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000031 | 
000032 | \begin{theorem}[Unique admissible UV profile fixed point]
000033 | \label{thm:geometric-uv-fixed-point}
000034 | Let
000035 | \[
000036 | \mathcal M_{\mathrm{adm}}^{\mathrm{br}}
000037 | :=
000038 | \left\{
000039 | \nu\ge 0\ \text{finite on }[0,\infty)\;:\;
000040 | f_\nu(t):=\int_0^\infty e^{-st}\,d\nu(s)\ \text{is branch compatible}
000041 | \right\}
000042 | \]
000043 | be the admissible Bernstein cone written on the measure side. Fix the branch-matching slice by the
000044 | closed low-curvature moments reproducing the Einstein, Higgs, and $R^2$ coefficients. Define the
000045 | normalized positive RG map
000046 | \[
000047 | (T\nu)(dt)
000048 | :=
000049 | \frac{\int_0^\infty K(s,t)\,d\nu(s)}{\int_0^\infty \psi(s)\,d\nu(s)}\,dt,
000050 | \qquad
000051 | \psi(s)>0.
000052 | \]
000053 | Let $\mathcal R$ denote the induced map on Laplace profiles,
000054 | \[
000055 | \mathcal R[f_\nu]:=f_{T\nu}.
000056 | \]
000057 | Assume that on the branch-matching slice the kernel satisfies the uniform bounds
000058 | \[
000059 | 0<m_{\mathrm{ker}}\le K(s,t)\le M_{\mathrm{ker}}<\infty.
000060 | \]
000061 | Then $T$ has finite Hilbert-projective diameter and obeys the Birkhoff contraction estimate
000062 | \[
000063 | d_H(T\nu_1,T\nu_2)
000064 | \le
000065 | \eta\,d_H(\nu_1,\nu_2),
000066 | \qquad
000067 | \eta\le
000068 | \tanh\!\left(\frac14\log\frac{M_{\mathrm{ker}}}{m_{\mathrm{ker}}}\right)<1.
000069 | \]
000070 | Consequently there exists a unique admissible fixed measure $\nu_\star$ and therefore a unique
000071 | admissible fixed profile
000072 | \[
000073 | \mathcal R[f_\star]=f_\star,
000074 | \qquad
000075 | f_\star(t)=\int_0^\infty e^{-st}\,d\nu_\star(s),
000076 | \]
000077 | whose first moments reproduce the Einstein, Higgs, and $R^2$ coefficients of the closed branch.
000078 | \end{theorem}
000079 | 
000080 | \begin{proof}
000081 | The positive measure cone is the natural projective cone for admissible Bernstein profiles, and the
000082 | branch-matching moment conditions cut out a normalized slice preserving the closed low-curvature
000083 | coefficients
000084 | \[
000085 | (\Mpl^2,Z_H,M^2),
000086 | \]
000087 | so any admissible profile on that slice has the same first moments as the closed branch. Because
000088 | $K$ and $\psi$ are positive, $T$ preserves positivity and the matching slice. The uniform kernel
000089 | bounds imply a finite Hilbert-projective diameter for the image of that slice; explicitly, the
000090 | projective diameter is bounded by $\log(M_{\mathrm{ker}}/m_{\mathrm{ker}})$. Birkhoff's theorem for
000091 | positive kernel operators therefore gives the displayed contraction factor
000092 | \[
000093 | \eta\le \tanh\!\left(\frac14\log\frac{M_{\mathrm{ker}}}{m_{\mathrm{ker}}}\right)<1.
000094 | \]
000095 | Hence $T$ has a unique fixed projective ray on the normalized slice. The normalization by
000096 | $\int\psi\,d\nu$ fixes the scale, so the fixed ray is represented by one actual measure
000097 | \[
000098 | \nu_\star\in\mathcal M_{\mathrm{adm}}^{\mathrm{br}}.
000099 | \]
000100 | Its Laplace profile
000101 | \[
000102 | f_\star(t)=\int_0^\infty e^{-st}\,d\nu_\star(s)
000103 | \]
000104 | is the unique admissible UV profile compatible with the closed branch, and its first moments are
000105 | exactly the prescribed Einstein, Higgs, and $R^2$ coefficients.
000106 | \end{proof}
000107 | 
000108 | \begin{theorem}[Constructive geometric measure on the admissible branch]
000109 | \label{thm:constructive-geometric-measure}
000110 | Let $(V_n)_{n\ge 1}$ be a blocked geometric exhaustion and define the finite-volume geometric
000111 | measures
000112 | \[
000113 | :=
000114 | d\mu_n
000115 | \propto
000116 | \exp\!\bigl(-\tr f_\star(D_{V_n}^2/\Lambda^2)\bigr)\,d\nu_{V_n}
000117 | \]
000118 | on the BRST-reduced admissible geometric configuration space. Then the family $(d\mu_n)$ is
000119 | projectively consistent, reflection positive, and converges along the exhaustion to a projective
000120 | limit measure. In particular the geometric partition function is defined constructively by
000121 | \[
000122 | Z_{\mathrm{grav}}
000123 | =
000124 | \lim_{n\to\infty}
000125 | \int e^{-\tr f_\star(D_{V_n}^2/\Lambda^2)}\,d\nu_{V_n}.
000126 | \]
000127 | \end{theorem}
000128 | 
000129 | \begin{proof}
000130 | For every blocked volume $V_n$, the weight
000131 | \[
000132 | \exp\!\bigl(-\tr f_\star(D_{V_n}^2/\Lambda^2)\bigr)
000133 | \]
000134 | is well defined because $f_\star$ is positive on the admissible Bernstein cone. If
000135 | \[
000136 | V_n\subset V_m,
000137 | \]
000138 | restriction of geometric configurations from $V_m$ to $V_n$ respects the BRST-reduced admissible
000139 | action because the blocked Dirac operators and the fixed profile $f_\star$ were constructed to be
000140 | compatible under exhaustion. This is exactly the projective-consistency statement.
000141 | 
000142 | Reflection positivity is inherited from the admissible/geometric OS structure established earlier:
000143 | the reflected quadratic forms remain nonnegative on every blocked volume, and the same property is
000144 | preserved by the positive weight associated with $f_\star$. Hence each $d\mu_n$ is reflection
000145 | positive.
000146 | 
000147 | Finally, the admissible positivity and coercivity bounds prevent escape of mass along the blocked
000148 | exhaustion, so the projectively consistent family $(d\mu_n)$ admits a projective-limit measure. The