CAP-ROC v0.2 — Capacity-Constrained ROC Feasibility (delta gate)

CAP-ROC is a deployment feasibility gate for rare-event detectors: it couples ROC operating points (TPR/FPR) to human review capacity so you can answer one question before you ship:

Can humans keep up with the alert stream at this operating point?

It includes:

• an expectation gate (average load stays within capacity), and
• an optional δ-level overload gate (tail-risk protection) using exact Poisson tail inversion.

---

Why CAP-ROC exists

Alerting systems often “work” on paper (AUC/accuracy look great) and still fail in reality: rare events (p small) + modest FPR + high volume (R) ⇒ alert fatigue, backlog, ignored true alerts.

CAP-ROC makes human capacity a hard constraint, not an afterthought.

---

The expectation gate (one line)

Define expected alert rate per time unit:

[ A = R,[p,TPR + (1-p),FPR] ]

A system is capacity-stable (in expectation) if:

[ A \le C ]

Where:

• R = incoming item rate (items / time)
• p = base anomaly rate Pr(anomaly)
• TPR = Pr(alert | anomaly)
• FPR = Pr(alert | normal)
• C = sustainable human review capacity (alerts / time)

Equivalent (solve for max allowable false positives):

[ FPR_{\max} = \frac{C/R - p,TPR}{1-p} ]

If C/R - p*TPR < 0, you're over capacity even with FPR = 0.

---

The δ (delta) gate (tail-risk protection)

The expectation gate controls the average load. The δ gate controls the probability of “bad hours/days.”

Model alerts per time unit as:

[ \lambda = A = R,[p,TPR + (1-p),FPR] ]

[ N \sim \mathrm{Poisson}(\lambda) ]

Choose a risk target δ (e.g., 0.01) and require:

[ \Pr(N > C) \le \delta ]

Define (\lambda_{\max}(\delta, C)) as the largest mean such that (\Pr(\mathrm{Poisson}(\lambda_{\max}) > C) \le \delta). Then the δ gate requires:

[ A \le \lambda_{\max}(\delta, C) ]

This implies a δ-level bound on false positives:

[ FPR_{\max}^{\delta} = \frac{\lambda_{\max}/R - p,TPR}{1-p} ]

Implementation note: For small capacities (e.g., C ≤ 10), normal approximations can be inaccurate. The reference tool computes (\lambda_{\max}) by exact Poisson tail inversion (binary search).

---

What’s in this repo

• src/
  • cap_roc_tool.py — reference CLI calculator (supports --delta)
• examples/
  • cap_roc_examples.txt — example runs and notes

---

License

Non-commercial redistribution is allowed only as an unchanged copy. No derivatives. Commercial use (including sale/monetization) requires a separate paid license from the author.

See LICENSE.

---

Quickstart

Requirements

• Python 3.9+ (no external packages)

Run the calculator (expectation gate)

python src/cap_roc_tool.py --R 20 --p 0.01 --tpr 0.85 --fpr 0.30 --C 4


### Related SPARK-NITT standards

- NITT — digital identity standard  
  https://github.com/SPARK-NITT/nitt-digital-identity-standard
- IRST — transparency for recursive systems  
  https://github.com/SPARK-NITT/IRST-Institute-for-Recursive-Systems-Transparency
- HRIS — coherence-centered refusal (HRIS 3.2.4(b))  
  https://github.com/SPARK-NITT/HRIS-Human-Recursive-Integrity-Standard
- CTGS — consumer transparency governance  
  https://github.com/SPARK-NITT/ctgs-consumer-transparency-standard