# Correlation Access and Causal Efficiency

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**Author:** U. Warring\
**Affiliation:** Institute of Physics, University of Freiburg\
**Version:** 0.1.3\
**Last updated:** 2025-12-22\
**License:** CC BY 4.0\
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**Status:** Proposed framework (Stage 1) — validation pending\
**Scope:** Comparison geometry and access constraints only\
**Exclusions:** Oscillator physics; relativistic modelling (IAU/IERS); quantum gravity
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> **Relation to Causal Clock Unification (CCUF):**\
> Our framework builds on CCUF’s core causal boundary condition (τ ≥ c L), using the same causal efficiency η metric. We augment CCUF with a **physical correlation access layer** (ξ and Constraint C1), which is a design-level hypothesis and subject to empirical validation. CCUF itself remains the authoritative causal geometry constraint.

***

### 0. Reader Contract (Protective Framing)

* This document **proposes** an architectural lens.
* It does **not** claim field consensus.
* Cited literature provides **empirical foundations**, not proof of the constructs introduced here.
* Validation of proposed constraints and predictions remains open.

***

### 1. Motivation and Context

* Optical clocks now outperform many comparison channels.
* Stability gains increasingly **saturate without diagnostic clarity**.
* This framework introduces an **access-geometry lens** to diagnose such plateaus.

**Context anchors:**\
Allan (1966); Rutman (1978); Lisdat et al. (2016); Zhang et al. (2022)

***

### 2. Core Quantities (Orthogonal by Design)

#### 2.1 Correlation length ξ — *Physical availability*

* **Definition:** Spacetime extent over which fluctuations remain correlated.
* **Properties:**
  * Set by environment, medium, turbulence, noise processes
  * Independent of interrogation or protocol
* **Examples:** fibre acoustic/thermal noise; atmospheric turbulence; ionospheric phase correlations

**Anchors:**\
Allan (1966); Rutman (1978); Coddington et al. (2008)

***

#### 2.2 Causal efficiency η(τ) — *Architectural utilisation*

<p align="center"><span class="math">\eta(\tau)=\frac{L_{\mathrm{comparison}}(\tau)}{c\,\tau} </span></p>

* **Definitions:**
  * $$L\_{\mathrm{comparison}}(\tau)$$: maximum spatial extent that must act coherently to produce one comparison sample
* **Properties:**
  * Dimensionless **operational control parameter**
  * Not a thermodynamic efficiency
* **Regimes:**
  * η ≪ 1: under-utilised causal envelope
  * η ≈ 1: causal limit for single round-trip
  * η > 1: delayed or multi-cycle closure

> **Relation to CCUF geometry:** η(τ) is defined by CCUF's causal boundary condition τ ≥ L\_comparison/c. This document treats η as a design variable by recognizing that practitioners choose both L\_comparison (network geometry) and τ (averaging time) within causal constraints. You cannot violate η ≤ 1 for single round-trip comparison, but you can optimize the (L\_comparison, τ) pair for your measurement goals.

**Anchors:**\
Ma et al. (1994); Calonico et al. (2014); IEEE 1139-2008

***

#### **2.3 Bridging Physics and Protocol —&#x20;*****Why two quantities?***

The separation between ξ and η(τ) may seem artificial at first. Why not combine them into a single figure of merit?

**The answer lies in control:**

**Correlation length ξ** responds to physical intervention:

* Improve fibre isolation → longer acoustic correlation times
* Reduce atmospheric turbulence → larger spatial coherence
* Stabilise environment → longer thermal correlation scales

You cannot change ξ by redesigning your measurement protocol.

**Causal efficiency η(τ)** responds to architectural choices:

* Faster synchronisation → shorter τ for same L\_comparison
* Tighter comparison geometry → smaller L\_comparison for same τ
* Multi-site coordination → optimised causal closure paths

You cannot change η by improving the physical medium alone.

**Worked intuition (fibre link example):**

Consider a 100 km phase-stabilised optical fibre:

* **Physical layer:** Active noise suppression extends ξ from \~1 km (passive fibre) to \~100 km (stabilised)
* **Protocol layer:** Round-trip stabilisation with τ = 1 ms gives:
  * L\_comparison = 2 × 100 km = 200 km (full round-trip path for causal closure)
  * η(τ) = 200 km / (c × 1 ms) ≈ 0.67

Both layers contribute to performance:

* If ξ were only 10 km → physical bottleneck (C1 violated)
* If η = 0.01 → architectural inefficiency (most causal envelope wasted)

Success requires **matching both**.

**Why this matters:**

When optical clock comparisons saturate in performance:

* **Diagnosis:** Is it physics (insufficient ξ) or architecture (low η)?
* **Intervention:** Physical upgrade (stabilisation, isolation) vs protocol redesign (faster sampling, geometry change)

Traditional stability metrics (Allan deviation, modified Allan deviation) measure aggregate instability but cannot distinguish between physical correlation limits and architectural inefficiencies. The ξ vs η separation provides this **diagnostic clarity**.

> **Non-claim:** This framework does not predict absolute performance limits. It identifies which layer constrains performance in a specific architecture.

***

### 3. Proposed Interface Condition (C1)

#### 3.1 Constraint C1 — *Proposed access condition*

<p align="center"><span class="math"> \xi \;\lesssim\; L_{\mathrm{comparison}}(\tau) </span></p>

* **Epistemic status:** Framework prediction (testable; not yet validated across architectures)
* **Meaning:** Correlations not jointly accessible within one comparison cycle cannot contribute coherently.
* **Non-claims:** Not a new physical law; not a causality bound.

**Context anchors (not proofs):**\
Lisdat et al. (2016); Riehle et al. (2015); Zhang et al. (2022)

***

### 4. Architectural Intuition (Non-prescriptive)

#### 4.1 Phase-stabilised fibre links

* Two-way / round-trip causal closure
* Active suppression **extends the accessible correlation length**
* Enables high-η operation without violating C1

**Anchors:**\
Ma et al. (1994); Calonico et al. (2014); Lisdat et al. (2016)

***

#### 4.2 Free-space optical transfer

* Physical correlation length limited by turbulence and reciprocity
* η-optimisation alone eventually saturates performance
* **Framework prediction:** C1 may fail in this regime (validation required)

**Anchors:**\
Coddington et al. (2008); Zhang et al. (2022)

***

### 4.5 Worked Example (Minimal, Numerical)

#### Case study: GPS carrier-phase comparison *(illustrative)*

* **Approximate parameters (order-of-magnitude estimates):**
  * ξ\_iono ≈ 300 km (ionospheric correlation scale; typical mid-latitude conditions)
  * L\_comparison ≈ 20 000 km (global GNSS network baseline)
* **Regime:**
  * C1 satisfied (ξ ≪ L\_comparison)
  * η ≪ 1 due to global geometry and timing latency
* **Prediction:** Synchronisation improvements should yield gains; environmental mitigation is secondary.

**Anchors:**\
Allan (1966); Rutman (1978); IEEE 1139-2008

***

### 5. Falsifiable Prediction

#### S1 — Saturation Test

* **Statement:** At fixed physical conditions, reducing η(τ) ceases to improve measured instability once C1 fails.
* **Observable:** Plateau in σ\_y(τ) versus protocol timing.
* **Status:** Architecture-specific; data-driven validation required.

**Anchors:**\
Allan (1966); NIST SP 1065 (2008; 2014 update)

***

#### 5.1 Validation Pathway to Handbook Status (Stage 2)

**Minimum evidence required:**

* C1 tested in ≥2 distinct architectures (e.g. fibre and free-space)
* η(τ) calculation method specified and reproduced independently
* S1 saturation behaviour documented with published datasets
* ≥1 η\_opt prediction confirmed or falsified

**Current status:** None of these criteria yet met.

***

### 6. Conditional Design Guidance (Guarded)

> **Applies only where C1 and S1 are testable**

* If performance saturates under η-optimisation → investigate physical limits on ξ
* If ξ is abundant → optimise synchronisation, geometry, and closure latency

*No universal optimisation strategy is implied.*

***

### 7. Scope and Aspirational Outlook

**Current scope:** This framework is developed for optical time and frequency comparison networks, where both ξ and η(τ) can be operationally defined and measured.

**If validated**, this framework could re-centre comparison science on interrogation geometry and provide diagnostic tools for saturation regimes in optical clock networks.

**Potential future extensions (speculative, not validated):**

The ξ vs η separation may be architecturally relevant to other comparison geometries:

* **VLBI (Very Long Baseline Interferometry):** Radio-frequency phase comparison across continental/global baselines; correlation limited by ionospheric and tropospheric coherence.
* **Pulsar Timing Arrays (PTAs):** Gravitational wave detection via correlated timing residuals; correlation extended by astrophysical stability but limited by terrestrial clock comparison.

However:

* These systems have fundamentally different physical processes determining ξ
* Operational definitions of L\_comparison(τ) require domain-specific protocols
* Constraint C1 and prediction S1 must be independently validated in each regime

**Extension criterion:** Before claiming applicability to non-optical systems, we require worked examples showing how to calculate ξ and η(τ), and empirical tests of C1 in those architectures.

**Present status:** Proposal awaiting multi-architecture testing within optical frequency transfer domain.

***

### 8. Note on References (Protective)

The references below establish the empirical foundations of time-frequency metrology and optical transfer.\
The ξ/η separation, C1, and S1 are **novel architectural constructs** introduced here to unify and interrogate these results.

***

### 9. References

* Allan, D. W. (1966). *Proc. IEEE* **54**, 221–230.
* Barnes, J. A. et al. (1971). *IEEE TIM* **IM-20**, 105–120.
* Calonico, D. et al. (2014). *Appl. Phys. B* **117**, 979–986.
* Coddington, I., Swann, W. C., & Newbury, N. R. (2008). *Phys. Rev. Lett.* **100**, 013902.
* Lisdat, C. et al. (2016). *Nat. Commun.* **7**, 12443.
* Ma, L.-S., Ye, J., & Hall, J. L. (1994). *Opt. Lett.* **19**, 1777–1779.
* Newbury, N. R. (2011). *Nat. Photonics* **5**, 186–188.
* Riehle, F. et al. (2015). *C. R. Physique* **16**, 506–515.
* Rutman, J. (1978). *Proc. IEEE* **66**, 1048–1075.
* Zhang, H. et al. (2022). *Nature* **607**, 687–692.
* IEEE 1139-2008. *Definitions of Frequency and Time Metrology.*
* NIST SP 1065. *Handbook of Frequency Stability Analysis* (2008; 2014 update).

***
