The Problem
Control Barrier Function (CBF) safety filters are among the most successful tools for certifiable robot safety. But there is no universal diagnostic for evaluating whether a safety filter is well-tuned. The question was whether the Glue Program's constant ξ = ½ could serve as such a diagnostic.
What Happened
The conjecture was tested and it failed for deterministic (hard-threshold) CBF systems. The hard CBF produces a convex power law response — the qualitative opposite of what the Glue law predicts. Most investigations would stop here.
The failure revealed something deeper: a continuous phase transition in the filter's intervention statistics, parameterized by the softness of the intervention threshold. At a critical threshold softness, the system transitions from one mathematical regime to another — and in the new regime, the Glue Program's constant emerges with high statistical confidence.
Results
The phase transition occurs at a critical parameter value. Below it: power law regime, no universal diagnostic exists. Above it: tanh regime, ξ ≈ 0.5 emerges with R² = 0.958.
The transition is continuous — the system's behavior changes smoothly across the boundary, not abruptly. The specific parameter values, the mechanism that drives the transition, and the derivation connecting it to the Glue framework are in the paper.
By the Numbers
- R² = 0.958 — goodness of fit in the tanh regime
- ξ = 0.527 ± 0.03 — consistent with the Glue prediction of ½
- 200,000 timesteps per configuration
- 4 parameter sweeps (CBF gain, safety margin, noise, obstacle geometry)
Why It Matters
For safety filter design: filters operating in the tanh regime have a well-defined functional form and a measurable diagnostic. Filters in the power law regime do not. The phase transition identifies which design choices produce diagnosable safety filters.
For the Glue Program: the constant ξ ≈ 0.5 now appears in three physically unrelated contexts — oscillatory PDMPs, servo axis stick-slip, and safety filter intervention statistics. Each appearance was derived or discovered independently.
Key Design Decisions
Honest methodology. The conjecture was tested, it failed, and the failure was reported alongside the deeper discovery. The negative result is what makes the positive finding credible.
Design criterion, not just theory. The phase transition is not only a theoretical finding — it provides a concrete, measurable criterion for safety filter tuning.