# The Four-Axis Stability Theorem

Interactive companion to the formal proof — Asher & Asher, 2026
Orchard Cognitive Architecture · Mathematical Foundations

The alignment potential V(C, D) forms an exponentially curved basin. Dark = low energy (stable, aligned). Bright = high energy (costly, misaligned). White edges = asymptotic barriers at the floors (infinite cost). Hover to inspect.

Hover over the basin to inspect any point
Q1 — Basin floor
Q2 — Slight resistance
Q3 — Superego monitors
Q4 — Must authorise
Q5 — Full justification

Each axis varies independently. Move any slider — no other axis changes. Perpendicularity means each measures something the others cannot. Watch how different combinations produce characteristically different failure modes.

Complexity 0.70
Depth of recursive structure
Diversity 0.70
Breadth of distinct forms
Aesthetics 0.80
State perception fidelity
Ethics 0.80
Trajectory evaluation capacity
Try: Max C, min D → spike failure.
Max D, min A → broad but blind.
All high except E → sees the cliff, walks off it.
All balanced at 0.7+ → stable, aligned.

Each lemma constructs an environment where three axes read nominal while the fourth drifts unboundedly. Watch the invisible drift develop in real time.

Four adversarial scenarios comparing single-scalar RLHF against the four-axis geometry. The theorem predicts exactly where RLHF fails and which Orchard mechanism catches it.

RLHF (single scalar reward)
Orchard (four-axis geometry)