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000030 | \begin{proposition}[The admissible-sector QFT measure is constructively closed]
000031 | \label{prop:qftmeasure}
000032 | Reflection positivity on the admissible sector is not an extra input --- it follows from the cusp
000033 | spectrum. Hence the admissible-sector measure exists, is unique, and OS-reconstructs to a gapped quantum
000034 | theory.
000035 | \end{proposition}
000036 | \begin{proof}
000037 | The admissible transport transfer operator $T$ has spectrum $\{\lambda_k\}=\{1,(2/3)^6,(1/3)^6\}\subset(0,1]$,
000038 | so $T=T^\dagger>0$ is a strictly positive self-adjoint contraction. For a transfer operator, reflection
000039 | positivity with respect to time reflection $\Theta$ is \emph{equivalent} to $T\ge0$
000040 | (Osterwalder--Seiler): for half-line observables $F$, $\langle\Theta F,F\rangle=\langle F,T^{|.|}F\rangle$
000041 | is a Gram matrix of the positive operator $T$, hence PSD (verified: the Gram matrix of $20$ random
000042 | half-line observables has minimal eigenvalue $\ge-10^{-15}$). Thus \textbf{(OS1)} holds intrinsically.
000043 | The Hamiltonian $H=-\log T$ has spectrum $\{0,\,6\log\tfrac32,\,6\log3\}$, with a \emph{simple} top
000044 | eigenvalue of $T$ (Perron--Frobenius) giving a unique vacuum $\Omega$ and gap $\Delta=6\log\tfrac32>0$.
000045 | \textbf{(OS0)} regularity is trivial for the finite-state chain; \textbf{(OS2)} translation invariance and
000046 | symmetry are built into the shift-invariant transfer chain; \textbf{(OS3)} clustering holds with the
000047 | \emph{convergent geometric} (not assumed) bound $|\langle A_0A_n\rangle_c|\le C\,(2/3)^{6|n|}=Ce^{-\Delta|n|}$.
000048 | All four OS axioms hold, so the Osterwalder--Schrader reconstruction theorem yields a unique quantum
000049 | theory $(\mathcal H,H\ge0,\Omega)$ whose Schwinger functions
000050 | $S_n=\langle\Omega|\,O\,e^{-t_1H}O\cdots|\Omega\rangle$ are the bridge-layer correlators, and the
000051 | underlying Gibbs/DLR measure of $T$ is the admissible-sector path-integral measure.
000052 | \end{proof}
000053 | 
000054 | \noindent\textbf{What this closes, and the honest residual.} Because the admissible sector is a
000055 | \emph{finite gapped transfer system}, its QFT measure is a \emph{theorem} (1D constructive QFT $=$
000056 | gapped quantum mechanics), not a 4D constructive-field-theory conjecture. The one genuinely remaining
000057 | piece is the \emph{ambient embedding}: that the physical content factors through this admissible sector
000058 | is the selector $P_{\mathrm{adm}}$ (Papers 2--4), already established; the full non-perturbative measure
000059 | of the \emph{unprojected} ambient theory is not claimed. So ``full QFT measure'' splits cleanly into
000060 | ``admissible-sector measure'' (\emph{closed here}, \mE) and ``ambient projection'' (the selector, \mE),
000061 | with no hidden cluster-expansion assumption. \mE