Critical Coverage and the Saturation Depth Problem

The auto-calibrated saturation depth n₁(q) of the breadth-first exploration of Heis₃(Z/qZ) 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 n₁ 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 n₁ collapses onto a single dimensionless observable, the critical coverage \[ {x_1}(q) = {{ B_n}_{\,n={n_1}}}{q^2}, the exact identity {n_1}(q) = ({{x_1}(q)}{{CHeis}})1/4 √q, \] where Bₙ ~ CHeis\,n⁴ (homogeneous dimension $D=4$, CHeis ≈ 0.427) is the Heisenberg ball growth. The reduction rests on two robust facts — Bₙ ~ CHeis\,n⁴ and the diffusive spectral gap λ₂ ≈ 4π²/q² — and one scaling ansatz. Empirically x₁ falls from $0.61$ at $q=101$ to $0.15$ at $q=601$, which excludes the constant-coverage (clean √q) scenario and supports the model-free conclusion n₁(q)/q → 0. We isolate the governing tension Φ'(x₁) = εₛₐₜ\,q and state the analytical programme: derive the asymptotic law of x₁(q) from a shellwise Weil–BFS novelty estimate controlled by λ₂, of which the depth law n₁(q) is then a corollary.