G–Causal Clock Unification — A Design Framework for Unified Timekeeping Across Scales

Timekeeping as Causal Geometry

Author: U. Warring Affiliation: Institute of Physics, University of Freiburg Version: 1.0.0 Last updated: 2025-12-16 License: CC BY 4.0 Status: Citation-stable conceptual layer Companion: CSP (Causal Steering Protocols) — operational details forthcoming Disclaimer — This document defines a boundary-condition design framework. Its authority derives from transparency, not finality.

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0. Orientation

Modern timekeeping operates across integration times spanning more than fifteen orders of magnitude. No single oscillator covers this range. Clocks that agree at one timescale may diverge at another, not because they fail, but because their comparison geometries impose different causal constraints.

This framework introduces a single architectural lens: the causal boundary condition governing phase comparison. It does not replace existing metrology tools, redefine time standards, or propose new physics. It offers a design language for reasoning about why different clock architectures exhibit different stability regimes.

How to read this document. Figure 1 provides visual orientation across architectures. Table 1 provides quantitative parameters with sources. Each section states its structural contribution. Hypotheses are labeled; predictions are testable. Open questions are marked as such.

Document structure. Sections 1–4 constitute the citation-stable core. Sections 5–9 address synthesis, open questions, and near-term applications; these may evolve in future versions.

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Figure 1 — Five-Gear Clockwork Across τ and η

                                    CAUSAL EFFICIENCY η = L/(cτ)
                    ←— Deep Interior (η≪0.01) —|— Transition (0.01–0.5) —|— Moderate (η~1) —→

    10⁻² s  ┤                                   
            │  ┌─────────────┐                  
    10⁰ s   ┤  │   OPTICAL   │←————————————————————————————————————┐
            │  │   CLOCKS    │  steering                           │
    10² s   ┤  │  (Gear 1)   │————————┐                            │
            │  └─────────────┘        │                            │
    10³ s   ┤         ↑               ↓                            │
            │         │        ┌─────────────┐    ┌─────────────┐  │
τ   10⁴ s   ┤    frequency     │  FOUNTAINS  │    │    GNSS     │  │ decadal
            │    comparison    │   + TAI     │←——→│  (Gear 3)   │  │ stability
    10⁵ s   ┤          │       │  (Gear 2)   │    └──────┬──────┘  │ check
            │          │       └──────┬──────┘           │         │
    10⁶ s   ┤          │              │                  │         │
            │          │              │  Earth           ↓         │
    10⁷ s   ┤          │              │  orient.  ┌─────────────┐  │
            │          │              └——————————→│    VLBI     │  │
    10⁸ s   ┤          │                          │  (Gear 4)   │  │
            │          │                          └──────┬──────┘  │
    10⁹ s   ┤          │                                 │         │
            │          │           long-baseline         ↓         │
    10¹⁰ s  ┤          │           correlation    ┌─────────────┐  │
            │          └←—————————————————————————│   PULSAR    │——┘
    10¹¹ s  ┤                                     │   TIMING    │
            │                                     │  (Gear 5)   │
                                                  └─────────────┘

            ├─────────────────┼─────────────────┼─────────────────┤
           10⁰              10³               10⁶              10⁹   L_comparison (km)

Caption: The five gear-trains of precision timekeeping across integration time (τ) and comparison baseline (L_comparison), with causal efficiency regimes indicated. Arrows show phase-information coupling. This is a design lens for reasoning about comparison geometry, not an architecture mandate. VLBI and PTA are included to demonstrate that the causal constraint spans the full range; this structural inclusion does not imply that operational integration is necessary or beneficial. Data ranges correspond to Table 1.

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Quickstart: Compute η and Classify

Step	Action
1. Inputs	Determine L_comparison (or L_path for complex topologies) and operating τ
2. Compute	η = L_comparison / (c × τ)
3. Classify	Deep interior (η ≪ 0.01) · Transition (0.01–0.5) · Boundary (η ~ 1)
4. Interpret	See recommended focus below

η Range	Regime	Primary Constraint	Recommended Focus
η ~ 1	Boundary	Comparison geometry	Topology, L_path optimization
0.01–0.5	Transition	Mixed	η_opt characterization (§9.1)
η ≪ 0.01	Deep interior	Oscillator/systematics	Not comparison-limited

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CORE FRAMEWORK (Citation-Stable)

The following four sections constitute the immutable core of CCUF. They are frozen for citation and will not change in future versions.

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1. The Causal Boundary as a Design Constraint

Every comparison of two oscillators requires information exchange. The propagation time along the longest required signal path sets an irreducible lower bound on the timescale at which their phases can be correlated.

1.1 The Realisability Horizon

Definition 1.1 (Realisability Horizon). For any phase comparison operation requiring causal information exchange, with the speed of light $c$, over a longest required signal path L_path, the integration time must satisfy:

$\tau \ge \frac{L_{\text{path}}}{c}$

For mutual verification (bidirectional / round-trip causal closure):

$\tau \ge \frac{2 L_{\text{path}}}{c}$

L_path reduces to L_comparison in direct two-node comparisons; for star/mesh topologies, L_path is the maximum required path for the chosen protocol.

This inequality is a boundary condition, not a dynamical law. It does not describe how clocks evolve; it describes which comparisons are possible. Any architecture proposing agreement over baseline L in time τ < L/c violates causality. No oscillator improvement, no signal processing technique, and no statistical method can circumvent this bound.

1.2 Three Orthogonal Length Scales

Clocks involve three distinct spatial scales that must not be conflated:

Scale	Symbol	Definition	Examples
Source	L_source	Spatial extent of oscillator	Ion trap: ~1 mm; Lattice: ~1 cm; Pulsar magnetosphere: ~10 km
Apparatus	L_apparatus	Interrogation/control region	Vacuum chamber: ~1 m; Radio antenna: ~10 m
Comparison	L_comparison	Baseline for phase agreement	Lab: ~1 m; Continental: ~10³ km; Galactic: ~10¹⁹ m

Only L_comparison enters the causal constraint. Advances in L_source (better oscillators) or L_apparatus (better interrogation) do not relax the realisability horizon.

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2. Causal Efficiency as an Architectural Control Parameter

2.1 Definition

For any clock architecture operating at integration time τ with comparison baseline L_comparison, define the causal efficiency:

$\eta(\tau) = \frac{L_{\text{comparison}}}{c \, \tau}$

η Value	Interpretation	Regime
η = 1	Causal limit: all integration time consumed by propagation	Boundary
0.01 < η < 0.5	Propagation significant relative to integration	Transition
η ≪ 0.01	Propagation negligible	Deep interior

Causal efficiency is not an optimization target. Different architectures may exhibit optimal noise performance at different values of η, determined by their dominant noise sources and comparison geometries.

2.2 The η_opt Hypothesis

Hypothesis 2.1 (Optimal Causal Efficiency). Each clock architecture exhibits a noise-regime transition at some characteristic efficiency η_opt, determined by comparison geometry and dominant noise sources.

Firewall: The causal boundary condition (Definition 1.1) remains valid independent of whether η_opt is confirmed for any architecture. Confirmation of η_opt adds predictive resolution; non-confirmation does not invalidate the boundary-condition analysis.

This hypothesis is testable: for a given architecture, measure stability as a function of τ while holding L_comparison fixed. If a transition in noise scaling correlates with a characteristic η value, the hypothesis is supported. If no such transition is observed, the hypothesis is falsified for that architecture.

The experimental protocol is specified in §9.1.

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3. Clocks as Architectures, Not Devices

A clock, in this framework, is not an atomic transition or a cavity mode. It is a complete architecture comprising:

1. Oscillator — the frequency source

2. Interrogation apparatus — the local control system

3. Comparison baseline — the infrastructure for phase agreement

No universal clock metric. Two clocks cannot be ranked by a single figure of merit without specifying L_comparison and τ.

Comparison is constitutive. A clock achieves its stated performance only relative to a specified comparison architecture. Stability numbers quoted without comparison context are incomplete.

Modularity. Oscillators, apparatus, and comparison links can be designed and upgraded independently. Improvement in one component improves the architecture only if no other component becomes limiting.

Reader orientation. In this framework, σᵧ(τ) denotes the measured stability of a complete clock architecture at averaging time τ, while η(τ) characterises how close that architecture operates to the causal limit imposed by its comparison geometry. σᵧ(τ) answers “how stable?” η(τ) answers “under what geometric constraints?” Both are required for architectural reasoning; neither replaces the other.

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4. Agreement Requires Causal Closure

4.1 Closure Condition

Two oscillators at separation L achieve causal closure at integration time τ if and only if:

$2L \le c\, \tau$

This is the bidirectional special case of Definition 1.1.

Without causal closure, clocks may be individually stable but cannot verify mutual agreement. They run independently, not in concert.

4.2 Closure Requirements by Scale

Scale	L (one-way)	τ for closure	Implication
Laboratory	10 m	67 ns	Instantaneous on human scales
Metropolitan	10 km	67 μs	Sub-ms closure feasible
Continental	1000 km	6.7 ms	Limits short-τ comparison
Intercontinental	10,000 km	67 ms	~100 ms minimum epoch
Earth–Moon	4×10⁸ m	2.7 s	Multi-second epochs
Earth–pulsar	10¹⁹ m	10¹¹ s	Decades for round-trip

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SYNTHESIS AND APPLICATIONS

The following sections address operational synthesis, open questions, and near-term applications. They may evolve in future versions while the core (§1–4) remains stable.

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5. Strongly-Coupled Clockwork (Operational Synthesis)

Modern precision timekeeping operates as a coupled system, though this coupling is rarely explicit. The framework identifies five principal gear-trains spanning fifteen orders of magnitude in baseline and integration time.

Table 1 — Clock Architectures: Parameters and Sources

Architecture	L_comparison	τ_min	Typical τ	η Range	Dominant Noise	σ_y(τ)	Source
Optical lattice (local)	~1 m	~3 ns	1–10⁴ s	10⁻¹²–10⁻⁸	Quantum projection	~1×10⁻¹⁸ @ 10⁴ s	[1]
Optical ion (local)	~1 mm	~3 ps	1–10⁴ s	10⁻¹⁵–10⁻¹¹	Quantum projection	~1×10⁻¹⁸ @ 10⁴ s	[2]
Optical network	10²–10³ km	0.3–3 ms	10³–10⁵ s	10⁻²–10⁻¹	Fiber phase noise	~1×10⁻¹⁸ @ 10⁴ s	[3]
Microwave fountain	~10⁴ km	~30 ms	10⁴–10⁶ s	10⁻³–10⁻²	Dick effect	~1×10⁻¹⁶ @ 10⁵ s	[4]
Hydrogen maser	~10⁴ km	~30 ms	10²–10⁵ s	10⁻³–10⁻¹	Cavity drift	~1×10⁻¹⁵ @ 10⁴ s	[5]
GNSS	~2×10⁴ km	~70 ms	1–10⁴ s	10⁻²–10⁰	Ionosphere	~1×10⁻¹⁴ @ 10³ s	[6]
TWSTFT	~10⁴ km	~30 ms	10²–10⁵ s	10⁻³–10⁻¹	Troposphere	~5×10⁻¹⁶ @ 10⁴ s	[7]
VLBI	6–10×10³ km	20–30 ms	10³–10⁵ s	10⁻²–10⁻¹	Troposphere	~1×10⁻¹⁵ @ 10⁴ s	[8]
Pulsar timing†	~10¹⁹ m	~10¹¹ s	10⁷–10⁹ s	≪10⁻²	Timing noise, ISM	~1×10⁻¹⁵ @ 10⁸ s	[9]
PTA correlation†	~10¹⁶ m	~10⁸ s	10⁸–10⁹ s	0.03–0.3	Red noise	~1×10⁻¹⁵ @ 10⁸ s	[10]

Table 1 Sources: [1] Bothwell et al. 2022, Metrologia 59:065009 [2] Brewer et al. 2019, PRL 123:033201 [3] Lisdat et al. 2016, Nat. Commun. 7:12443 [4] Guéna et al. 2012, IEEE TUFFC 59:391 [5] Parker 2012, Metrologia 49:S86 [6] Petit & Jiang 2008, Metrologia 45:S35 [7] Fujieda et al. 2014, Metrologia 51:253 [8] Schuh & Behrend 2012, J. Geodyn. 61:68 [9] Hobbs et al. 2020, MNRAS 491:5951 [10] NANOGrav 2023, ApJL 951:L8

Notes:

• σ_y(τ) values are representative of demonstrated performance; actual values vary by implementation

• ISM = interstellar medium scattering/dispersion

†PTA semantics: For single-pulsar timing, L_comparison is Earth–pulsar distance. For array correlation (gravitational-wave detection), L_comparison represents inter-pulsar angular separation projected onto the gravitational-wave wavelength; ~10¹⁶ m is representative for nHz-frequency spatial correlations. See §9.3 for baseline-dependence predictions.

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Table 2 — Five-Gear Clockwork: Coupling Structure

Gear	Systems	L_comparison	τ Domain	Role	Couples To
1	Optical clocks	≲10³ km	10³–10⁵ s	Highest stability	Gear 2
2	Fountains/TAI	≲10⁴ km	10⁴–10⁶ s	SI realization	Gear 1, 3
3	GNSS	~2×10⁴ km	1–10⁴ s	Global distribution	Gear 2, 4
4	VLBI	~10⁴ km	10³–10⁵ s	Earth orientation	Gear 3, 5
5	Pulsar timing	10¹⁶–10¹⁹ m	10⁷–10⁹ s	Decadal stability	Gear 4, 1

Structural note: VLBI and PTA systems are included to demonstrate that the causal constraint spans the full range of baselines and integration times. This structural inclusion does not imply that operational integration across all gears is necessary or beneficial for any particular application.

5.1 TA-OPT: Architectural Example

TA-OPT — a timescale realized exclusively from optical clocks, serving as a high-stability reference.

TA-OPT is introduced solely to illustrate how gear coupling could be expressed in a timescale architecture. No operational recommendation or institutional implication is intended or implied. Implementation would require NMI coordination and BIPM processes that lie outside this framework's scope.

5.2 Candor Weighting (Principle)

Candor weighting: Clocks with statistically significant unexplained deviations are downweighted relative to demonstrated consistency.

This is a design principle. Specific update laws are implementation-dependent and belong in CSP.

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6. What This Framework Is — and Is Not

This framework defines a conceptual design layer. Algorithms, operational protocols, and governance mechanisms are explicitly out of scope.

What This Framework Is

• A design language for reasoning about clock architectures in terms of comparison geometry

• A boundary condition (L_comparison ≤ cτ) constraining all phase-comparison operations

• An architectural control parameter (η) characterizing proximity to the causal limit

• A synthesis making explicit the coupling between existing timekeeping systems

What This Framework Is Not

Exclusion	Clarification
Not a new physical theory	Causality assumed, not derived
Not a replacement for Allan variance	Noise characterization remains essential
Not a redefinition of TAI/UTC/SI	Existing standards retain authority
Not a policy proposal	Governance decisions lie outside scope
Not a claim of optimality	η_opt is a hypothesis, not a prescription
Not an implementation specification	Algorithms belong in CSP
Not a substitute for relativistic corrections	IERS conventions assumed

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7. Open Design Questions

The framework's value lies in making questions explicit and comparable, not in closing them.

7.1 Architecture: Topology and Layering

The framework does not prescribe a single network topology. Different scales may legitimately use different architectures.

Scale	Candidate Topologies	Open Questions
Local (lab)	Point-to-point	Protocol optimization
Regional (100 km)	Star, ring	Topology effect on η_opt?
Continental (1000 km)	Mesh, small-world	Resilience vs. complexity
Intercontinental	Hybrid satellite+fiber	Cost-performance tradeoffs
Planetary	Hierarchical federation	Governance interoperability

Small-world topology (high local clustering with sparse long-range links) is one candidate meriting systematic evaluation for continental-scale networks: it may offer resilience with bounded complexity. This framework identifies it as a candidate for testing, not as an optimal solution.

7.2 Steering, Weighting, and Authority

The framework separates how clocks are compared from how a timescale is steered. Steering remains an institutional decision, informed—but not dictated—by verification geometry.

7.3 Compatibility with Future Quantum Enhancements

This framework does not rely on entanglement or quantum-enhanced links. It is fully applicable with classical optical links and existing technologies.

Quantum correlations may modify comparison strategies without violating causal bounds; such effects are deferred to future work.

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8. Near-Term Impact

Near-term impact (next 3–5 years)

• Can be implemented using existing clocks, fiber links, satellite transfer, VLBI

• Requires documentation and coordination, not new physics

• Aligns with current funding and infrastructure cycles

Minimum Viable Adoption (6–12 months)

Action	Effort	Output
Publish η for existing links	Low	Annotated comparison reports
Document L_path in specifications	Low	Updated technical documentation
Run one pre-registered η_opt scan	Medium	Technical note (even null result)

Table 3 — Actor Entry Points

Actor	Entry Point	Immediate Action
NMIs (PTB, NIST, NPL, etc.)	Optical networks	Document η at operating points
BIPM	Timescale algorithms	Evaluate candor weighting principle
Space agencies	Satellite timing	Add L_path to mission specs
VLBI community	IVS/IERS	Cross-validate with optical links
PTA consortia	IPTA/NANOGrav	Report baseline-dependent systematics

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9. Testable Predictions (Non-Circular)

9.1 Prediction: η_opt Location for Optical Clock Networks

Setup: Fixed baseline L_comparison = 1000 km.

Protocol: Measure modified Allan deviation while scanning τ from 10² s to 10⁵ s, covering η ∈ [0.01, 0.5].

Prediction: A transition in noise scaling occurs at some characteristic η_opt within this range.

Falsification criterion: No changepoint observed over the pre-registered interval. (This does not invalidate the boundary-condition core; see Firewall in §2.2.)

9.2 Prediction: Architecture-Specific η_opt

Setup: Optical network (1000 km baseline) vs. VLBI (6000 km baseline) at matched η values.

Prediction: Different η_opt values reflecting different dominant noise sources (fiber phase noise vs. tropospheric delay).

Falsification criterion: Identical η_opt despite different noise environments falsifies architecture-specificity.

9.3 Prediction: PTA Geometry Insensitivity

Setup: Pulsar timing array with L_comparison ~ 10¹⁶ m, τ ~ 10⁸ s (η ≪ 0.01 for some configurations).

Prediction: At deep-interior η values, stability is limited by intrinsic timing noise and ISM effects, not comparison geometry.

Falsification criterion: Systematic baseline dependence at fixed τ would require revision of the geometry-insensitivity prediction for this regime.

9.4 Non-Circularity Safeguards

• η_opt values are measured, not assumed

• Noise sources are characterized independently of the framework

• Search intervals are pre-registered to prevent post-hoc adjustment

• The framework predicts correlations; it does not define the observables

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Special thanks to the intellectual and institutional environments that shaped this work.

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References

1. D. W. Allan, Statistics of atomic frequency standards, Proc. IEEE 54, 221–230 (1966).

2. C. Lisdat et al., A clock network for geodesy and fundamental science, Nat. Commun. 7, 12443 (2016).

3. P. Delva et al., Test of special relativity using a fiber network of optical clocks, PRL 118, 221102 (2017).

4. G. Petit & B. Luzum (Eds.), IERS Conventions (2010), IERS Technical Note 36.

5. S. M. Brewer et al., ²⁷Al⁺ quantum-logic clock, PRL 123, 033201 (2019).

6. E. Bothwell et al., JILA SrI optical lattice clock, Metrologia 59, 065009 (2022).

7. W. McGrew et al., Atomic clock performance enabling geodesy below the centimetre level, Nature 564, 87–90 (2018).

8. G. Hobbs et al., The International Pulsar Timing Array: second data release, MNRAS 491, 5951 (2020).

9. NANOGrav Collaboration, The NANOGrav 15-year gravitational-wave background, ApJL 951, L8 (2023).

10. H. Schuh & D. Behrend, VLBI: A fascinating technique for geodesy and astrometry, J. Geodyn. 61, 68 (2012).

11. J. Guéna et al., Progress in atomic fountains at LNE-SYRTE, IEEE TUFFC 59, 391 (2012).

12. M. Fujieda et al., Carrier-phase two-way satellite frequency transfer, Metrologia 51, 253 (2014).

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Document Metadata

Field	Value
Version	1.0
Status	Citation-stable conceptual layer
Core sections	§1–4 (frozen)
Evolving sections	§5–9 (may be revised)
Canonical DOI	https://doi.org/10.5281/zenodo.17948436
Citation	Warring, U. (2025). Causal Clock Unification (v1.0).
Companion	CSP (Causal Steering Protocols) — operational details separate

Relationship to Companion Documents

Firewall Statement: This document (CCUF) defines the conceptual framework. The core (§1–4) is frozen for citation. CSP and implementation notes may evolve without requiring core revision. If companion development reveals gaps in the core, amendments require a new major version.

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Version History

Version	Date	Summary
0.1–0.3	2025-12–14	Initial formulation through Five-Gear synthesis
0.5	2025-12-15	Parameter tables, open questions, near-term impact
0.6	2025-12-15	Trust repairs, quickstart, provisional example, small-world
1.0	2025-12-16	Merged structure: citation-stable core (§1–4) separated from evolving synthesis (§5–9). Restrained VLBI/PTA framing. Auditable Table 1 with sources. TA-OPT as architectural example only. Retained actionability elements (Quickstart, MVA, actor entry points).

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This framework defines a conceptual design layer. Algorithms, operational protocols, and governance mechanisms are explicitly out of scope.