Structural properties of the implicit function defined by an integral self-consistency equation

arXiv:2606.04243v1 Announce Type: new Abstract: We study the integral equation $\int_0^m \eta\rho(\eta)/(C-\eta)\,d\eta = 1$ with $C>m$, where $\rho$ is a $C^1$ probability density on $[0,M]$ vanishing polynomially at $\eta=M$. Setting $\mathcal{I}^+(m) := \lim_{C \downarrow m}\int_0^m \eta\rho(\eta)/(C-\eta)\,d\eta$ and $\Omega := \{m \in (0,M) : \mathcal{I}^+(m) > 1\}$, the equation determines $C$ implicitly as a function of $m$ on $\Omega$, and our object of study is the dimensionless ratio $\beta(m) := C(m)/m$. Writing $h(\eta) := \eta\rho(\eta)$, our main theorem establishes openness of $\Omega$, $C^1$-smoothness of $\beta$, a sign formula identifying $\beta'(m)$ with a positively-weighted integral of $dh/d\ln\eta$, transfer of monotonicity from $h$ to $\beta$, and existence of an interior critical point of $\beta$ when $h$ is unimodal and two technical hypotheses hold. Numerically, $\beta$ has a single critical point in seven log-concave test densities (mostly Beta-type), in support of a separate uniqueness conjecture. A bimodal density that violates both unimodality and log-concavity exhibits three critical points; this shows that dropping the two hypotheses jointly admits multiple critical points, but does not separate their roles.