Synthesizing to a target. You asked for one model combining all three behaviors, tuned to 30 intrinsic dimensions at 3.2% curvature. Two honest constraints shaped it. First, dimensionality is bounded by token count minus one — 30 dims is impossible at 28 tokens, so this runs at N=48. Second, dimensionality and curvature aren't dials; they're measured outputs of the mixture, so hitting a target means searching the tree/noise weights until the geometry lands on it. The search found tree=0.09, noise=0.52 → dim 30.5, curvature 3.25%. The orange dot in the plane is the current mix; the crosshair is your target; hit SYNTHESIZE to snap there, or drag the sliders and watch the dot travel the frontier.
The caveat the search exposed. Curvature and dimensionality pull against each other: dimensionality wants noise, curvature (real, hierarchical) wants structure. Worse, the frontier has two branches. The green clean branch (low noise) is where 3.2% reflects genuine mild hyperbolic structure from the hierarchy. The red branch (high noise) shoots curvature up to 30%+ — but that's distortion, not hierarchy: noisy distance estimates simply refuse to lie flat, and the negative-eigenvalue metric can't tell meaningful curvature from measurement noise. The target sits on the clean branch, which is why the synthesis is honest — 3.2% there is real, mild, structural curvature, not inflated noise.
Amber tier: modeled attention, but the geodesics, MDS, and eigendecomposition are exact and run live on every slider change. The 3D shows the synthesized manifold — a genuinely high-dimensional (~30-D) fuzzy cloud with faint hierarchical structure, of which only the top 3 dimensions are drawable. That it looks like a blob rather than a clean shape is the honest render of a 30-dimensional geometry: it can't be faithfully shown in 3, and pretending otherwise would be the lie the whole thread has been avoiding.