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A single adaptive noise window can confirm its own active label. A fast window reacts to recent changes; a slow window represents the baseline. Their disagreement can trigger a change-point alarm, but an unlock certificate should also use evidence that did not create the threshold being challenged.
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Chapter contents
Choose the route closest to the decision, or read from Chapter 1. The chapters remain open; these grouped routes are a navigable index.
Route 1 · Define the decision
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE DECISION
1. Two windows answer different questions
The fast estimator asks what variability looks like now. The slow estimator asks what the stable baseline has been. A large ratio can signal a noise burst, a real manifold transition or contamination from a rejected local step.
Neither window proves a change by itself. The estimates may share observations and therefore be strongly dependent; thresholds must be calibrated to that design.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. An alarm is not yet an unlock
A change-point alarm says the residual process no longer looks stable under the monitoring model. Unlocking abandons or suspends a local manifold label and can change the optimisation path.
Require a second gate from full-space stationarity, feasibility, held-out residuals or a disjoint data stream. This makes the escape less vulnerable to the same feedback loop that distorted the noise window.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. A dual-window ledger
Suppose the fast window uses 8 accepted steps and the slow window uses 40 earlier accepted global steps.
| Iteration | Fast sigma | Slow sigma | Ratio | Independent gate |
|---|---|---|---|---|
| 120 | 0.003 | 0.004 | 0.75 | Pass |
| 128 | 0.006 | 0.004 | 1.50 | Pass |
| 136 | 0.011 | 0.0042 | 2.62 | Fail |
| 144 | 0.013 | 0.0045 | 2.89 | Fail |
The slow window should exclude or downweight the same transition segment; otherwise both estimates can drift together and hide the change.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Worked unlock certificate
Use an illustrative certificate: fast-to-slow ratio above 2.5 twice, cumulative-sum statistic above 0.020 and independent gate failure twice. All conditions must occur within a six-check horizon.
| Check | Ratio condition | CUSUM condition | Independent gate | Action |
|---|---|---|---|---|
| 1 | No | No | Pass | Stay local |
| 2 | Yes | No | Pass | Watch |
| 3 | Yes | Yes | Fail once | Quarantine acceleration |
| 4 | Yes | Yes | Fail twice | Unlock and reset |
The constants are design values to validate. A safety-critical feasibility failure may justify immediate fallback without waiting for the full certificate.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Calibrate alarms under dependence
Overlapping windows, autocorrelated iterates and stochastic gradients change the false-alarm rate. Use block resampling, simulation under a stable regime or a time-uniform boundary matched to the dependence assumptions.
Do not treat repeated checks as independent tests. If thresholds were tuned on the same run, evaluate performance on held-out problems or fresh seeds.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Reset without erasing the audit trail
On unlock, freeze local acceleration, quarantine the fast window and rebuild the baseline from accepted global steps. Preserve the old estimates and trigger path for diagnosis.
A warm restart can keep safe problem information while clearing self-reinforcing label memory. State exactly which variables survive the reset.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: sparse supports
Transfer the method to support identification in sparse optimisation. A fast residual window may jump when one coefficient should leave the active set, while the slow window still reflects the previous support.
Use a full-gradient optimality residual on inactive coordinates as the independent gate. After unlock, require fresh global steps before certifying the revised support.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect the identification branch
The false-lock guide defines the feedback loop and basic escape state. The adaptive-hysteresis guide develops normalised widths, floors and ceilings.
Use the degenerate-manifold guide when the assumptions behind finite identification may fail.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
Two windows are not independent merely because their lengths differ. Shared data, shared iterates and shared model error can make agreement misleading.
A change-point alarm detects a statistical departure under its model; it does not identify which manifold is correct or prove that a topological transition occurred.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
Archive window membership, robust scale estimators, overlap, baseline updates, alarm statistics, calibration simulations, independent gates, quarantine events, reset state and post-unlock recovery.
Measure false alarms, missed transitions, detection delay, time spent in fallback and objective progress across stable, switching, burst-noise and degenerate cases.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Should the windows overlap?
They may, but overlap increases dependence. A lagged or disjoint baseline is easier to interpret when enough data exist.
Is CUSUM required?
No. It is one cumulative change detector. Any alarm needs a calibrated null model and a clear relationship to the unlock gate.
What makes a gate independent enough?
It should not be mechanically determined by the same filtered residuals and label. Full-space or held-out evidence is often more informative than nominal statistical independence.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Page introduced continuous inspection schemes underlying cumulative-sum change detection. Adams and MacKay develop Bayesian online change-point detection. Sun and colleagues give finite manifold-identification conditions for proximal methods. The dual-window certificate above is an original safeguard and does not replace those statistical or convergence assumptions.
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