Metrology
The claim that the uncertainty of a time-averaged measurement reaches a minimum set by the ratio of averaging time to drift time.
Does averaging a measurement have a floor?
Test of hypothesis 158, Volume XII — "Averaging has a floor" · run 30 October 2026 Published proposal: <https://incoprea.com/research-library/original-theories-vol12-20261030/>
The result in one paragraph
No. Over 30 years of BIPM Circular T the Allan deviation of [UTC-UTC(k)] keeps falling for most laboratories: 63 of the 90 that qualify (70.0%) reach their smallest value at the longest averaging time the record allows and never turn up (Figure 1). Another 27 (30.0%) do turn up, but usually as a shallow bend — only 5 rise by as much as a factor of two — and where a turn occurs it sits between one and seven years of averaging, not at one characteristic time (Figure 2). The refutation criterion fixed in advance, "sigma(tau) falls monotonically to the longest available tau for most laboratories", is met. What survives is narrower: among the ten laboratories where a post-minimum slope can be fitted at all, all ten are positive (median +0.85), and the ordering of the minimum across laboratories tracks the drift time predicted from each clock's own drift rate (Spearman rho = 0.74, p = 0.0002, n = 24) — but that predicted time is on average 6,800 times longer than the observed one and, for all 24, longer than the record (Figure 3). Drift orders the minima; it does not set them.
Design
Claim verbatim, predictions and every criterion are in hypothesis.md, fixed before the first number was computed: P1 the deviation falls, reaches a minimum, then rises; P2 the minimum sits at a drift-related averaging time; P3 the post-minimum slope is positive.
Data: every BIPM Circular T issue from 100 (1996 March) to 465 (2026 September), 366 issues fetched sequentially, parsed, reduced to one series table; raw text was hashed and discarded. That is 166,767 distinct (laboratory, epoch) values across 117 laboratory codes, of which 90 have the required 500 five-day points.
Estimator: the overlapping Allan deviation of phase data, sigma_y^2(tau) = (1/2 n tau^2) * sum_k [x(t_k+2tau) - 2x(t_k+tau) + x(t_k)]^2, over every triple present in the record, on a log-spaced grid from 5 to 3,000 days. sigma_y is fractional frequency. Second differences cancel a constant frequency offset, so the estimator is blind to a linear trend in phase by construction; S1 demonstrates that rather than assuming it.
Parse check. 359 of 365 consecutive issue pairs share an epoch, giving 23,111 values quoted twice. 22,318 agree exactly — 96.57% — and 22,322 within 0.1 ns. All 793 disagreements sit at the boundary epoch that BIPM revises when later data arrive (median revision 1 ns, 99th percentile 524 ns, largest 1,345 ns); the later value is kept.
P1 — refuted for the majority
| interior minimum (turns up before the end) | 27 (30.0%) |
|---|---|
| falls to the longest available tau | 63 (70.0%) |
| interior minimum rising at least 2× | 5 (5.6%) |
Median tau_min 1,915 days (IQR 1,305–2,320; range 340–2,320), median sigma_min 3.5 × 10⁻¹⁵. For the 63 falling curves the median log-log slope over the whole curve is −0.65 (IQR −0.78 to −0.54): averaging keeps paying, though more slowly than the τ⁻¹ of a perfectly bounded phase.
NIST, USNO, PTB, OP and SU all fall to about 2,800 days; four of the five have no interior minimum, and NIST's ends only 4.7% above its minimum. Their minima are lower than everyone else's (Mann–Whitney p = 0.032 on sigma_min) but occur at the same late times (p = 0.15 on tau_min).
P2 — the ordering survives, the level does not
For white frequency noise plus linear drift D, sigma_y^2 = a/tau + D^2 tau^2/2 gives a minimum at tau_opt = (a/D^2)^(1/3). With D fitted per series and a from the shortest tau, the 24 laboratories with well-determined drift show a strong rank association (rho = 0.74, permutation p = 0.00015) but a median observed:predicted ratio of 1 : 6,800; none is within a factor of three and all 24 predictions exceed the record. A minimum "set by" the drift time would sit on the identity line in Figure 3. None does.
P3 — positive, unresolved
All ten laboratories with enough points past the minimum to fit a slope have a positive one (median +0.85), as drift would produce. None has a 95% CI excluding zero: with a handful of correlated points past the minimum and df_effective = span/tau_min, every interval spans zero, the closest being MTC at p = 0.059. The pre-specified support criterion — at least half of all included laboratories showing an interior minimum, a positive slope and a CI excluding zero — is met by 0 of 90 (exact binomial p = 1.6 × 10⁻²⁷), and no test in the 10-member Benjamini–Hochberg family (the per-laboratory post-minimum slopes, the only declared family) survives.
Sensitivity
- S1 linear phase trend removed: identical, 27 of 90. The estimator never saw it.
- S2 quadratic trend removed (drift removed): minima 27 → 20, falling 63 → 70.
Some minima are drift; most are not.
- S3 first-difference estimator: 8 of 90. S4 rise ≥ 2×: 5 of 90.
- S5 sub-nanosecond era only (issues 184–465; the 84 earlier issues print whole
nanoseconds): 25 of 84 interior (29.8%), 59 falling — the verdict does not depend on the 1 ns quantisation of 1996–2003.
- S6 classification on a three-point median of log σ, which suppresses a single
noisy end point: identical, 27 of 90.
Limits
The test measures the steered residual [UTC-UTC(k)], not a free-running clock. A laboratory steering towards UTC keeps its phase bounded, and a bounded phase makes the Allan deviation fall roughly as τ⁻¹ however bad the underlying oscillator is. That is why 70% of curves fall, with slopes near −0.65: the record shows the control loop more than the physics of averaging, and cannot settle the claim for a free-running clock. Nothing beyond about 2,800 days is observable, so a floor further out is invisible. Step changes are endemic (73 of 90 series contain a five-day change above 100 ns) and inflate long-τ deviation in a way that mimics a rising floor without any smooth drift.
What would change the answer
Free-running data — TAI-TA(k) before steering, or a laboratory's own frequency measurements — showing minima at the drift-predicted time would convert P2's rank association into the level agreement it lacks. Removing the controller is the decisive experiment.
run.py rebuilds every table and figure from the cached parse, and --refetch re-downloads all 366 issues; SOURCE.md records every URL and hash.
Designed and executed with AI assistance (DeepSeek, via its coding agent); not peer reviewed. Observational data: associations only, no causal claim.
Figures
