More Frequent Is Not Always More Accurate

An exact Navier–Stokes benchmark for delayed sample-and-hold observations

St0kes research manuscript · version 0.3 · 18 September 2026

Working manuscript for author review. Authorship, affiliation and venue have not been assigned. This is a reproducible benchmark note, not a submission-ready claim of a new estimation algorithm. The statements below are proved here; priority relative to the complete sampling literature remains unestablished.

Abstract

Reporting frequency and delivery delay are distinct aspects of observation quality. We construct an exactly solvable family of forced, incompressible two-dimensional Navier–Stokes shears to isolate their effects on a fixed sample-and-hold estimator. For the unit-frequency shear, fourfold denser nested sampling increases time- and space-averaged squared velocity error for every source phase whenever the nondimensional delay divided by π lies between approximately 0.43742 and 1.06258. At delay π, the dense estimator's error is 1/2 + 1/π, whereas the sparse estimator's error ranges from 1/4 to 3/4. The deterioration therefore does not require a specially chosen source phase. A frozen synthetic protocol evaluates 648 configurations spanning three exact shear families, twelve seeds, three levels of reporting-period jitter and six delays. At delay π all 108 configurations exhibit deterioration, including the jittered and multifrequency cases; quadrature agrees with an independent analytic integral within 2.45 × 10⁻¹⁵. These observations concern a prescribed hold estimator, not the value of additional information to an optimal estimator. A separate market-data audit finds no whole-window deterioration in 450 fixed-delay comparisons, restricting empirical transfer of the construction. The contribution is an auditable diagnostic benchmark with explicit estimator and reference assumptions, rather than a claim about typical turbulent flows or financial markets.

1. Introduction

A recipient of timestamped observations must decide how to turn an arriving measurement of the past into an estimate of the present. Increasing reporting frequency gives that recipient more information. It need not improve a particular rule that automatically replaces its current estimate with every arriving value. This distinction becomes visible when a source oscillates on a timescale comparable to the delivery delay.

We ask a narrow question: under identical fixed delivery delay and nested acquisition schedules, can a denser schedule systematically worsen the integrated error of sample-and-hold reconstruction? The word systematically is given a precise, limited meaning: the comparison holds for every source phase in an explicit family, over a nonzero interval of delivery delays. We then test finite perturbations of the reporting schedule and the source spectrum.

Navier–Stokes provides an exact, spatially resolved benchmark rather than a financial analogy. The chosen shear subspace eliminates nonlinear advection and permits independent verification of every reported error. Consequently the result does not depend on numerical discretization of a PDE. This advantage is also a restriction: the construction says little about nonlinear transfer between modes in turbulence. The observation effect can occur in a scalar oscillator; embedding it in an exact fluid solution does not by itself constitute new fluid mechanics.

The paper provides an all-phase comparison with proof, a reproducible perturbation benchmark, and a boundary on interpretation illustrated by an existing oracle-data audit. Earlier St0kes experiments on nonlinear assimilation and reduced-cost replay are separate exploratory studies; they are not pooled with this benchmark as evidence for its theorem.

2. Related work and positioning

Age-of-information research distinguishes timeliness from task-specific estimation objectives and studies sampling under communication constraints [1]. We hold the delivery delay fixed, excluding queue-induced congestion as an explanation. Remote estimation through queues is an established subject [2]; neither delay-induced error nor sampling optimization is claimed as a new research direction here.

Discrete data assimilation for the Lorenz and two-dimensional Navier–Stokes equations has established conditions on observation intervals for convergence [3]. Time-delay nudging with noisy observations also has a dedicated NS literature [4]. Those methods evolve an approximate dynamical state and inject observational information. Our hold estimator evolves no dynamical model, so its failure does not contradict their guarantees or establish superiority over assimilation methods.

Pricing-oracle research, including Ormer [5], addresses accuracy, delay and on-chain cost. Our market audit is not an implementation or comparison of such algorithms. It is included to prevent an analytic possibility from being mistaken for an observed general financial phenomenon. No gas efficiency, manipulation resistance or deployable oracle improvement is asserted.

These references establish the scope of relevant work; this targeted review is not an exhaustive novelty search. The elementary harmonic integration and all-phase inequality should be evaluated as benchmark specifications until a fuller priority review establishes otherwise.

3. Source, channel and loss

Let Ω = [0, 2π)² have periodic boundary conditions, and let ν > 0. For distinct positive integers kⱼ, define

u(x,y,t) = ( Σⱼ Aⱼ sin(ωⱼt + φⱼ) sin(kⱼy), 0 ),
p = constant,
f(x,y,t) = ( Σⱼ Aⱼ[ωⱼ cos(ωⱼt + φⱼ)
                      + νkⱼ² sin(ωⱼt + φⱼ)] sin(kⱼy), 0 ).

Then ∇·u = 0 and (u·∇)u = 0, and direct differentiation gives ∂ₜu = νΔu + f. These are smooth exact forced NS solutions for all time. Viscosity can be any positive value because the prescribed forcing compensates its dissipative term; no Reynolds-number dependence follows from this construction.

An acquisition at sᵢ delivers the full exact field u(sᵢ) at sᵢ + ℓ, where ℓ ≥ 0 is constant. The recipient uses the most recently delivered acquisition:

û(t) = u(sᵢ),   i = max{j : sⱼ + ℓ ≤ t}.
J = (1 / T) ∫[t₀,t₀+T] ||u(t) − û(t)||²_Ω dt,
||v||²_Ω = (1 / |Ω|) ∫Ω |v|².

All comparisons use the same source, delay and scoring interval. A fine acquisition set contains its coarse counterpart. No future observation is used. Spatial orthogonality makes J the sum of one half of each modal amplitude's integrated squared error. J is mean squared velocity error, not its square root and not a relative percentage.

4. Exact comparison across every phase

Proposition 1. Phase-averaged harmonic loss

For a single unit-amplitude, unit-frequency shear, periodic report spacing h and a source phase uniform on [0,2π), the phase-averaged loss over a complete reporting cell is

J̄(h,ℓ) = 1/2 − [sin(ℓ+h) − sin(ℓ)] / (2h)
        = 1/2 − (1/2) cos(ℓ+h/2) sinc(h/2),
where sinc(z) = sin(z)/z and sinc(0) = 1.

Proof. At an elapsed time q after delivery, the true and held amplitudes are sin(θ+ℓ+q) and sin θ. Averaging their squared difference over θ gives 1−cos(ℓ+q). The spatial mean of sin²y is 1/2. Integration over q in [0,h] and division by h yields the expression. This is an elementary covariance calculation. It is an ensemble statement; arbitrary fixed phases at resonant spacings need not equal their phase average.

Proposition 2. All-phase deterioration over a delay interval

Take u = (sin(t+φ) sin y, 0), coarse acquisitions sᶜₖ = 2πk and fine acquisitions sᶠₖ = (π/2)k. Score over any integer number of complete 2π periods after the first delivery. Then for every phase φ,

Jcoarse(φ) = 1/4 + (1/2) sin²φ,
Jfine(ℓ)  = 1/2 + [sin ℓ − cos ℓ] / π.

Let δ = arcsin(π / (4√2)). If

π/4 + δ < ℓ < 5π/4 − δ   (modulo 2π),

then Jfine(ℓ) > Jcoarse(φ) for every φ. In units of π, one such interval is (0.4374196643, 1.0625803357). The inequality is strict; at an endpoint equality is possible for a maximizing coarse phase.

Proof. Every coarse sample equals sin φ. Over a full source period the true amplitude has mean zero and mean square 1/2, giving the first expression after spatial averaging. For the four fine samples, let θₖ = φ+kπ/2. For each q in [0,π/2], the four-point averages obey

(1/4) Σₖ sin²θₖ = 1/2,
(1/4) Σₖ sin²(θₖ+ℓ+q) = 1/2,
(1/4) Σₖ sin θₖ sin(θₖ+ℓ+q) = (1/2) cos(ℓ+q).

Thus the fine error is phase-independent and equals Proposition 1 at h = π/2. The maximum coarse error is 3/4. Hence every-phase deterioration is equivalent to sin ℓ − cos ℓ > π/4. Writing the left side as √2 sin(ℓ−π/4) gives the interval. ∎

At ℓ = π the fine error is approximately 0.8183098862. The ratio Jfine/Jcoarse is between 1.0910798482 and 3.2732395447 across all phases. The earlier zero-phase example attains the upper endpoint; it is not representative of the worst coarse phase. The phase-averaged error ratio is 1 + 2/π ≈ 1.63662, which is a ratio of averaged errors, not the average of phasewise ratios.

Proposition 3. Additional information does not hurt the optimal estimator

When observations under one schedule contain those under another, and the richer estimator is allowed to ignore observations without a resource penalty, its minimum achievable risk cannot be larger. Every decision rule available under the smaller information set remains available under the larger set.

In this benchmark, discarding the extra reports exactly reproduces the coarse trajectory. With exact dynamics, forcing and the state at a delivered acquisition time, model propagation reconstructs the present exactly. This comparison has stronger model knowledge than the prescribed hold policy, but makes the information distinction explicit.

A particularly simple baseline underscores the limitation: a constant zero estimate has J = 1/4 for the unit shear over a full period. At delay π it outperforms both fine hold and every nonzero coarse phase. The benchmark exposes a poor reconstruction rule; it does not recommend sparse reporting as an optimal policy.

5. Perturbed schedules and multimode shears

The protocol was saved locally before its first execution; it was not externally preregistered. The analytic construction and expected single-mode reversal were already known. This is a stress test of that construction, not a blind discovery study.

We used three source families: a unit-frequency mode; three spatial modes with amplitudes (1, 1/2, 1/4) and temporal frequencies (1,2,3); and the same amplitudes with temporal frequencies (1,√2,√3). Spatial wavenumbers are 1,2,3. Phases are independently drawn from the uniform distribution by a deterministic generator for twelve seeds, 91001–91012. Each run spans forty coarse acquisition intervals.

Coarse interval lengths are 2π(1+εξᵢ), with ξᵢ uniform on [−1,1] and ε in {0,0.05,0.15}. Each coarse interval is divided into four equal fine intervals. Thus nesting holds even with jitter. Delays are ℓ/π in {0,0.5,0.8,1,1.05,1.25}. The common integration window begins with delivery of the acquisition at zero and ends with delivery of the final coarse acquisition. Identical random draws are reused across paired conditions to isolate changes; the 648 configurations are correlated and are not 648 independent replications.

The primary calculation uses 32-point Gauss–Legendre quadrature on every interval between deliveries. The prescribed 16-point comparison differed by at most 2.44 × 10⁻⁷. An analytic antiderivative check added after that run agreed with the 32-point results within 2.45 × 10⁻¹⁵. Its implementation and code hash are included. This check assesses integration accuracy; no PDE solver error is present because the source is analytic.

At delay π, all 108 configurations had Jfine > Jcoarse. The table gives ratios over the twelve seeds for each source and jitter setting. It does not assert that arbitrary jitter or arbitrary spectra preserve the theorem. The incommensurate, unjittered minimum ratio is only 1.000392, showing that the margin can become small.

All six delay settings, including zero delay, are retained in the machine-readable results and Figure 2. No unfavorable settings are omitted from the artifact. The twelve random seeds describe a small deterministic reproducibility ensemble, not a calibrated statistical population.

The complete delay-wise reversal counts are:

6. Boundary of transfer: a pricing-oracle audit

The exact source family was motivated during an audit of proposed claims about reporting frequency in pricing oracles. That provenance does not establish that prices follow the fluid equations. We report the audit as negative evidence against that transfer.

For ETH, BTC and LINK, existing caches each contain 65,000 minute bars. Nested periodic hold policies were compared at fine/coarse periods 1/2, 2/4, 4/8 and 8/16 minutes, imposed delays 0,1,2,4,8 minutes, and all coarse phases. Each comparison shares a 32-minute burn-in. Integrated squared log-price error is exact under linear interpolation of the Binance minute opens. There were zero whole-window deteriorations among 150 comparisons per asset, or 450 total. Shorter daily reversals existed for ETH and BTC; they did not imply whole-window worsening. The caches had been inspected previously and are development data.

A separate event-level calculation compared old and newly published Chainlink answers against every price in the corresponding Binance minute high–low band. Among 4,940 retained events, 428 worsened squared log-price error throughout that band. Widening it by ±10 log-price basis points reduced the count to 19; ±25 reduced it to 2. These are reference-assumption sensitivity calculations, not confidence intervals.

For old answer a, new answer b and reference x, the change in half squared error is (b−a)((a+b)/2−x). It is affine in x, so its extrema on a reference interval occur at the endpoints. The resulting classifier is exact conditional on the interval. A retrospective single-exchange OHLC band is not a certified contemporaneous fair-value interval. Binance USDT quotes and Chainlink USD aggregation can differ, and acquisition timestamps are unavailable. We therefore cannot label these events objectively harmful or attribute them to transport delay. The historical inputs are not redistributed; hashes and source-processing code accompany the audit. This empirical appendix is not fully reproducible from the package alone without those caches.

7. Limitations and implications

The all-phase result still uses a resonant coarse acquisition period. The jitter experiments relax exact periodicity in a finite set of cases; they do not prove uniform robustness to all schedule perturbations. Multimode shears remain noninteracting, and no chaotic or turbulent generalization follows. Known forcing is designed to sustain the source, so its role cannot be hidden when comparing model-based observers.

The positive conclusion is specific: decreasing observation age pointwise need not decrease trajectory loss for a fixed hold rule, even with nested, noiseless observations and unchanged delivery delay. An information-aware estimator can always decline an update. Reporting systems should therefore specify the estimator alongside the channel when claiming that frequency improves accuracy.

The mathematical ingredients are elementary. A standalone publication must be judged on the usefulness and originality of the benchmark specification and validation, not on inflated theorem language. Stronger research would establish a distinctive contribution relative to sampling theory, extend beyond the shear invariant subspace with controlled model error, or develop an estimator with a demonstrated benefit against competent baselines. Those tasks are not claimed complete by this manuscript.

8. Reproducibility and research integrity

The package contains the frozen protocol, executable benchmark, complete results, figures and tests. Six tests check phasewise trajectory integration against the closed forms, the all-phase interval boundaries, limiting behavior, temporal convergence at zero delay, the forced PDE identity and spatial orthogonality. An additional regression check verifies the analytic integral for all stored configurations. Input code and protocol hashes are recorded in the output.

This draft was prepared with substantial AI assistance in derivation, coding and writing. The eventual human authors must independently review the mathematics, verify source attribution, approve the results and adopt a disclosure appropriate to the selected venue. No author identity, affiliation, registration, acceptance or external peer review is implied.

References

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[2] T. Z. Ornee and Y. Sun. Sampling for Remote Estimation through Queues: Age of Information and Beyond. WiOpt, 2019. https://dl.ifip.org/db/conf/wiopt/wiopt2019/1570516672.pdf

[3] K. Hayden, E. Olson and E. S. Titi. Discrete Data Assimilation in the Lorenz and 2D Navier–Stokes Equations. Physica D, 2011. https://arxiv.org/abs/1010.6105

[4] E. Celik and E. Olson. Data Assimilation using Time-Delay Nudging in the Presence of Gaussian Noise. arXiv:2201.12244, version 2, 2023. https://arxiv.org/abs/2201.12244

[5] D. Bai, J. Cao, Y. Cao, L. Wen and M. Stojmenovic. Ormer: A Manipulation-resistant and Gas-efficient Blockchain Pricing Oracle for DeFi. arXiv:2410.07893, version 2, 2025. https://arxiv.org/abs/2410.07893