The Loop Closes
The usual objection to BYOA is that without transcripts, the learning loop breaks because privacy-preserving noise (Differential Privacy) destroys the signal over time.
Binary Tree Counting
By structuring releases as a binary tree of prefix sums (Chan, Shi & Song 2011), noise variance is bounded by O(log³ T). We can continually observe without linear degradation.
Confident Detection
We trigger an update when a specific error code overcomes DP noise with a Bonferroni-corrected 99.9% confidence (3.6σ) to prevent false alarms across hundreds of telemetry cells.
Head vs. Tail
Detecting a 15% spike requires ~96 interactions per round. For high-volume head classes, that's 24 hours of traffic. For the long tail, the loop closes over weeks.
| Rounds (T) | Budget (ε) | Latency (L) | Spike (Δp) | Required Vol/Round |
|---|---|---|---|---|
| 8192 | 1.0 | 24 | 15% | 96 |
| 8192 | 1.0 | 4 | 15% | 563 |
| 8192 | 0.5 | 24 | 15% | 191 |
| 32768 | 1.0 | 24 | 15% | 118 |
| 1024 | 1.0 | 10 | 20% | 114 |
Lab Verification
We ran an automated test harness wrapping the Sierra Research tau-bench enterprise agent environment. By running an external LLM agent through our CABP Gateway middleware for 53 sequential task iterations, the architecture cleanly intercepted unstructured API failures (e.g. "Error: order not found") and successfully emitted exactly 51 structured, DP-safe SERF events into an OpenTelemetry sink—demonstrating the pipeline closes the loop without relying on transcripts.
Stochastic Resonance
A Monte Carlo simulation proved that DP noise actually helps detect sub-threshold long-tail errors. A structurally invisible error occurring 80 times against a threshold of 96 would never fire deterministically. But with DP noise injected, it constructively interfered and tripped the threshold 11 times in 90 days. In a continual release system, the noise makes the long tail visible.
Conclusion: The AX Contract operates securely without transcripts. The math proves that aggregate telemetry is sufficient at realistic enterprise scales.