Not in atlas — F-stats ≤ 2.8 on all four candidate metrics at the framework's signal lengths, plus an internal r = +0.80 between axial_excess and wedge_concentration. Square mode is the live atlas lens; Sacks ships as a diagnostic embedding only.
What it measures: The same threshold-marked-positions question as Ulam Spiral (Square), but on the Sacks parametrization — sample k placed at radius √k and angle 2π√k. Sacks puts perfect squares on the +x ray; quadratic polynomials trace spirals.
Sacks mode is computed by UlamSpiralGeometry(spiral_type='sacks') and exposed through the source guide's per-source render (perfect squares pop out as a stripe on the +x ray — genuinely informative for human eyes). The four scalar metrics below are documented for the embedding-discovery pipeline and for direct use; they're not in the atlas profiles JSON.
Gini coefficient of mark counts across 60 angular wedges of the Sacks plane. 0 = perfectly uniform angular distribution; → 1 = marks concentrate on a few wedges. The Sacks signature for an integer sequence is high axial_excess (perfect squares fall on a single ray; quadratic polynomials fall on a few discrete rays).
Incremental quadratic R² on the marked indices inside the densest wedge. Up to 1.0 if the marked subsequence fits a quadratic better than a linear; near 0 otherwise. The Sacks analogue of the Square mode's polynomial_concentration, restricted to the dominant angular direction.
Peak wedge count / (median wedge count + 1). Captures whether marks pile up on a single dominant ray (the Ulam/Sacks signature: perfect squares on +x, prime quadratics on specific spirals). Strongly correlated with axial_excess (intra-Sacks r ≈ +0.80), which is the redundancy that justifies the atlas exclusion.
A sister metric fft_anisotropy (diagonal-vs-cardinal 2D FFT wedge ratio on the rasterized Sacks grid) is class-only — kept in compute_metrics for direct callers but dropped from the atlas by the same class-level atlas_exclude that applies to Square mode (r = +0.90 with Spectral Analysis:high_freq_ratio).
Practically: it doesn't, at the framework's signal lengths. The geometry sits in the codebase as a renderer-only lens — the Sacks render is genuinely informative for human eyes (perfect squares pop out as a stripe on the +x ray), but the four scalar metrics don't separate sources strongly enough to earn an atlas slot. If the framework ever moves to longer signal lengths where the Sacks density grows enough to fill more angular wedges, these metrics become candidates again.