Critical Coverage and the Saturation Depth Problem

The auto-calibrated saturation depth ${n_1}(q)$ of the breadth-first exploration of ${\mathrm{Heis}}_3(\mathbb{Z}/q\mathbb{Z})$ was the central open asymptotic variable of O25–O28. This note makes two points. First, the depth problem is not a vector-space filling problem: at the window edge the cumulative Weil–fingerprint rank is far from full, so ${n_1}$ is the depth at which the creation of new directions per explored vertex drops below a fixed threshold — a stop of marginal novelty. Second, and consequently, the entire asymptotic content of ${n_1}$ collapses onto a single dimensionless observable, the critical coverage \[ {x_1}(q) = \frac{{\lvert B_n\rvert}_{\,n={n_1}}}{q^2}, \qquad\text{through the exact identity}\qquad {n_1}(q) = \left(\frac{{x_1}(q)}{{C_{\mathrm{Heis}}}}\right)^{1/4} \sqrt{q}, \] where ${\lvert B_n\rvert} \sim {C_{\mathrm{Heis}}}\,n^4$ (homogeneous dimension $D=4$, ${C_{\mathrm{Heis}}} \approx 0.427$) is the Heisenberg ball growth. The reduction rests on two robust facts — ${\lvert B_n\rvert} \sim {C_{\mathrm{Heis}}}\,n^4$ and the diffusive spectral gap $\lambda_2 \approx 4\pi^2/q^2$ — and one scaling ansatz. Empirically ${x_1}$ falls from $0.61$ at $q=101$ to $0.15$ at $q=601$, which excludes the constant-coverage (clean $\sqrt{q}$) scenario and supports the model-free conclusion ${n_1}(q)/q \to 0$. We isolate the governing tension $\Phi'({x_1}) = {\varepsilon_{\mathrm{sat}}}\,q$ and state the analytical programme: derive the asymptotic law of ${x_1}(q)$ from a shellwise Weil–BFS novelty estimate controlled by $\lambda_2$, of which the depth law ${n_1}(q)$ is then a corollary.