TFPT scientific source excerpt
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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.