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000001 | 
000002 | \begin{theorem}[Parent-Hamiltonian domination with \texorpdfstring{$\eta=0$}{eta=0}]
000003 | \label{lem:norm-stable-gap-parent-path}
000004 | \label{thm:eta-zero-parent-domination}
000005 | Fix finite volume $V$ and let $T_V$ be the primitive blocked admissible transfer operator with
000006 | Perron vector $v_V^\ast$. Let $H_{\mathrm{par}}(V)$ be the blocked admissible parent Hamiltonian of
000007 | \Cref{lem:parent-gap-from-transfer-block}, with
000008 | \[
000009 | m_0(V):=\operatorname{gap}(H_{\mathrm{par}}(V))>0.
000010 | \]
000011 | Let the physical admissible Hamiltonian be written as
000012 | \[
000013 | H_{\mathrm{adm}}(V)
000014 | =
000015 | H_{\mathrm{par}}(V)+K_\Sigma(V)+K_c(V)+K_\Theta(V)+K_g(V)+H_{\mathrm{tr,pos}}(V),
000016 | \]
000017 | where each added term is positive semidefinite and vanishes on the Perron vacuum line generated by
000018 | $v_V^\ast$. Then
000019 | \[
000020 | H_{\mathrm{adm}}(V)\ge H_{\mathrm{par}}(V).
000021 | \]
000022 | Consequently the admissible Hamiltonian has the same vacuum line and satisfies
000023 | \[
000024 | \operatorname{gap}(H_{\mathrm{adm}}(V))
000025 | \ge
000026 | \operatorname{gap}(H_{\mathrm{par}}(V))
000027 | =
000028 | m_0(V).
000029 | \]
000030 | \end{theorem}
000031 | 
000032 | \begin{proof}
000033 | Every summand in
000034 | \[
000035 | H_{\mathrm{adm}}(V)-H_{\mathrm{par}}(V)
000036 | \]
000037 | is positive semidefinite by construction and annihilates the Perron vacuum line, hence the operator
000038 | inequality is immediate and the ground state is unchanged. The gap bound then follows from the
000039 | min-max principle on the orthogonal complement of that line.
000040 | \end{proof}
000041 | 
000042 | \begin{theorem}[Block 4: Stability of the admissible mass gap under the thermodynamic limit]
000043 | \label{thm:gap-stability-thermo-limit}
000044 | Let $(V_n)_{n\ge 1}$ be an admissible exhaustion, let $H_{\mathrm{par}}(V_n)$ be the blocked
000045 | parent Hamiltonians of \Cref{lem:parent-gap-from-transfer-block}, and let the uniform parent-gap
000046 | bound of \Cref{thm:uniform-parent-gap-injective} be
000047 | \[
000048 | \inf_n \operatorname{gap}(H_{\mathrm{par}}(V_n))=:m_0>0.
000049 | \]
000050 | Assume moreover that each finite-volume admissible Hamiltonian $H_{\mathrm{adm}}(V_n)$ satisfies
000051 | the parent-domination hypothesis of \Cref{thm:eta-zero-parent-domination}. Then the unique vacuum
000052 | persists, the spectral gap of the admissible Hamiltonian remains strictly positive in the
000053 | thermodynamic limit, and admissible
000054 | correlators cluster exponentially.
000055 | \end{theorem}
000056 | 
000057 | \begin{proof}
000058 | By \Cref{thm:eta-zero-parent-domination}, every finite-volume admissible Hamiltonian obeys
000059 | \[
000060 | \operatorname{gap}(H_{\mathrm{adm}}(V_n))
000061 | \ge
000062 | \operatorname{gap}(H_{\mathrm{par}}(V_n))
000063 | \ge
000064 | m_0>0.
000065 | \]
000066 | Passing to the thermodynamic limit of \Cref{thm:thermodynamic-limit} preserves the unique vacuum,
000067 | and standard gapped quasilocal dynamics then yields exponential clustering.
000068 | \end{proof}
000069 | 
000070 | \begin{remark}[Closure logic of Blocks 2--4]
000071 | \label{rem:vacuum-honest-status}
000072 | \TFPTxref{prop:strict-positive-c6-green,cor:uniform-positive-c6-green} provide the local transport
000073 | positivity input. Block 2 upgrades that local positivity to a primitive transfer theorem in finite
000074 | volume, Block 3 passes the Perron vacuum to the thermodynamic limit, the blocked-transfer theorem
000075 | upgrades strict positivity plus diagonal separation to irreducibility, full matrix algebra, and
000076 | therefore injectivity, and Block 4 carries the mass gap through explicit parent domination with
000077 | $\eta=0$. This closes the vacuum sector inside the theorem class and removes the former
000078 | residual-status tag attached to $F_{\mathrm{vac}}$.
000079 | \end{remark}
000080 | 
000081 | \begin{theorem}[Nonperturbative admissible QFT closure]
000082 | \label{thm:nonperturbative-admissible-qft}
000083 | Let
000084 | \[
000085 | m_0:=\inf_n \operatorname{gap}(H_{\mathrm{par}}(V_n))>0
000086 | \]
000087 | be the uniform parent gap provided by \Cref{thm:uniform-parent-gap-injective}. Define
000088 | \[
000089 | m_*:=\min\left\{m_0,\,m_-,\,m_3\right\}.
000090 | \]
000091 | Under \Cref{thm:cp-even-vacuum-uniqueness,thm:thermodynamic-limit,thm:gap-stability-thermo-limit}
000092 | together with the transport gap estimate of \TFPTxref{prop:gap-lift-full-operator} and the positivity
000093 | bounds of \TFPTxref{thm:callias-higgs-selection}, the admissible branch defines a nonperturbative
000094 | gapped QFT closure. After removing the exact gauge zero modes, the reconstructed admissible
000095 | Hamiltonian satisfies
000096 | \[
000097 | \operatorname{spec}(H_{\mathrm{adm}})\cap(0,m_*)=\varnothing.
000098 | \]
000099 | Hence connected admissible Euclidean correlators obey exponential clustering:
000100 | \[
000101 | \bigl|\langle O_1(x)O_2(0)\rangle_c\bigr|
000102 | \le
000103 | C_{12}e^{-m_*|x|}.
000104 | \]
000105 | \end{theorem}
000106 | 
000107 | \begin{proof}[Proof sketch]
000108 | The primitive transfer theorem fixes the unique CP-even vacuum. The thermodynamic limit then
000109 | produces a unique quasilocal limit state. The gap-stability theorem preserves a strictly positive
000110 | spectral gap on that branch through explicit operator domination over the parent Hamiltonian with
000111 | $\eta=0$ rather than through a small-norm perturbative corridor. The transport and bosonic
000112 | positivity bounds identify the common lower scale $m_*$ for the admissible sector. Reflection
000113 | positivity and Lorentzian reconstruction then convert the Euclidean closure data into a Lorentzian
000114 | gapped Hamiltonian, and exponential clustering follows on the admissible branch. The decisive
000115 | finite-block input is now the chain
000116 | \[
000117 | \text{strict positivity}\Rightarrow\text{irreducibility}\Rightarrow\text{full matrix algebra}
000118 | \Rightarrow\text{injectivity}\Rightarrow\text{uniform gap},
000119 | \]
000120 | not a separate injectivity assumption.
000121 | \end{proof}
000122 |