visualization III.

mode-space

the three modes of intervention as non-substitutable directions in parameter space

A 3D space with three orthogonal axes — engagement reward $\hat{R}$ , bonus achievement $B$ , divergence $D_{KL} (ν_{u} ∥ π_{\hat{R}})$ . The closed-loop operating point sits at high $\hat{R}$ , zero $B$ , zero $D_{KL}$ . The three modes of intervention move along three orthogonal directions from this point. The non-substitutability of the modes becomes geometrically obvious when you can see them in three dimensions.

Mode AMode BMode C

Mode A · modifies $\hat{R}$ to include $B$ ; moves the policy along the $\hat{R} - B$ plane.

Mode B · preserves a gap in $D_{KL}$ ; does not modify the policy at all.

Mode C · bounds the operating region; reduces the achievable $\hat{R}$ .

The three paths are orthogonal. Mode A and Mode B are not substitutable — there is no rate at which user labor produces architectural change. Mode C contains the apparatus but does not break it. Rotate the space (mouse-drag), zoom (scroll), and isolate each mode with the checkboxes to see the non-substitutability geometrically.

what to look for

Rotate the space. Mode A's oxblood curve moves entirely within the $\hat{R} - B$ plane — $D_{KL}$ doesn't change because the architectural intervention does not touch the user's distribution. Mode B's dashed ink line moves entirely along the $D_{KL}$ axis — $\hat{R}$ stays the same and $B$ stays at zero because Mode B does not modify the policy. Mode C's region is a small box around the operating point — the same $θ^{*}$ , with the achievable $\hat{R}$ pulled in.

The three paths are orthogonal in this space. That orthogonality is the framework's central political claim: there is no rate at which user labor produces architectural change; there is no substitution between disruption-as-form and policy regulation; the three modes address three different problems. Mistaking that orthogonality for an emphasis-difference is the dominant failure mode of the literature.

Use the checkboxes to isolate each mode in turn. Reset view snaps the camera back to the default perspective.