The SI second, still defined by the caesium hyperfine frequency, is scheduled for redefinition on an optical transition or set of transitions, with a decision targeted for the 29th General Conference on Weights and Measures (CGPM) in 2030. One candidate form ('option 2') defines the unit as the weighted geometric mean of several optical clock transitions, realisable by any single clock through measured frequency ratios. The 2019 proposal left its specification incomplete: no optimal weight rule, no robust update protocol, no treatment of fundamental-constant sensitivity, and only sketched inclusion criteria. A 2025 follow-up by the original author resolved the weight-optimality component, and the CIPM's 2021 global adjustment of frequency ratios now accounts for input correlations. What remains open, as explicitly recorded in the roadmap of the Consultative Committee for Time and Frequency (CCTF) and the 2026 draft CGPM resolution, is to complete the remaining quantitative specification: (i) a globally consistent, redundant, outlier-resistant frequency-ratio network and adjustment at the few-10−1810^{-18} level (current inconsistencies are on record); and (ii) a pre-declared, convergent weight-update protocol with quantitative continuity safeguards against abruptly improved transitions and sporadic outlier ratios. Solving this cluster would make the ensemble definition fully specified and decision-ready for the redefinition of the second.

Background

Since 1967 the SI second has been defined by fixing the numerical value of the caesium-133 ground-state hyperfine transition frequency, ΔνCs=9 192 631 770\Delta\nu_{\mathrm{Cs}} = 9\,192\,631\,770 Hz; the best caesium fountain clocks realise it with a fractional uncertainty of a few parts in 101610^{16}. Optical atomic clocks — clocks based on narrow optical transitions in trapped ions or neutral-atom lattices — now reach systematic uncertainties near 10−1810^{-18}, two orders of magnitude better, and several species (87^{87}Sr, 171^{171}Yb, 27^{27}Al+^+, 171^{171}Yb+^+, 199^{199}Hg, and others) are near-equal contenders, so no single transition stands out as the obvious replacement.

The measurable quantities that connect these standards are frequency ratios ρij=νi/νj\rho_{ij} = \nu_i/\nu_j between transitions, which are dimensionless and independent of any unit definition. Because measured ratios carry statistical and systematic uncertainty, a worldwide set of ratio measurements generally violates transitivity (ρij=ρikρkj\rho_{ij} = \rho_{ik}\rho_{kj}); metrology institutes therefore adjust the measured ratios by least squares (or an equivalent graph-theory closure procedure) to obtain a best-estimate transitive ratio matrix together with its covariance matrix. This machinery already underlies the 'list of recommended values of standard frequencies' maintained by the International Committee for Weights and Measures (CIPM), from which Secondary Representations of the Second (SRS) — transitions that may realise the second with a stated uncertainty in addition to caesium — are designated.

Lodewyck (Metrologia, 2019) proposed to resolve the 'which species?' impasse by defining the frequency unit directly as a weighted geometric mean of a set CC of clock transitions,

ν=1N∏i∈Cνi wi,∑i∈Cwi=1,\nu = \frac{1}{N}\prod_{i\in C}\nu_i^{\,w_i}, \qquad \sum_{i\in C} w_i = 1,

with defining constants NN and the weights wiw_i. The unit can then be realised with a single clock based on any transition ii through a recommended frequency Ni=N∏kρˉikwkN_i = N\prod_k \bar{\rho}_{ik}^{w_k} constructed from the adjusted ratio matrix ρˉ\bar{\rho}, incurring an overhead uncertainty δNi/Ni\delta N_i/N_i beyond the clock's own uncertainty uiu_i. The definition generalises the current caesium definition (a single transition with unit weight): it is 'universal' (any constituent clock realises it), close to 'optimal' (the overhead is bounded by roughly a factor 2\sqrt{2} over the best clock), and 'evolutive' — the weights can be updated as clocks improve while a recalculated NN keeps the unit backward compatible. Such updates perform a random walk of the unit; by Kolmogorov's three-series theorem (a classical criterion for the convergence of random series) the walk converges provided the series ∑mvm\sum_m v_m of per-update variances converges, where vmv_m is built from the weight changes and clock uncertainties at update mm.

The proposal also has a fundamental-physics dimension. Each transition frequency can be written as a constant of dimension frequency (the Rydberg constant times cc) times a dimensionless function Fi(α,me/mp,… )F_i(\alpha, m_e/m_p, \dots) of constants such as the fine-structure constant α\alpha; the composite unit inherits F=∏iFiwiF = \prod_i F_i^{w_i}. Because different transitions have different, and partly opposite, sensitivities to α\alpha (e.g. a large negative sensitivity for the Yb+^+ octupole transition versus positive sensitivities for Sr, Yb, Al+^+, Hg), the weights could in principle be constrained so that FF is first-order insensitive to a hypothetical drift of α\alpha — at the price of moving weights away from the values that minimise realisation uncertainty. This possibility, raised as an optional remark in section 11 of the 2019 proposal, is not among the completion requirements recorded for the redefinition process.

Within the redefinition process coordinated by the Consultative Committee for Time and Frequency (CCTF), this ensemble approach is 'option 2': option 2a fixes the weights once at redefinition, while option 2b lets the CIPM update weights and NN under pre-declared rules that guarantee convergence. Option 1 replaces caesium by one chosen optical transition; option 3 (fixing another fundamental constant such as the electron mass) is currently impractical because that constant is known only to about 10−1010^{-10}. The 27th General Conference on Weights and Measures (CGPM, 2022), in its Resolution 5, asked the CIPM to bring proposals for 'the choice of the preferred species, or ensemble of species' to the 28th CGPM (2026), with adoption of a new definition targeted for the 29th CGPM (2030), subject to mandatory criteria covering clock accuracy, redundancy and consistency of frequency-ratio measurements, continuity with caesium, and contributions to international time scales.

Problem Statement

Complete the quantitative specification of the ensemble definition of the SI second — the unit ν=(1/N)∏i∈Cνiwi\nu = (1/N)\prod_{i\in C}\nu_i^{w_i} over a set CC of optical clock transitions — so that it is fully specified and decision-ready for the redefinition process targeted at the 29th General Conference on Weights and Measures (CGPM) in 2030. All components serve this single objective: an adoptable ensemble definition requires, jointly, a consistent data foundation and a safe evolution rule. Specifically, the following components, sketched in the 2019 proposal and not made rigorous, pre-declared or validated in subsequent literature, must be supplied by the solver's own construction:

(1) Consistent ratio network and inclusion criteria. Construct and validate a globally consistent, correlation-aware adjustment of the worldwide frequency-ratio measurements, with explicit, pre-stated inclusion criteria for transitions and ratio measurements (minimum number of independent measurements, repeatability requirements, treatment of outliers and of the currently documented inconsistencies in the global ratio set), that supports realisation overheads δNi/Ni\delta N_i/N_i (the additional uncertainty incurred when realising the unit with a single clock, beyond that clock's own uncertainty) at or below the contributing clock uncertainties (a few 10−1810^{-18}) and quantifies the control of common-mode systematic correlations.

(2) Pre-declared update protocol. Specify the revision rules for the dynamical variant of the definition (option 2b, in which the weights and the normalisation constant NN may be updated after redefinition under pre-declared rules): the quantitative trigger criterion for updating weights and the normalisation constant, with proofs or quantitative bounds guaranteeing (a) convergence of the unit under the update sequence (the Kolmogorov three-series condition, i.e. convergence of the sum ∑mvm\sum_m v_m of per-update variances of the unit, and its generalisations to arbitrary covariance structures), and (b) continuity — each update shifting the unit by no more than the accuracy of the best realisation of the unit at the moment of the update — under both gradual clock improvement and abrupt introduction of a radically better transition, together with safeguards against sporadic outlier ratio measurements (e.g. thresholding, minimum independent repeatability) and a quantitative risk analysis thereof.

The international adoption decision itself (single species versus ensemble; final weights and wording) is a governance act of the CGPM and the CIPM, not part of this problem; completion means a complete, internally consistent, quantitatively validated specification together with the evidence base (consistent ratio network, validated update rules) required for that decision. The weight-optimality component of the original question is excluded as resolved: weights proportional to inverse squared uncertainty, with uncertainties derived deterministically from the adjusted ratio covariance matrix, were proven optimal for this purpose by Lodewyck and Ido (2025). The possibility noted in section 11 of the 2019 proposal — constraining the weights so that the composite F(α,me/mp,… )\mathcal{F}(\alpha, m_e/m_p, \dots) is first-order insensitive to hypothetical drifts of fundamental constants — is an optional source remark contingent on such drifts being found; it is not recorded as a required open item in the CCTF roadmap's mandatory criteria or in the 2026 draft CGPM resolution, and it is not part of the required completion set.

Completion conditions. Sufficiency is fixed by degree, not only by kind. The two components are required jointly — a package that closes one of them in full while only sketching the other is a partial contribution — and each decomposes into independently adjudicable sub-claims, each closed by the solver's own artefact at the depth stated here. Restating, quoting or transcribing a procedure, formula or numerical value already published in the cited sources does not close a sub-claim: where a sub-claim overlaps material merely sketched in those sources, closure requires the solver's own derived, validated or pre-declared version of it.

  • (1a) Inclusion and outlier-rejection criteria. The pre-stated criteria list for admitting transitions and ratio measurements (minimum number of independent measurements, repeatability, outlier treatment, treatment of the currently documented inconsistencies). Closed by the list itself, fixed before the adjustment and applied.
  • (1b) Correlation-aware adjustment. The adjustment procedure, specified concretely enough to be reproduced, together with its output: a transitive best-estimate ratio matrix and its covariance, with the common-mode systematic correlations quantified. Closed by the solver's own reproduced procedure and by that procedure's output on a named input set.
  • (1c) Realisation-overhead target. Overheads δNi/Ni\delta N_i/N_i at or below the contributing clock uncertainties (a few 10−1810^{-18}). Depth bar: the target must be exhibited numerically by the solver's own adjustment on a named published input set (the input data of the 2021 CIPM update, ref. 4, is the reference choice; a later published global adjustment may be substituted if that set is named). If the target is not attained on the data available, the shortfall must be shown explicitly and the overhead bound re-expressed as an explicit function of the ratio uncertainties, together with the condition under which the 10−1810^{-18} level would be reached; an unquantified claim that the target is data-limited closes neither route.
  • (2a) Update trigger. The quantitative criterion for updating the weights and the normalisation constant, pre-declared and stated together with the argument that fixes it.
  • (2b) Convergence. A proof or explicit quantitative bound that the update sequence converges (the Kolmogorov three-series condition on ∑mvm\sum_m v_m), with its generalisation to arbitrary covariance structures; the generalisation may be given as explicitly stated sufficient conditions rather than a full proof; closure requires establishing it for the update sequence of the protocol supplied here.
  • (2c) Continuity. A proof or explicit bound that each update shifts the unit by no more than the accuracy of the best realisation at the moment of the update, under both gradual clock improvement and abrupt introduction of a radically better transition.
  • (2d) Outlier safeguards and risk analysis. The safeguards against sporadic outlier ratios (thresholding, minimum independent repeatability) and their quantitative risk analysis.

Analytic or conditional treatment, with the condition of validity stated explicitly, is acceptable for (2a)-(2d). The three data-facing sub-claims (1a)-(1c) require the artefacts above, with (1c) numerical unless the explicit conditional bound is given instead. Each sub-claim is checkable on its own, so a partial package can be adjudicated sub-claim by sub-claim.

References

  1. Jérôme Lodewyck, 'On a definition of the SI second with a set of optical clock transitions', Metrologia 56(5), 055009 (2019). DOI: 10.1088/1681-7575/ab3a82, https://doi.org/10.1088/1681-7575/ab3a82 (arXiv:1911.05551, https://arxiv.org/abs/1911.05551)
  2. Jérôme Lodewyck and Tetsuya Ido, 'Properties of a definition of the SI second based on several optical transitions', Metrologia 62(6), 065003 (2025). DOI: 10.1088/1681-7575/ae033f, https://doi.org/10.1088/1681-7575/ae033f (arXiv:2503.13278, https://arxiv.org/abs/2503.13278)
  3. N. Dimarcq, M. Gertsvolf, G. Mileti, S. Bize, C. W. Oates, E. Peik, D. Calonico, T. Ido, P. Tavella, F. Meynadier, G. Petit, G. Panfilo, J. Bartholomew, P. Defraigne, E. A. Donley, P. O. Hedekvist, I. Sesia, M. Wouters, P. Dubé, F. Fang, F. Levi, J. Lodewyck, H. S. Margolis, D. Newell, S. Slyusarev, S. Weyers, J.-P. Uzan, M. Yasuda, D.-H. Yu, C. Rieck, H. Schnatz, Y. Hanado, M. Fujieda, P.-E. Pottie, J. Hanssen, A. Malimon, N. Ashby, 'Roadmap towards the redefinition of the second', Metrologia 61(1), 012001 (2024). DOI: 10.1088/1681-7575/ad17d2, https://doi.org/10.1088/1681-7575/ad17d2 (arXiv:2307.14141, https://arxiv.org/abs/2307.14141)
  4. H. S. Margolis, G. Panfilo, G. Petit, C. W. Oates, T. Ido, S. Bize, 'The CIPM list "Recommended values of standard frequencies": 2021 update', Metrologia 61(3), 035005 (2024). DOI: 10.1088/1681-7575/ad3afc, https://doi.org/10.1088/1681-7575/ad3afc
  5. Boulder Atomic Clock Optical Network (BACON) Collaboration (Kyle Beloy, Martha I. Bodine, Tobias Bothwell, Samuel M. Brewer, Sarah L. Bromley, Jwo-Sy Chen, Jean-Daniel Deschênes, Scott A. Diddams, Robert J. Fasano, Tara M. Fortier, Youssef S. Hassan, David B. Hume, Dhruv Kedar, Colin J. Kennedy, Isaac Khader, Amanda Koepke, David R. Leibrandt, Holly Leopardi, Andrew D. Ludlow, William F. McGrew, William R. Milner, Nathan R. Newbury, Daniele Nicolodi, Eric Oelker, Thomas E. Parker, John M. Robinson, Stefania Romisch, Stefan A. Schäffer, Jeffrey A. Sherman, Laura C. Sinclair, Lindsay Sonderhouse, William C. Swann, Jian Yao, Jun Ye, Xiaogang Zhang), 'Frequency ratio measurements at 18-digit accuracy using an optical clock network', Nature 591, 564-569 (2021). DOI: 10.1038/s41586-021-03253-4, https://doi.org/10.1038/s41586-021-03253-4
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  7. Claudio E. Calosso and Nils Nemitz, 'An Accessible Formulation for Defining the SI Second Based on Multiple Atomic Transitions', 2025 Joint Conference of the European Frequency and Time Forum and IEEE International Frequency Control Symposium (EFTF/IFCS), pp. 1-3 (2025). DOI: 10.1109/EFTF/IFCS64367.2025.11194471, https://doi.org/10.1109/EFTF/IFCS64367.2025.11194471 (arXiv:2503.01778, https://arxiv.org/abs/2503.01778)
  8. Jérôme Lodewyck, 'A definition of the SI second based on several optical transitions', Journal of Physics: Conference Series 2889, 012026 (2024). DOI: 10.1088/1742-6596/2889/1/012026, https://doi.org/10.1088/1742-6596/2889/1/012026
  9. 27th CGPM, Resolution 5, 'On the future redefinition of the second' (2022). DOI: 10.59161/CGPM2022RES5E, https://www.bipm.org/en/cgpm-2022/resolution-5
  10. BIPM, 'Draft Resolutions of the 28th meeting of the CGPM (2026)', Draft Resolution B 'On the future definition of the second' (2026). https://www.bipm.org/en/cgpm-2026/documents
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