Minimum Conditioning, Rational Laplace Transforms, and Universal Exponential Tails in the Continuous Gilbreath Model
Authors: Liu, Zongmin
Type: Preprint
Published: 2025-12-27

<div>
<div>Let $F_n(x_0,\ldots,x_n)$ be the apex obtained after $n$ iterations of</div>
<div>adjacent absolute differences, and let $X_0,\ldots,X_n$ be independent</div>
<div>standard exponential random variables. The associated Gilbreath expectation</div>
<div>is $c_n=\E F_n(X_0,\ldots,X_n)$; even boundedness of $(c_n)$ is presently</div>
<div>open.</div>
<br>
<div>We prove two structural results for the full distribution. First, subtracting</div>
<div>the minimum gives an exact distributional decomposition: the law of $F_n(X)$</div>
<div>is the uniform mixture of the $n+1$ laws obtained by fixing one top coordinate</div>
<div>to zero and leaving the other coordinates independent exponential. Second,</div>
<div>the moment-generating function $\E e^{zF_n}$ is a rational function over</div>
<div>$\Q$. Its possible poles are positive rational numbers, its abscissa of</div>
<div>convergence is exactly $1$, and the pole at $1$ is simple. Equivalently, for</div>
<div>every fixed $n$ there are a rational number $K_n>0$ and $\delta_n>0$ such that</div>
<div>\[</div>
<div>\Pp(F_n(X)>t)=K_ne^{-t}+O_n(e^{-(1+\delta_n)t}).</div>
<div>\]</div>
<div>More precisely the survival function is a finite rational</div>
<div>exponential-polynomial.</div>
<br>
<div>We derive two exact cone formulas for $K_n$, including a one-large-spike</div>
<div>formula indexed by the odd entries of the $n$th row of Pascal's triangle. At</div>
<div>an endpoint spike the eventual affine offset is completely explicit: if</div>
<div>\[</div>
<div>S_n(y_1,\ldots,y_n)=\sum_{j=0}^{n-1}F_j(y_1,\ldots,y_{j+1}),</div>
<div>\]</div>
<div>then $F_n(T,y_1,\ldots,y_n)=T-S_n(y)$ for $T>S_n(y)$. Hence the endpoint</div>
<div>residue is $A_{n,0}=\E e^{-S_n}$, it decreases strictly with $n$, and</div>
<div>$K_{2^m}=2A_{2^m,0}$ decreases strictly along the powers of two. This identity</div>
<div>also identifies the exact endpoint contribution behind the Chase--Hunter--Tao</div>
<div>lower bound. An exact polyhedral implementation gives</div>
<div>\[</div>
<div>\begin{array}{c|rrrrrrr}</div>
<div>n&0&1&2&3&4&5&6\\ \hline</div>
<div>K_n&1&1&5/9&2/3&274999/1155000&0.3252524027\ldots&0.2250748011\ldots.</div>
<div>\end{array}</div>
<div>\]</div>
<div>The values through $n=5$ are reproduced by two independent rational fan</div>
<div>representations. At depth six, both spike components have a second exact</div>
<div>integration audit; the smaller component is also regenerated by the independent</div>
<div>rational-vertex fan.</div>
<br>
<div>Finally we compute the entire minimum-conditioned expectation profile at depth</div>
<div>six and prove that depth four is the first failure of unimodality and</div>
<div>log-concavity; the property returns at depth five and fails again at depth six.</div>
<div>We also prove the exact stochastic domination</div>
<div>\[</div>
<div>\Pp(F_n(X)>t)\le e^{-t}\qquad(t\ge0)</div>
<div>\]</div>
<div>for every $0\le n\le5$. The depth-five proof reconstructs the full rational</div>
<div>survival law and reduces monotonicity of $e^t\Pp(F_5>t)$ to two exact Abel-prefix</div>
<div>positivity certificates. These results do not prove boundedness. They isolate</div>
<div>a new uniformity problem: boundedness cannot be decided by the fixed-depth tail</div>
<div>exponent, which is always one, but requires a structural all-depth explanation</div>
<div>for the cumulative coefficient positivity or another uniform control of the</div>
<div>non-asymptotic distribution.</div>
</div>