TFPT scientific source excerpt
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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
000149 | geometric partition function is therefore defined constructively by the stated limit, rather than
000150 | only heuristically read off from low-curvature coefficients.
000151 | \end{proof}