The Height-Conductor Ratio on Curve Towers: Closed Forms for the Fermat Tower, the One-Third Barrier, and the Attainable Set

For a tower of curves $X_n$ over a number field with fixed bad-reduction set and $g_n \to \infty$, we study the ratio of the stable Faltings height to the conductor of its Jacobian, $\kappa_n := h_F(\mathrm{Jac}(X_n))/\log \mathrm{cond}(\mathrm{Jac}(X_n))$. We prove four groups of results. (I) Closed forms for the Fermat tower: slope $\frac{\log p}{4}\,n$ and second-order intercept $-6\log A$ ($A$ the Glaisher-Kinkelin constant), yielding $\kappa_{\mathrm{Fermat}} = \frac{1}{8}$; the orbit stratification is strictly increasing and converges to $\frac{1}{8}$ from below. (II) A one-third barrier $\kappa \leq \frac{1}{3}$ for $\mathrm{GL}_2$-type towers with $\sigma \to 1$, conditional on arithmetic BMY in general (Problem 10.1) and unconditional on modular towers, where the BMY input is a theorem (Dolce-Mercuri, Mayer) and where $X_0(N)$ attains equality asymptotically, so the constant cannot be improved. (III) The barrier is strictly stronger than BMY: $\kappa \leq \frac{1}{3} \Rightarrow R \leq \frac{4}{\sigma} - 1 \leq 3$. (IV) The closure of the attainable set in the abelian-variety tower world is $[\frac{1}{12}, \frac{1}{3}]$: the lower end is unconditional (Zhang, Wilms) and attained by Heisenberg towers, the upper end is the barrier and is attained by modular towers, and the interior is densely (countably) filled by reindexed product convexity; the curve-tower interior filling remains open. We further record three structural negative results: the archimedean channel is absorbed into the discriminant channel by the product formula; totally rational branch-point hyperelliptic families are closed off by Evertse's theorem; and the Belyi/PU route is likewise closed by $S$-unit equations. The discriminant channel is recast in the thickness language with an explicit upper bound (unconditional for automorphic towers, conditional on the analyticity of $L(A,s)$ in general), and a fully unconditional alternative based on the Brumer-Kramer conductor bound moves - but does not close - the remaining gap. All conditional dependencies are explicitly flagged in the body; all corrections are recorded in the revision history, Appendix B.