Every one of the 27,830 symbol-instances in the notation atlas is projected per-instance onto the 7-foundational basis using non-negative least squares, then catalogued along three independent axes: identity (which foundationals participate, in what dominance order), ratio (the percentage each carries, binned in 5% steps), and purity tier (the dominant share, binned P1–P5). Activity threshold is 1% — every contribution above the noise floor is preserved. The catalogue lets the geometry and signal speak; no external interpretation is imposed on the mix.
Vocabulary at every layer
(3D ∪ 4D) — the participants in dominance order, ignoring exact ratios. 53 in 3D, 49 in 4D, 47 shared.
Breadth distribution — k = number of foundationals active per instance
How many foundationals carry ≥ 1% of the signal in a single instance. Both dimensions show the same shape: most instances express three. The empirical ceiling is k=5; no instance in 3D or 4D expresses 6 or all 7 simultaneously. Pending higher-dimensional sweeps (5D, 6D) to determine whether k=6 or k=7 ever fill.
Purity tier — coherence of the dominant foundational
The fraction of total signal carried by the largest active foundational. P1 is a near-pure isolated expression. P5 is heavily disrupted, no clear leader. Same identity + same ratio can sit in different purity tiers — coherence is an independent axis from breadth.
Leaders and followers
Top identity signatures (3D vs 4D)
The participants in dominance order, ignoring exact percentages. Both dimensions are dominated by the same five signatures, in the same order, holding ~91% of all instances.
Stable behaviors — the empirical core
All 98 behaviors with ≥ 50 instances, ranked by population. Each row is a distinct (identity · ratio · purity) signature, with how many instances carry it, how clean the dominant signal is on average, the mean breadth (k), the typical decoherence position where it lives, and how many distinct shapes express it.
| # | Signature | n | 3D | 4D | Mean dom % | k | Mean dec | Topos |
|---|
3D ↔ 4D persistence
The dimension-specific behaviors are predominantly low-count rare combinations. The bulk of the vocabulary — the structural skeleton — is dimension-agnostic at 3D and 4D. This is held separately so the pattern can be tracked as 5D and 6D data arrive.