TFPT scientific source excerpt
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000111 | \begin{tcolorbox}[colback=tfptblue!2, colframe=tfptblue!40, breakable,
000112 |     title={Primitivity lemma underlying Block 2}, colbacktitle=tfptblue!60]
000113 | Let $G_V$ be the directed graph on the retained CP-even admissible basis vectors, with a directed
000114 | edge
000115 | \[
000116 | u\to v
000117 | \]
000118 | iff the corresponding matrix element of the blocked transfer operator $T_V$ is strictly positive.
000119 | Then:
000120 | \begin{enumerate}[label=(\roman*), leftmargin=1.5em, topsep=3pt, itemsep=2pt]
000121 |     \item by \TFPTxref{prop:strict-positive-c6-green}, every local admissible $C_6$ transport step is
000122 |     strictly positive on the retained basis;
000123 |     \item by the admissible word grammar and the finite blocked transport grammar, every retained
000124 |     basis vector reaches every other one after finitely many blocked steps, so $G_V$ is strongly
000125 |     connected;
000126 |     \item the diagonal admissibility weights and blocked selector preserve positive return
000127 |     amplitudes on retained basis states, so $G_V$ is aperiodic.
000128 | \end{enumerate}
000129 | Hence there exists $N_V<\infty$ such that
000130 | \[
000131 | T_V^{N_V}
000132 | \]
000133 | has strictly positive matrix entries on $\mathcal K_{+,\mathrm{adm}}^{(V)}$. Equivalently, $T_V$
000134 | is primitive.
000135 | \end{tcolorbox}
000136 | 
000137 | \begin{theorem}[Block 2: Uniqueness of the CP-even admissible vacuum from the primitive positive transfer]
000138 | \label{thm:cp-even-vacuum-uniqueness}
000139 | Fix finite volume or a declared cutoff. Let $\mathcal K_{+,\mathrm{adm}}^{(V)}$ be the cone of
000140 | CP-even admissible vectors in the retained basis, and let $T_V$ be the blocked admissible
000141 | transfer operator obtained from the local finite $C_6$ kernels
000142 | $\mathcal Y_y^{(\varepsilon)}$, the diagonal admissibility weights, and the two-stage selector
000143 | $\Padm$. Then $T_V$ is primitive on $\mathcal K_{+,\mathrm{adm}}^{(V)}$. Hence $T_V$ has a unique
000144 | strictly positive Perron vector $v_V^\ast$, the finite-volume CP-even admissible vacuum
000145 | $\rho^\ast_V$ of \Cref{thm:admissible-vacuum-finite-volume} is unique, and the gap above that
000146 | vacuum is strictly positive.
000147 | \end{theorem}
000148 | 
000149 | \begin{proof}[Proof sketch]
000150 | The primitivity lemma stated immediately above turns the transport positivity of
000151 | \TFPTxref{prop:strict-positive-c6-green} and the admissible word connectivity into an explicit
000152 | graph-theoretic criterion for $T_V$. Perron--Frobenius / Krein--Rutman then gives a unique
000153 | strictly positive Perron vector $v_V^\ast$ in the cone
000154 | $\mathcal K_{+,\mathrm{adm}}^{(V)}$, which yields the unique CP-even finite-volume vacuum. The
000155 | simplicity of the Perron root is precisely the finite-volume gap statement.