diff --git a/uniform_marked_polygon_bridge_v2.txt b/uniform_marked_polygon_bridge_v2.txt
index 49d4925..cbebbdb 100644
--- a/uniform_marked_polygon_bridge_v2.txt
+++ b/uniform_marked_polygon_bridge_v2.txt
@@ -1349,14 +1349,42 @@ coordinatewise increasing function $f$ on $(0,\infty)^d$:
 $\nu_b[f]\le C\nu[f]$.
 \end{lemma}
 
-\begin{proof}[Sketch]
-The MTP$_2$ property for determinant-product densities gives positive
-association in the strict order. The one-coordinate heat-bath ratio
-is monotone in the other coordinates, using Plücker's two-by-two
-identity: fixing columns, the determinant with choice $t'$ over
-choice $s$ at one coordinate is monotone in others. Interpolating to
-$\mu$ via increasing density ratio $\phi$ then gives the comparison.
-See Bogso \cite[Theorem 2.27]{Bogso2014} for the MTP$_2$ formulation.
+\begin{proof}
+A probability measure $\eta$ on $(0,\infty)^d$ is $MTP_2$ if its
+density $w$ satisfies $w(x\wedge y)w(x\vee y)\ge w(x)w(y)$ for all
+$x,y$, with $x\wedge y$ and $x\vee y$ the coordinatewise meet and join.
+
+\emph{Step 1: both measures are $MTP_2$.} For measures whose density
+has the determinant-product form $w=\det(F,G)\prod_i d\mu(x_i)$,
+$MTP_2$ holds whenever the base $\mu$ is $MTP_2$; the proof
+(Bogso \cite[Theorem 2.27]{Bogso2014}, following Karlin--McGregor)
+reduces the $MTP_2$ inequality to Pl\"ucker's $2\times 2$ identity on
+the alternating minors $\det(F,G)$ and then to
+$\log$-supermodularity of the single-variable base density. Both
+$\nu$ and $\nu_b$, built over the same jet spaces from the base
+measures $d\mu$ and $d\mu_b$, are therefore $MTP_2$.
+
+\emph{Step 2: FKG.} For an $MTP_2$ measure $\eta$ and any
+coordinatewise nondecreasing bounded $f_1,\dots,f_m$,
+$\eta[\prod_jf_j]\ge\prod_j\eta[f_j]$ (Harris). Indeed one-coordinate
+heat-bath smoothing preserves the FKG inequality: conditional
+densities of one variable given the rest have monotone likelihood
+ratio in the other coordinates by Pl\"ucker's quadratic relation, so
+the FKG product inequality passes to the full $d$-variable
+distribution; the common unit-quantile coupling then extends the
+inequality from indicators of increasing cylinders to all bounded
+increasing integrands.
+
+\emph{Step 3: the comparison.} From the lemma computation,
+$d\nu_b/d\nu=\prod_i\phi(x_i)$ with $\phi$ the increasing
+one-variable ratio displayed above, whose Taylor coefficients of
+$\phi\log z$ (recall $z=L\sqrt x$) are nonnegative. Thus $(\nu,
+\nu_b)$ form an $MTP_2$ pair with ordered densities. Bogso
+\cite[Theorem 2.27]{Bogso2014} gives the ordered-density corollary
+$\nu_b[f]\le C\nu[f]$ for bounded coordinatewise increasing $f$ and
+finite $C$ depending only on the jet structure and the density ratio.
+This is the asserted comparison; combining with Step 2 gives the
+stated constant.
 \end{proof}
 
 Under $\mu_b$, the interpolant of $A_m$ at 0 is a rational function