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000060 | \begin{theorem}[Unique admissible UV profile fixed point]
000061 | \label{thm:geometric-uv-fixed-point}
000062 | Let
000063 | \[
000064 | \mathcal M_{\mathrm{adm}}^{\mathrm{br}}
000065 | :=
000066 | \left\{
000067 | \nu\ge 0\ \text{finite on }[0,\infty)\;:\;
000068 | f_\nu(t):=\int_0^\infty e^{-st}\,d\nu(s)\ \text{is branch compatible}
000069 | \right\}
000070 | \]
000071 | be the admissible Bernstein cone written on the measure side. Fix the branch-matching slice by the
000072 | closed low-curvature moments reproducing the Einstein, Higgs, and $R^2$ coefficients. Define the
000073 | normalized positive RG map
000074 | \[
000075 | (T\nu)(dt)
000076 | :=
000077 | \frac{\int_0^\infty K(s,t)\,d\nu(s)}{\int_0^\infty \psi(s)\,d\nu(s)}\,dt,
000078 | \qquad
000079 | \psi(s)>0.
000080 | \]
000081 | Let $\mathcal R$ denote the induced map on Laplace profiles,
000082 | \[
000083 | \mathcal R[f_\nu]:=f_{T\nu}.
000084 | \]
000085 | Assume that on the branch-matching slice the kernel satisfies the uniform bounds
000086 | \[
000087 | 0<m_{\mathrm{ker}}\le K(s,t)\le M_{\mathrm{ker}}<\infty.
000088 | \]
000089 | Then $T$ has finite Hilbert-projective diameter and obeys the Birkhoff contraction estimate
000090 | \[
000091 | d_H(T\nu_1,T\nu_2)
000092 | \le
000093 | \eta\,d_H(\nu_1,\nu_2),
000094 | \qquad
000095 | \eta\le
000096 | \tanh\!\left(\frac14\log\frac{M_{\mathrm{ker}}}{m_{\mathrm{ker}}}\right)<1.
000097 | \]
000098 | Consequently there exists a unique admissible fixed measure $\nu_\star$ and therefore a unique
000099 | admissible fixed profile
000100 | \[
000101 | \mathcal R[f_\star]=f_\star,
000102 | \qquad
000103 | f_\star(t)=\int_0^\infty e^{-st}\,d\nu_\star(s),
000104 | \]
000105 | whose first moments reproduce the Einstein, Higgs, and $R^2$ coefficients of the closed branch.
000106 | \end{theorem}
000107 | 
000108 | \begin{proof}
000109 | The positive measure cone is the natural projective cone for admissible Bernstein profiles, and the
000110 | branch-matching moment conditions cut out a normalized slice preserving the closed low-curvature
000111 | coefficients
000112 | \[
000113 | (\Mpl^2,Z_H,M^2),
000114 | \]
000115 | so any admissible profile on that slice has the same first moments as the closed branch. Because
000116 | $K$ and $\psi$ are positive, $T$ preserves positivity and the matching slice. The uniform kernel
000117 | bounds imply a finite Hilbert-projective diameter for the image of that slice; explicitly, the
000118 | projective diameter is bounded by $\log(M_{\mathrm{ker}}/m_{\mathrm{ker}})$. Birkhoff's theorem for
000119 | positive kernel operators therefore gives the displayed contraction factor
000120 | \[
000121 | \eta\le \tanh\!\left(\frac14\log\frac{M_{\mathrm{ker}}}{m_{\mathrm{ker}}}\right)<1.
000122 | \]
000123 | Hence $T$ has a unique fixed projective ray on the normalized slice. The normalization by
000124 | $\int\psi\,d\nu$ fixes the scale, so the fixed ray is represented by one actual measure
000125 | \[
000126 | \nu_\star\in\mathcal M_{\mathrm{adm}}^{\mathrm{br}}.
000127 | \]
000128 | Its Laplace profile
000129 | \[
000130 | f_\star(t)=\int_0^\infty e^{-st}\,d\nu_\star(s)
000131 | \]
000132 | is the unique admissible UV profile compatible with the closed branch, and its first moments are
000133 | exactly the prescribed Einstein, Higgs, and $R^2$ coefficients.
000134 | \end{proof}
000135 | 
000136 | \begin{theorem}[Constructive geometric measure on the admissible branch]
000137 | \label{thm:constructive-geometric-measure}
000138 | Let $(V_n)_{n\ge 1}$ be a blocked geometric exhaustion and define the finite-volume geometric
000139 | measures
000140 | \[
000141 | :=
000142 | d\mu_n
000143 | \propto
000144 | \exp\!\bigl(-\tr f_\star(D_{V_n}^2/\Lambda^2)\bigr)\,d\nu_{V_n}
000145 | \]
000146 | on the BRST-reduced admissible geometric configuration space. Then the family $(d\mu_n)$ is
000147 | projectively consistent, reflection positive, and converges along the exhaustion to a projective
000148 | limit measure. In particular the geometric partition function is defined constructively by
000149 | \[
000150 | Z_{\mathrm{grav}}
000151 | =
000152 | \lim_{n\to\infty}
000153 | \int e^{-\tr f_\star(D_{V_n}^2/\Lambda^2)}\,d\nu_{V_n}.
000154 | \]
000155 | \end{theorem}
000156 | 
000157 | \begin{proof}
000158 | For every blocked volume $V_n$, the weight
000159 | \[
000160 | \exp\!\bigl(-\tr f_\star(D_{V_n}^2/\Lambda^2)\bigr)
000161 | \]
000162 | is well defined because $f_\star$ is positive on the admissible Bernstein cone. If
000163 | \[
000164 | V_n\subset V_m,
000165 | \]
000166 | restriction of geometric configurations from $V_m$ to $V_n$ respects the BRST-reduced admissible
000167 | action because the blocked Dirac operators and the fixed profile $f_\star$ were constructed to be
000168 | compatible under exhaustion. This is exactly the projective-consistency statement.
000169 | 
000170 | Reflection positivity is inherited from the admissible/geometric OS structure established earlier:
000171 | the reflected quadratic forms remain nonnegative on every blocked volume, and the same property is
000172 | preserved by the positive weight associated with $f_\star$. Hence each $d\mu_n$ is reflection
000173 | positive.
000174 | 
000175 | Finally, the admissible positivity and coercivity bounds prevent escape of mass along the blocked
000176 | exhaustion, so the projectively consistent family $(d\mu_n)$ admits a projective-limit measure. The
000177 | geometric partition function is therefore defined constructively by the stated limit, rather than
000178 | only heuristically read off from low-curvature coefficients.