Delay-Constrained Anomaly-Aware Consensus in Heterogeneous Clock Networks
View the Project on GitHub threehouse-plus-ec/admec-clock-consensus
Status: Work Package 1 complete. DG-1 closed. No outstanding criteria.
WP1 defined and calibrated a pointwise observable — information content (IC) — for quantifying distributional inconsistency in clock data, established its behaviour under controlled null conditions, and positioned it relative to established figures of merit. IC is model-free at the signal level but calibrated under specific null noise assumptions.
Information content (IC). A single clock produces N readings x₁, …, x_N over time, each with a declared uncertainty σ₁, …, σ_N. The Gaussian-mixture background density is constructed from this clock’s own readings:
P(y) = (1/N) Σᵢ 𝒩(y; xᵢ, σᵢ)
Each component 𝒩(y; xᵢ, σᵢ) is a Gaussian centred on reading i with width set by its declared uncertainty. The mixture P(y) represents the clock’s empirical self-distribution — the background against which each individual reading is evaluated.
The interval probability for reading k is the probability mass assigned by P(y) to the interval [x_k − σ_k, x_k + σ_k]:
p_k = ∫ from x_k − σ_k to x_k + σ_k of P(y) dy
evaluated analytically via the Gaussian CDF. The information content is:
I_k = −log₂(p_k) [bits]
High IC indicates that reading k is unlikely given the clock’s own track record — the interval around x_k captures little probability mass under the mixture background. The interval width is set by σ_k, so IC depends both on the mixture background (which determines the density being integrated) and on the declared local uncertainty scale (which determines the integration window). This dual dependence is the structural reason for the σ-sensitivity results reported below.
IC requires no parametric model for the anomaly itself. The Gaussian-mixture form of P(y) is part of the IC definition; what is established through null calibration is the expected baseline AIPP and the percentile thresholds under specific noise models.
Mathematical note on scope. The function compute_ic(values, sigmas) is mathematically agnostic: it takes any array of N values with N declared uncertainties and returns IC per point. The same computation applies whether the N points are temporal readings from one clock (longitudinal — the WP1 calibration mode) or simultaneous readings from N clocks (cross-sectional — as used in the Entry 005 comparison example). WP1 defines and calibrates IC in the longitudinal mode. Entry 005 applies it cross-sectionally to demonstrate its properties relative to χ² and Huber; this is a legitimate use of the same mathematics but operates in the mode that WP2 will develop further. The extension to cross-clock comparisons requires a causal-ordering layer. In a distributed clock network, physical clocks produce readings (frequency measurements with declared uncertainties) — IC operates on these. But clocks do not share a global time axis; before comparing their IC profiles, one must establish which readings were available to which node and in what order. Lamport timestamps (Lamport 1978) provide this causal scaffold: they are logical counters that enforce “A happened before B” ordering across distributed events, without reference to physical time. They do not produce data that IC analyses; they organise the exchange of IC-classified physical clock data across the network. This causal-ordering layer is a WP2 concern.
AIPP (average information per point). The ensemble-level aggregate over one clock’s N readings:
AIPP = (1/N) Σ_k I_k
Under the Gaussian null with σ_data = σ_declared = 1 — where σ_data is the standard deviation of the true data distribution and σ_declared is the per-point declared uncertainty — as N → ∞ the mixture P(y) converges to 𝒩(0, σ = √2) — the convolution of the data distribution with the Gaussian kernel — giving AIPP → 1.25 bit. The earlier estimate of 0.55 bit (based on the self-contribution alone) was incorrect; see logbook Entry 001 for the derivation and correction.
δ_min (effect-size thresholds). Minimum temporal-structure effect sizes that separate structured from unstructured anomalies, computed over a trailing window of W = 20 time steps within a single clock’s reading sequence. Two statistics:
| Trailing variance slope: linear regression of log(running variance) over sub-windows of size W/4 within the trailing window. δ_min = 0.2105, calibrated as 3 × median( | slope | ) under the hardest null (Student-t with ν = 3 — heavy tails create sporadic variance spikes in sub-windows). |
| Trailing lag-1 autocorrelation: sample autocorrelation at lag 1 over the trailing window. Rising autocorrelation is a classical early-warning indicator of critical slowing down (Scheffer et al. 2009, Dakos et al. 2012); here it is used as a classification heuristic, not a detection theory. δ_min = 0.8703, calibrated as the 95th percentile of | autocorrelation | under the hardest null (random walk — inherently non-stationary, with strong serial dependence). |
The two statistics use different calibration methods because the variance slope is unbounded (so a multiplier-based threshold is natural), while autocorrelation is bounded to [−1, 1] (so a percentile-based threshold avoids exceeding the domain).
Calibration parameters. The AIPP threshold calibrations (Entries 001, 003) and σ-sensitivity analysis (Entry 002) use N i.i.d. draws per realisation (no temporal structure), with N ranging from 10 to 1000 and 200–300 realisations per condition. The δ_min calibrations (Entry 004) use single time series of length T = 200 with trailing window W = 20, at 300 realisations per noise model. These are different experimental designs reflecting the different statistical quantities being calibrated.
Calibration under nulls. AIPP converges to the theoretical limit within 5% at N ≥ 100 and within 1% at N ≥ 200. The 95th-percentile anomaly-detection threshold is stable within a factor of 1.24× across all ten tested null models: Gaussian i.i.d., heteroscedastic Gaussian, Student-t (ν = 3, 5), AR(1) (ρ = 0.7, 0.9), symmetric Pareto (α = 2.5, 3.0), fractional Gaussian noise (Hurst exponent H = 0.9, via Davies-Harte circulant embedding), and random walk (cumulative sum of i.i.d. Gaussian increments). The pre-registered ×1.5 stability criterion is satisfied with margin. Finite-N bias follows AIPP(N) ≈ 1.248 − 0.913/N + 1.02/N² and is below 1% for N ≥ 75; for smaller samples, the correction −0.9/N bit is available.
Sensitivity analysis. IC is robust under random ±20% perturbation of declared uncertainties (AIPP shift +1.2%) and under systematic overestimation (+20% → AIPP shift −12.7%). Systematic underestimation (−20%) produces a 19.3% AIPP shift, exceeding the pre-registered 15% bound. This is the single identified failure in WP1. IC is not invariant under rescaling or misestimation of declared uncertainties; it inherits any systematic bias present in σ. The asymmetry is expected: underestimating σ narrows the integration interval, capturing less probability mass from the mixture background, so AIPP rises. The mitigation adopted for WP2 is worst-case threshold calibration — procedural, not intrinsic.
Temporal-structure thresholds. The δ_min values were calibrated from null distributions of variance slope and lag-1 autocorrelation across all ten noise models. Sanity checks confirm that realistic injected signals — sinusoidal drift (amplitude 2σ, period 50 steps) and linearly growing variance (σ(t) = 1 + 0.03t) — exceed their respective δ_min values. The three-way classification rule is fully specified.
Positioning against established figures of merit. A controlled comparison (20 clocks, T = 200 steps, one clock with linear drift 0.02/step, seed 2026) showed that the per-point squared normalised residual (χ²) amplifies deviations quadratically, Huber loss (Huber 1981; tuning constant c = 1.345) bounds their influence linearly beyond its threshold, and IC compresses them logarithmically through the probability transform. Note: this comparison uses IC in the cross-sectional mode (N clocks at each time step), previewing the WP2 usage; see the mathematical note above.
| Property | χ² / residuals | Huber loss | Allan deviation | IC / AIPP |
|---|---|---|---|---|
| Scale behaviour | quadratic | linear beyond threshold | N/A (variance scaling) | logarithmic (via probability) |
| Dependence on σ | explicit, quadratic | explicit | implicit (via noise model) | explicit, nonlinear |
| Additivity | yes (sum of squares) | yes | no (windowed statistic) | yes (information units) |
| Tail sensitivity | very high | bounded | not designed for outliers | distribution-dependent |
| Temporal structure | none | none | scaling only | none (pointwise) |
IC does not replace these measures. It provides a complementary quantity: a scale-normalised, additive measure of self-consistency with respect to declared uncertainties. Unlike χ², it is directly interpretable in probabilistic terms; unlike robust losses, it does not encode a decision policy; and unlike Allan deviation, it does not rely on specific noise-scaling assumptions. Allan deviation (Allan 1966) characterises temporal frequency stability — it is not a pointwise diagnostic and is not directly comparable to IC, χ², or Huber on a per-sample basis. IC’s primary role is to separate the detection of inconsistency from its subsequent interpretation.
WP1 does not demonstrate that IC improves timekeeping or enables better consensus performance. It does not provide a complete taxonomy of anomaly types beyond the binary structured/unstructured distinction defined by δ_min. It does not resolve whether the separation of detection and interpretation — the architectural choice being tested — yields measurable advantage at the network level. These are WP2 questions.
IC as defined and calibrated in WP1 is a single-clock, longitudinal observable. Extending it to a network — where IC profiles from multiple clocks are compared — requires a causal-ordering layer that determines which physical clock readings were available to which node at which moment. Lamport timestamps provide this ordering without assuming a shared physical time axis; they structure the exchange of data, not the data itself. This is the conceptual step required for WP2. The clock-network context that motivates this extension is the emerging regime of optical clock networks with sub-10⁻¹⁸ instability (Lisdat et al. 2016, Bothwell et al. 2022), where effects previously negligible become detectable and the distinction between informative and uninformative anomalies becomes worth testing. WP1 establishes the per-clock foundation; WP2 tests whether the network-level architecture built on it adds value.
IC remains a pointwise observable. It does not distinguish between random and structured deviations — that distinction requires the temporal-structure layer (variance slope, lag-1 autocorrelation) calibrated via δ_min. Alone, IC can flag that a clock reading is anomalous but cannot say whether the anomaly is a one-off outlier or a persistent drift.
STABLE: IC < threshold_95
STRUCTURED ANOMALY: IC ≥ threshold_95 AND (|var_slope| > 0.2105 OR |autocorr| > 0.8703)
UNSTRUCTURED ANOMALY: IC ≥ threshold_95 AND |var_slope| ≤ 0.2105 AND |autocorr| ≤ 0.8703
where threshold_95 is the 95th-percentile AIPP value from the null calibration (Entry 001), recalibrated under worst-case σ conditions (systematic −20% underestimation, Entry 002). The variance slope and autocorrelation are computed over a trailing window of W = 20 time steps within a single clock’s reading sequence.
Note: the threshold_95 used here is derived from the AIPP distribution (average over N readings). When applied to classify individual readings in WP2, this threshold must be replaced by the corresponding percentile of the per-reading I_k distribution, which has higher variance. This recalibration is a WP2 task.
DG-1 closed with one identified limitation: systematic σ-underestimation exceeds the pre-registered 15% sensitivity bound (actual: 19.3%). The 15% criterion was not relaxed post hoc. The failure is recorded, and the mitigation (worst-case threshold calibration) is adopted for WP2. All other DG-1 sub-criteria pass.
WP2 will evaluate whether the separation of inconsistency detection (IC) and temporal-structure classification (δ_min) enables improved handling of anomalous clocks at the network level. This requires extending the single-clock IC observable to a network context with a causal-ordering layer (Lamport timestamps) that determines which physical clock readings are available to which node and when — structuring the exchange of IC-classified data without assuming a shared physical time axis. The comparison set includes frequentist weighted averaging, Huber M-estimation (Huber 1981), Bayesian online changepoint detection (Adams & MacKay 2007), and an interacting multiple model filter (Blom & Bar-Shalom 1988). Clock noise parameters are drawn from published hydrogen maser characterisations (Panfilo & Arias 2019). Both positive and negative results will be published.
| Artefact | Location |
|---|---|
| Logbook Entries 001–005 | logbook/ |
| IC implementation | src/ic.py |
| Noise generators | src/noise.py |
| Temporal-structure statistics | src/temporal.py |
| Comparison functions | src/comparison.py |
| Test suite (88 tests, 86 passing) | tests/ |
| Archived data (entries 001–005) | data/ |
| Outreach document | docs/outreach.md |
Author: U. Warring, Physikalisches Institut, Albert-Ludwigs-Universität Freiburg. AI tools (Claude, Anthropic) were used for structural editing and code prototyping. All scientific content, decisions, and claims are the sole responsibility of the author.