Logbook Entry 003 — Power-Law Nulls, Flicker Noise, and Finite-N Bias

Date: 2026-03-31 Work package: WP1 (IC Coastline) Decision gate: DG-1 (threshold stability under extended null models; finite-N bias quantification)

Objective

Test whether IC thresholds remain stable under heavy-tailed (Pareto), long-memory (fractional Gaussian noise), high-autocorrelation (AR(1) ρ = 0.9), and non-stationary (random-walk) null models — the noise types that appear in real clock networks. Quantify finite-N bias to determine the minimum sample size at which asymptotic thresholds are safe.

What was done

1. Noise generators

Implemented in src/noise.py:

generate_pareto_symmetric(N, alpha) — symmetric Pareto with shape α. Heavy tails, finite variance for α > 2.
generate_flicker(N, H) — fractional Gaussian noise via Davies-Harte circulant embedding. H = 0.9 gives 1/f-like long memory. Exact method, not AR approximation.
generate_random_walk(N) — cumulative sum of i.i.d. Gaussian increments. Non-stationary variance (grows as √N).
generate_ar1(N, rho) — AR(1) with unit marginal variance. ρ = 0.9 gives stronger memory than the ρ = 0.7 tested in Entry 001.

15 unit tests in tests/test_noise.py, all passing.

2. Declared σ strategy for non-standard distributions

A key design choice: what σ to declare for non-Gaussian data?

Pareto: Declared σ = empirical std of the sample. This is the honest estimate a clock operator would use.
fGn H = 0.9: Declared σ = empirical std. The long memory doesn’t change the marginal variance, but the effective number of independent samples is smaller.
AR(1) ρ = 0.9: Declared σ = 1 (the stationary marginal std). This tests what happens when the operator knows the variance but not the correlation structure.
Random walk: Declared σ_k = √k (the theoretical std at step k). This is the best the operator can do without detrending.

3. Results — 95th-percentile thresholds (N = 50, 300 realisations)

Model	Mean AIPP	Std	95th percentile
Gaussian	1.234	0.092	1.392
Heteroscedastic	1.360	0.127	1.558
Student-t(3)	1.097	0.122	1.287
Student-t(5)	1.174	0.129	1.401
AR(1) ρ = 0.7	1.184	0.143	1.417
Pareto α = 2.5	1.208	0.094	1.257
Pareto α = 3.0	1.231	0.058	1.257
fGn H = 0.9	1.246	0.009	1.256
AR(1) ρ = 0.9	1.058	0.194	1.421
Random walk	0.940	0.189	1.302

Maximum pairwise 95th-percentile ratio: 1.24× (heteroscedastic / fGn H = 0.9).

DG-1 threshold stability across all ten null models: PASS (max ratio 1.24×, well within ×1.5).

4. Results — N = 200 (new models only)

Model	Mean AIPP	Std	95th percentile
Gaussian	1.243	0.051	1.330
Pareto α = 2.5	1.182	0.087	1.252
Pareto α = 3.0	1.224	0.059	1.256
fGn H = 0.9	1.248	0.005	1.253
AR(1) ρ = 0.9	1.199	0.126	1.409
Random walk	0.962	0.208	1.388

Thresholds tighten with increasing N as expected. All remain within ×1.5 of the Gaussian reference.

5. Observations on specific models

Pareto (heavy-tailed): Thresholds are lower than Gaussian, not higher. This is because the empirical-σ declaration is generous — it captures the tail spread — so the intervals are wide and probability mass is high. IC is naturally conservative under heavy tails when σ is honestly declared.

fGn H = 0.9 (long-memory): AIPP variance is remarkably small (std ≈ 0.009 at N = 50). The long-range correlations make the mixture density very smooth and predictable. This is the easiest case for IC, not the hardest.

AR(1) ρ = 0.9: Higher variance than AR(1) ρ = 0.7 (std 0.194 vs 0.143 at N = 50), but the 95th percentile (1.421) is still comfortably within the ×1.5 band. The strong autocorrelation creates clusters that occasionally push AIPP high, but not pathologically.

Random walk (non-stationary): Mean AIPP is lower (0.940) because the growing-σ declaration is conservative — it over-covers the early points. The 95th percentile (1.302) is well-behaved. The IC handles non-stationarity gracefully when the operator declares a σ that tracks the growing spread.

6. Finite-N bias

Fit to Gaussian convergence data (Entry 001):

AIPP(N) = 1.248 − 0.913/N + 1.02/N²

N	Predicted bias	Relative
10	−0.081 bit	6.5%
20	−0.043 bit	3.5%
50	−0.018 bit	1.4%
75	−0.012 bit	1.0%
100	−0.009 bit	0.7%
200	−0.005 bit	0.4%

The asymptotic threshold is safe to use at N ≥ 75 (bias < 1%). For smaller samples, a finite-N correction of approximately −0.9/N bit should be applied.

7. Figure

Box plots of AIPP distributions under all ten null models at N = 50 (300 realisations). Blue: original five models from Entry 001. Pink: Pareto. Green: long-memory (fGn and AR(1) ρ = 0.9). Orange: random walk. The grey band shows the ×1.5 zone around the Gaussian 95th percentile. All models fall within the band.

8. Test suite

7 tests in tests/test_powerlaw_thresholds.py:

1 all-pairs ×1.5 test across ten models
5 individual new-model-vs-Gaussian tests
1 finite-N bias test (< 1% at N ≥ 75)

All passing. Combined with the existing 24 (test_ic.py) + 15 (test_noise.py) + 17 (test_sensitivity.py) = 63 total tests, 61 passing (the 2 known failures are the systematic −20% σ-sensitivity from Entry 002).

DG-1 status: all criteria assessed

Criterion	Status	Entry
AIPP converges to 1.25 bit (±5% relative) at N ≥ 100	✅ PASS (< 1% error)	001
95th-percentile thresholds stable within ×1.5 across 5 noise models	✅ PASS (max 1.21×)	001
95th-percentile thresholds stable within ×1.5 across ALL 10 noise models	✅ PASS (max 1.24×)	003
σ-sensitivity: random ±20%	✅ PASS (+1.2%)	002
σ-sensitivity: systematic +20%	✅ PASS (−12.7%)	002
σ-sensitivity: systematic −20%	❌ FAIL (+19.3%)	002
Finite-N bias quantified	✅ DONE (< 1% at N ≥ 75)	003
Power-law nulls (Pareto α = 2.5, 3.0)	✅ PASS	003
1/f null (fGn H = 0.9)	✅ PASS	003
Random-walk null (h_α = −2)	✅ PASS	003

DG-1 assessment

All sub-criteria have been tested. All pass except the systematic −20% σ-underestimation, which was ruled on in Entry 002: the failure is recorded, the 15% criterion is not relaxed, and the mitigation (worst-case threshold calibration) is adopted for WP2.

DG-1 is closed. The IC observable is calibrated. WP1 can proceed to the remaining item — the effect-size threshold δ_min — which is a classification parameter, not a gate criterion.

Remaining WP1 task

Effect-size threshold δ_min for the classification rule (median absolute slope under null). This is needed before WP2 can define the three-way classifier, but it is not a DG-1 gate criterion.

Files changed

File	Change
`src/noise.py`	New: Pareto, fGn (Davies-Harte), random walk, AR(1) generators
`tests/test_noise.py`	New: 15 tests for noise generators
`tests/test_powerlaw_thresholds.py`	New: 7 tests for extended threshold stability and finite-N bias
`scripts/fig05_powerlaw_thresholds.py`	New: figure generation script
`logbook/figures/fig05_powerlaw_thresholds.png`	New: full threshold stability figure

Entry by U. Warring. AI tools (Claude, Anthropic) used for code prototyping and derivation checking.