Problem 13 Goal — Exponential boundary elementary density

For every $b_0,c\in\mathbb R$ with $b_0>0$ and $c>0$, there exists a finite elementary-expression tree $e$, possibly depending on $b_0$ and $c$, whose evaluation $E:\mathbb R\to\mathbb R$ is continuous at every positive real number and satisfies $E(t)\ge 0$ for every $t>0$. The permitted expression trees are generated from the variable $t$, rational constants, arbitrary real constants, addition, subtraction, multiplication, division, negation, exponential, real logarithm, real square root, sine, cosine, and the operation $x\mapsto\phi(x)$, where $\phi(x)=(2\pi)^{-1/2}\exp(-x^2/2)$; each operation is applied pointwise to the evaluations of its subexpressions. These are total real operations: in particular, division by zero has value $0$, the real logarithm has value $0$ at zero and agrees with $\log|x|$ for negative $x$, and the real square root is $0$ on nonpositive inputs. With $B(u)=b_0\exp(-cu)$, the same function $E$ must satisfy, for every real $t>0$, the exact Abel identity $\int_0^t \frac{1}{\sqrt{t-s}}\,\phi\!\left(\frac{B(t)-B(s)}{\sqrt{t-s}}\right)E(s)\,ds=\frac{1}{\sqrt t}\,\phi\!\left(\frac{B(t)}{\sqrt t}\right)$. These are Mathlib’s oriented real interval integrals with respect to Lebesgue measure; for $t>0$, the left side uses the interval $(0,t]$. At the endpoint $s=t$, the reciprocal $1/\sqrt{t-s}$ and the quotient $(B(t)-B(s))/\sqrt{t-s}$ are interpreted using totalized real division, so both are $0$ where appropriate and the displayed integrand has value $0$ there; this endpoint has measure zero. No separate integrability hypothesis is stated, so the integrals use Mathlib’s total integral convention, under which a nonintegrable integrand is assigned integral $0$. Finally, as the real upper endpoint $T$ tends to $+\infty$, the oriented interval integrals $\int_0^T E(t)\,dt$ must converge to $1$. The assertion does not require a single expression working uniformly for all $b_0,c$, does not assert uniqueness of $e$, and imposes no continuity or nonnegativity requirement at $t=0$ or at negative times; values for negative $T$ do not affect the stated limit.