G–Causal Clock Unification — A Design Framework for Unified Timekeeping Across Scales
Timekeeping as Causal Geometry
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.
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.
Quickstart: Compute η and Classify
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 ~ 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
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.
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 , over a longest required signal path L_path, the integration time must satisfy:
For mutual verification (bidirectional / round-trip causal closure):
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:
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.
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:
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.
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:
Oscillator — the frequency source
Interrogation apparatus — the local control system
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.
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:
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
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
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.
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
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.
Table 2 — Five-Gear Clockwork: Coupling Structure
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.
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
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
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.
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.
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)
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
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
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
Special thanks to the intellectual and institutional environments that shaped this work.
References
D. W. Allan, Statistics of atomic frequency standards, Proc. IEEE 54, 221–230 (1966).
C. Lisdat et al., A clock network for geodesy and fundamental science, Nat. Commun. 7, 12443 (2016).
P. Delva et al., Test of special relativity using a fiber network of optical clocks, PRL 118, 221102 (2017).
G. Petit & B. Luzum (Eds.), IERS Conventions (2010), IERS Technical Note 36.
S. M. Brewer et al., ²⁷Al⁺ quantum-logic clock, PRL 123, 033201 (2019).
E. Bothwell et al., JILA SrI optical lattice clock, Metrologia 59, 065009 (2022).
W. McGrew et al., Atomic clock performance enabling geodesy below the centimetre level, Nature 564, 87–90 (2018).
G. Hobbs et al., The International Pulsar Timing Array: second data release, MNRAS 491, 5951 (2020).
NANOGrav Collaboration, The NANOGrav 15-year gravitational-wave background, ApJL 951, L8 (2023).
H. Schuh & D. Behrend, VLBI: A fascinating technique for geodesy and astrometry, J. Geodyn. 61, 68 (2012).
J. Guéna et al., Progress in atomic fountains at LNE-SYRTE, IEEE TUFFC 59, 391 (2012).
M. Fujieda et al., Carrier-phase two-way satellite frequency transfer, Metrologia 51, 253 (2014).
Document Metadata
Version
1.0
Status
Citation-stable conceptual layer
Core sections
§1–4 (frozen)
Evolving sections
§5–9 (may be revised)
Canonical DOI
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.
Version History
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).
This framework defines a conceptual design layer. Algorithms, operational protocols, and governance mechanisms are explicitly out of scope.
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