139 | Hence the primitive scaffold
000140 | \[
000141 | \Tprim^{\mathrm{kin}}
000142 | :=
000143 | (\Xbulk,\widetilde M,M,\Seam,\taudbl,\iC,D_{\mathrm{ref}},D_{\mathrm{rel}},\chiseed)
000144 | \]
000145 | is derived, not assumed. The derived doubled datum may be recorded as
000146 | \[
000147 | \Tmin^{\mathrm{cl}}:=(\mathcal A,\mathcal H,D,J,\Gamma,\taudbl,\iC),
000148 | \]
000149 | If one further chooses an almost-commutative factorization
000150 | \[
000151 | \mathcal H_{\mathrm{prim}}\cong L^2(\Xbulk,S)\otimes \Hint,
000152 | \]
000153 | then $S$ and $\Hint$ refer only to that chosen realization and are not additional
000154 | reconstruction outputs.