TFPT scientific source excerpt
Original repository path: documents/archive--paper-latex--tfpt-42/sections/002-section-big-picture-what-tfpt-actually-claims.tex
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000065 | dependency maps that follow: its job is to put the manuscript's decoder core on one page while
000066 | keeping the closed datum, theorem, and derived-consequence flow visible.}
000067 | \label{fig:tfpt-headline-claims}
000068 | \end{figure}
000069 | 
000070 | \begin{tcolorbox}[colback=tfptblue!3, colframe=tfptblue, breakable,
000071 |     title={TFPT master equation and closed indices}, colbacktitle=tfptblue]
000072 | \begin{equation}
000073 | \label{eq:tfpt-master}
000074 | \begin{aligned}
000075 | S_{\mathrm{TFPT}}
000076 | =
000077 | \operatorname{Tr}f\!\left(\frac{D_{\mathrm{rel}}+A_{\Sigma}^{\mathrm{seed}}}{\chiseed}\right)
000078 | -
000079 | \operatorname{Tr}f\!\left(\frac{D_{\mathrm{ref}}}{\chiseed}\right)
000080 | +\frac{i\pi}{2}\Delta\eta_\Sigma,
000081 | \qquad
000082 | A_{\Sigma}^{\mathrm{seed}}:=\mathcal B_{\mathrm{rel}}\oplus\nabla_{\chiseed}.
000083 | \end{aligned}
000084 | \end{equation}
000085 | \begin{equation}
000086 | \label{eq:tfpt-selectors}
000087 | \begin{aligned}
000088 | \operatorname*{Stat}_{\alpha,\chiseed,e,\wspin}S_{\mathrm{TFPT}}=0,
000089 | \qquad
000090 | \Padm:=\Pi_{\ker\Delta_{\mathrm{adm}}},
000091 | \qquad
000092 | \partial_\alpha\Gamma_{U(1)}(\alpha)=0,
000093 | \\
000094 | \operatorname{Ind}(D^+_{\mathrm{fam}})=3,
000095 | \qquad
000096 | \operatorname{Ind}(\mathcal B^+_{E_2})=1,
000097 | \qquad
000098 | \operatorname{Ind}(\mathcal B^+_{E_3})=0.
000099 | \end{aligned}
000100 | \end{equation}
000101 | \end{tcolorbox}
000102 | 
000103 | The hard carrier theorems below do not require the full theorem stack behind
000104 | \Cref{eq:tfpt-master,eq:tfpt-selectors}. In the present paper the master equation is the
000105 | organizing closed form: the carrier theorems are hard, the later closure theorems are internal
000106 | statements on the reconstructed branch, and the empirical interface is kept outside this theorem
000107 | box. At
000108 | this overview level the seed package is written as $A_{\Sigma}^{\mathrm{seed}}$; on the later
000109 | geometric branch it is reorganized as $A_{\Sigma}^{\mathrm{geo}}$ with
000110 | $\chiseed\mapsto\chigeo=\Lambda e^\sigma$.
000111 | 
000112 | \begin{remark}[Selector versus dynamics]
000113 | \label{rem:selector-versus-dynamics}
000114 | The admissibility projector $\Padm$ selects the physical sector but is not itself the full
000115 | Minkowski dynamics. The actual dynamics is carried by the relative generating functional
000116 | $Z_{\mathrm{rel}}$, its Schwinger distributions, the reconstructed local net on the admissible
000117 | Hilbert space, the running effective actions $\Gamma_k$ and
000118 | $\Gamma_{\mathrm{TFPT}}^{\mathrm{ren}}$, and the resulting scattering theory.
000119 | \end{remark}