Theorem of Temporal Resolution Limitation (v1.1)

This work establishes a fundamental limitation on temporal coordination in distributed systems that lack continuous physical coupling between local clocks.

1. Introduction

The scalability of distributed quantum computing, precision sensor networks, and aligned distributed artificial intelligence is constrained by the need for high-fidelity temporal coordination between independent nodes. Shared physical timing references (e.g., GNSS, optical links) can help, but may be costly, vulnerable, or unavailable. This work formalizes the core difficulty—the impossibility of maintaining absolute temporal alignment without persistent coupling—and introduces a solution that shifts coordination from absolute time to a shared cyclic phase convention.

2. Theorem of Temporal Coordination Limitation

Theorem 2.1: Asynchronous coordination limitation (variance identity and unbounded growth)

Let S = {N₁, N₂, ..., Nₖ} be a set of distributed nodes. Each node Nᵢ possesses a local clock whose indicated time Cᵢ(t) deviates from an ideal reference time t according to:

Cᵢ(t) = t + ∫₀ᵗ ϵᵢ(τ) dτ,

where ϵᵢ(t) is the instantaneous fractional frequency deviation.

Fix two nodes Nᵢ and Nⱼ and define the clock-offset process

Δᵢⱼ(t) := Cᵢ(t) - Cⱼ(t) = ∫₀ᵗ (ϵᵢ(τ) - ϵⱼ(τ)) dτ.

Assume:

ϵᵢ and ϵⱼ are zero-mean, uncorrelated, wide-sense stationary;
their autocorrelations Rᵢ(u) = E[ϵᵢ(t)ϵᵢ(t + u)] and Rⱼ(u) satisfy ∫₀^∞ |Rᵢ(u)| du < ∞ and ∫₀^∞ |Rⱼ(u)| du < ∞;
no continuous physical coupling exists between nodes after initialization.

Then

Var[Δᵢⱼ(t)] = 2 ∫₀ᵗ (t - u) [Rᵢ(u) + Rⱼ(u)] du, (1)

and if Rᵢ + Rⱼ ≠ 0 this variance grows without bound as t → ∞.

Consequently, fixed-tolerance coordination strategies that rely on targeting or comparing future absolute timestamps cannot provide a time-uniform guarantee of coordination within a fixed tolerance over arbitrarily long horizons without periodic recalibration or continuous coupling.

3. The Phase-Coordination Principle and Formal Definitions

Principle 3.1: Phase-triggered coordination: The limitation of Theorem 2.1 motivates a shift from absolute time targeting to shared temporal phase conventions. Coordination is achieved by agreeing on a target phase value within a cycle and triggering when a node’s locally computed phase enters a predefined acceptance window around that target.

Definition 3.1: Φ–normalized temporal phase (protocol convention): Let t₀ be a public reference epoch (a protocol convention) and let T_cycle > 0 be a fixed cycle duration (e.g., a day or an application-defined period). Define the global normalized temporal phase

Φ(t) = ( (t - t₀) / T_cycle ) mod 1 ∈ [0, 1) ≅ S¹.

A node Nᵢ computes a local phase estimate Φ̂ᵢ(t) := Φ(Cᵢ(t)) from its local clock Cᵢ(t).

Definition 3.2: Circular distance on S¹: For 𝜙, 𝜓 ∈ [0, 1) define

d_S¹(𝜙, 𝜓) := min{|𝜙 − 𝜓|, 1 − |𝜙 − 𝜓|} .

Definition 3.3: Phase window (wrap-around-safe, half-width convention): Fix a target phase Φ_target ∈ [0, 1) and tolerance ΔΦ ∈ (0, 1/2). The acceptance window is the arc subset

W(Φ_target, ΔΦ) := {𝜙 ∈ [0, 1) : dₛ₁(𝜙, ΦₜₐᵣGₑₜ) ≤ ΔΦ} .

Its physical duration is Δt_w = 2ΔΦ · T_cycle.

4. Theorem of Bounded Coordination via Phase Triggering

Theorem 4.1: Bounded coordination error via phase triggering (cycle-anchored)

Consider two nodes Nᵢ and Nⱼ with local clocks Cᵢ(t), Cⱼ(t) as in Theorem 2.1. Fix protocol conventions (t₀, T_cycle) and fix a coordination instruction consisting of:

an intended cycle index k ∈ Z,
a target phase Φ_target ∈ [0, 1),
a tolerance ΔΦ ∈ (0, 1/2) defining the window W = W(Φ_target, ΔΦ) and duration Δt_w = 2ΔΦ T_cycle.

Assume that, during the intended cycle, each node’s phase estimate t ↦ Φ̂ℓ(t) enters the window W at least once and that triggering occurs at the first entry time within that intended cycle.

Define the trigger times (cycle-anchored):

tᵢ := min{t ∈ [t₀+kTₖ, t₀+(k+1)Tₖ) : Φ̂ᵢ(t) ∈ W},

tⱼ := min{t ∈ [t₀+kTₖ, t₀+(k+1)Tₖ) : Φ̂ⱼ(t) ∈ W}.

Then for any operational horizon Tₒₚ that covers the intended cycle, the physical desynchronization satisfies

|tᵢ − tⱼ| ≤ Δt_w + κᵢⱼ(Δt_w; Tₒₚ). (2)

In particular, the bound depends on the chosen window duration and short-horizon relative clock variation, not on the total elapsed time since initialization.

5. Discussion and Implications

The Phase-Coordination Principle does not contradict Theorem 2.1; it targets a different objective. Theorem 2.1 concerns the impossibility of maintaining absolute clock alignment without continuous coupling. Phase-window triggering coordinates relative events inside a specified cycle; the resulting skew depends on window duration and short-horizon relative clock variation.

6. Conclusion

We established a variance-based limitation on absolute timestamp coordination under autonomous clocks without continuous coupling (Theorem 2.1). We then introduced a cycle-anchored, wrap-around-safe window-triggering model on S¹ and proved a bounded-skew statement under an explicit short-horizon stability term (Theorem 4.1).

References

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