Atomic clocks

Atomic clocks

Optical frequency-ratio measurements provide a useful test case for the physics bridge because the published observable is both precise and highly reduced. A reported ratio is obtained only after atomic interrogation, oscillator control, frequency transfer, correction, averaging, and uncertainty evaluation. The public table retains part of that measurement chain, but it does not retain the complete metrological procedure.

In this analysis, I use source-linked atomic-clock data to examine a metric-first version of the Sense bridge. The objective is not to identify the Sense coordinates with clock variables or to propose a new theory of time. The objective is to determine whether an auditable frequency-ratio corpus can constrain how q_S, c_S, phi_S, mu_S, and tau_S are used in one local physical context.

Here, metric attachment does not mean fitting five hidden parameters to the clock data. Each physical surface is used instead to narrow, separate, or block a model-side interpretation. A valid bridge should preserve the distinctions required by the measurement and should remain incomplete when the source does not support a stronger statement.

Why atomic-clock ratios are a discriminating test

An atomic clock realizes a frequency reference by stabilizing an oscillator to an atomic or ionic transition. A comparison between two clocks therefore reports a relation between two transition frequencies rather than an independent measurement of “time” as a substance.

For two clock species A and B, the primary observable may be written as

R_AB = nu_A / nu_B.

The reported value depends on more than the transitions themselves. It also depends on state preparation, interrogation, local oscillators, servo response, frequency-comb or transfer systems, environmental corrections, averaging, and the statistical model used to report the result.

This measurement chain is relevant to the bridge concept because it separates three roles.

The frequency ratio is the payload retained by the comparison.

The ions, oscillators, control loops, transfer chain, and reduction procedure form the physical carrier.

The public ratio value, uncertainty statement, timestamps, and released tables form the measurement-facing shell.

These roles are connected, but they are not interchangeable. A public table can preserve ratio values and temporal ordering while omitting the raw counter records, correction history, covariance structure, or phase-transfer logs that produced the final result. The bridge must therefore specify which parts of the carrier remain represented in the public shell and which parts are unavailable.

The local Sense calibration map

The atomic-clock route uses the following local map:

q_S → source identity, isotope branch, transition pair, and ratio-lane binding

c_S → supported residual, repeatability, coherence, and uncertainty envelopes

phi_S → direct phase, lock, comb, or transfer evidence

mu_S → ion-clock carrier, apparatus, correction, and systematic context

tau_S → timestamps, run structure, averaging dependence, repeatability, and stability evidence

These arrows denote calibration against physical metrics. They are not identities.

q_S is not an atomic transition.

c_S is not physical uncertainty.

phi_S is not physical phase.

mu_S is not an ion, trap, or oscillator.

tau_S is not clock time.

The expected contribution was therefore not a relabeling of standard metrology. I tested whether the mapping imposed non-mixing rules.

An isotope or transition mismatch should not be treated as additional statistical uncertainty. Temporal structure should not be removed by assuming that all rows are independent. A list of systematic effects should not be promoted to a propagated uncertainty budget. General knowledge that clocks are phase-sensitive should not substitute for source-linked phase records. A public ratio table should not be treated as a raw calibration chain.

This is the operational relation to the bridge program. The experiment supplies the metric surfaces; the Sense grammar proposes a local organization; the source audit determines which parts of that organization remain admissible.

Why two source layers were used

The source work was divided into a discovery layer and a primary numerical lane.

The discovery corpus included public optical-clock papers, supplementary tables, ratio records, uncertainty and systematics material, stability and Allan-deviation context, and phase- or coherence-related material from several clock experiments.

Its purpose was to identify the physical objects that might support a bridge. It was not used to combine unrelated experiments into one synthetic measurement. A ratio value, a systematic budget, a stability curve, and a phase-transfer record may be published in different source objects. Their presence in the same topic area does not establish that they belong to one device, one reduction chain, or one comparison interval.

The primary numerical corpus was the public Ar/Yb frequency-ratio dataset in Zenodo record 6901524. It contains two ratio lanes:

36Ar13+ / 171Yb+

40Ar13+ / 171Yb+

I selected this corpus because it provided a source-linked and auditable table-level record. The local files could be associated with public metadata and checksums. Individual rows included dates and UTC times. Both argon isotopes were compared with the same ytterbium reference. The source paper and extended tables supplied reported ratio targets and uncertainty context.

The corpus contained 53,682 numeric rows for 36Ar13+ / 171Yb+ and 102,888 rows for 40Ar13+ / 171Yb+. During active intervals, the cadence was often close to one second, while the full records also contained gaps and separated measurement runs.

These features made the corpus suitable for four tests:

source and isotope identity;

table-level residual and repeatability structure;

sensitivity to temporal ordering and averaging assumptions;

and the boundary between public-table evidence and raw metrology.

The use of two isotope branches provided an additional identity control. The reference clock remained the same while the argon isotope changed. A correct analysis therefore had to bind the ratio lane before applying any precision or stability interpretation.

A more ambitious optical-clock intercomparison would not have been a stronger bridge object if its reducer, covariance, phase-transfer records, or calibration chain were unavailable. For this stage, an auditable partial corpus was preferable to a nominally more precise but inaccessible measurement chain.

Expected reconstruction

The expected result was a local precision-envelope map, not a reproduction of the laboratory’s final ratio reducer.

The analysis proceeded in five steps.

First, each table had to be bound to the correct isotope and source target. This constrained q_S.

Second, the ratio rows were expressed as fractional residuals relative to a declared local reference, including the table median and the source target. This constrained c_S at the level of table scatter and comparison envelopes.

Third, the residuals were examined across days, blocks, averaging sizes, and ordered or shuffled sequences. This constrained tau_S at the level of table-based temporal diagnostics.

Fourth, the paper and supplementary material were used to identify the carrier and systematic context represented by the source. This constrained mu_S without converting that context into a final uncertainty budget.

Fifth, the primary lane was checked for direct phase, lock, comb, or transfer records. These objects were required before phi_S could advance beyond contextual association.

A positive result required the five questions to remain separate. A stronger result would have required a non-generic residual relation or a predeclared failure test specific to the Sense bridge. The present route did not reach that stronger level.

Source-anchored ratio replay

The source targets admitted for table comparison were

R(36Ar13+ / 171Yb+) = 1.05776646273518748(13)

R(40Ar13+ / 171Yb+) = 1.05776938758748094(11).

The corresponding public-table medians were

1.0577664627351873

and

1.0577693875874807.

The median-to-target differences were of order 10^-16. This supports the conclusion that the released tables belong to the reported Ar/Yb ratio lanes and can be used for a table-level replay.

It does not establish that the source-certified final reduction algorithm was reproduced.

The identity audit also detected an earlier mismatch in the 36Ar target assignment. A row-exact return to the source recovered the value

R(36Ar13+) = 1.05776646273518748(13).

The fractional separation between the 36Ar and 40Ar source targets is approximately 2.765 × 10^-6. This is many orders of magnitude larger than the reported 10^-16 uncertainty scale.

The consequence is direct: the wrong isotope branch is not a noisy realization of the correct branch. It is a different physical object. The correction therefore constrained q_S before any uncertainty analysis was admitted.

Residual statistics and temporal structure

The fractional residual standard deviations relative to the table medians were approximately

6.85 × 10^-15 for 36Ar13+ / 171Yb+

and

8.02 × 10^-15 for 40Ar13+ / 171Yb+.

If all rows are treated as independent and identically distributed samples, the corresponding standard errors are

2.96 × 10^-17

and

2.50 × 10^-17.

These values are smaller than the source-target uncertainty envelopes of approximately 1.2 × 10^-16 and 1.0 × 10^-16. The reduction follows directly from the large number of rows, but it depends on an IID assumption that is not source-certified.

The table-level temporal diagnostics did not support treating that assumption as a final uncertainty model. Daily-bin reduced chi-squared values were approximately 66.9 for the 36Ar lane and 26.6 for the 40Ar lane. Under the adopted daily-bin model, these values indicate that the observed day-to-day structure is not represented by the naive IID estimate.

The excess may contain temporal correlation, run-to-run variation, drift, overscatter, or effects already handled in the source reduction. The public tables alone do not determine which explanation is correct.

For this reason, the uncertainty analysis was retained as an assumption-sensitivity study.

For 36Ar, the source-target envelope was 1.20 × 10^-16. Adding the naive table standard error in quadrature gave 1.24 × 10^-16. Replacing the naive standard error with a daily-chi-squared-inflated diagnostic gave 2.70 × 10^-16.

For 40Ar, the corresponding values were 1.00 × 10^-16, 1.03 × 10^-16, and 1.63 × 10^-16.

A fourth policy used the single-sample residual scatter as a conservative diagnostic envelope, producing values at the 10^-15 scale. This policy was not interpreted as an uncertainty on the final mean ratio.

None of these scenarios is a source-certified uncertainty propagation. They quantify the dependence of the result on explicit assumptions. In particular, no covariance matrix, posterior object, or reproducible source statistical model was available.

The missing covariance was therefore retained as missing rather than set to zero.

Systematic and carrier context

The source corpus contained substantial context for effects relevant to optical-clock comparisons. Eleven component families were identified, including Zeeman shifts, isotope and mass shifts, electric-quadrupole effects, excess micromotion, Doppler and time-dilation effects, servo or line-pulling effects, blackbody-radiation shifts, and Stark or polarizability effects.

This material constrained mu_S by specifying the physical carrier and the classes of corrections associated with it.

It did not provide a complete source-certified component budget with typed values, units, signs, correlations, and a reproducible propagation rule. Numerical tokens and named effects in a paper or supplement are not sufficient to reconstruct a final uncertainty budget unless their roles in the reduction are explicitly bound.

The systematic information was therefore retained as carrier and context evidence only.

Stability and phase evidence

The discovery corpus contained source context for daily repeatability, averaging time, fractional instability, white-frequency-noise behavior, and Allan-deviation analysis. The Ar/Yb tables also supported table-level block-stability and time-order diagnostics.

These objects constrained tau_S by demonstrating that the interpretation depends on temporal organization and averaging policy.

They did not provide a source-certified Allan-deviation reducer or a complete measurement model from which the laboratory stability curve could be independently reproduced. The stability result therefore remained diagnostic.

The primary Ar/Yb lane also lacked the direct phase-transfer objects required for a source-native phi_S analysis. No phase-stabilized link logs, phase-locked-loop records, acousto-optic-modulator logs, frequency-comb phase records, or source-certified phase-noise transfer model were admitted.

Atomic-clock operation is phase-sensitive, but this general fact does not convert a public frequency-ratio table into a phase record. phi_S was therefore not advanced in the primary fork.

How the data calibrated the Sense Model

The atomic-clock analysis constrained the local Sense interpretation in five specific ways.

q_S became source- and branch-bound. It had to distinguish the 36Ar and 40Ar ratio lanes before any precision statement was evaluated. An identity error could not be represented as a larger uncertainty.

c_S became an envelope coordinate rather than a physical uncertainty variable. It could be compared with residual scatter, source-target uncertainty, and assumption-sensitive diagnostic envelopes, but it could not supply a final uncertainty in the absence of a certified propagation model.

phi_S became evidence-dependent. General phase sensitivity was insufficient. The coordinate could advance only when direct phase, lock, comb, or transfer records were present.

mu_S became carrier- and context-bound. It retained the ion, apparatus, correction, and systematics structure without treating a catalogue of effects as a completed budget.

tau_S became sensitive to temporal ordering, block structure, averaging, and stability context. A large number of rows did not justify an IID interpretation when the ordered diagnostics indicated additional structure.

This is the metric attachment obtained from the clock corpus. The physical observables did not become Sense variables. They restricted the operations and promotions that the Sense variables were allowed to support.

The analysis also clarified the specificity limit. A careful metrologist already distinguishes source identity, residual statistics, systematic corrections, phase-transfer evidence, and temporal correlation. The present result therefore does not establish a unique physical ontology for the Sense Model.

Its narrower contribution is a common non-absorption and abstention rule across the five coordinates: branch identity cannot be repaired by uncertainty, temporal structure cannot be absorbed into static scatter, systematic context cannot be promoted to a budget, and absent phase records cannot be replaced by generic clock terminology.

Current status

The atomic-clock route is frozen as a source-anchored, table-level precision-envelope bridge.

The admitted result includes two correctly identified Ar/Yb ratio lanes, public-table replay, fractional residual and repeatability diagnostics, assumption-sensitive uncertainty envelopes, table-level stability analysis, and partial carrier and systematics context.

The route does not admit a source-certified final reducer, a complete uncertainty budget, covariance or posterior information, a source-certified Allan or stability reducer, direct phase-transfer logs, or the raw counter and calibration chain. It does not establish a new clock ratio, a metrological theorem, or a physical identification of q_S, c_S, phi_S, mu_S, and tau_S.

A stronger test requires at least one new source-certified object: the final reduction algorithm, a typed component budget with covariance, a source-defined stability reducer, direct phase-transfer records, or a raw calibration chain.

The value of the atomic-clock case is therefore specific. It demonstrates that public ratio data can impose source identity, uncertainty, temporal, carrier, and phase-evidence constraints on a local Sense bridge. It also specifies where the interpretation must stop when the public table no longer represents the underlying measurement chain.