The Phase-Coordination Principle (v2.0)

From Fundamental Limitation to Implementation-Facing Coordination

Abstract

This paper formalizes phase-window coordination for distributed systems with autonomous local clocks. We recall a fundamental limitation: without continuous physical coupling, internode clock-offset uncertainty grows without bound over long horizons, preventing time-uniform fixed-tolerance coordination based on absolute timestamps. We then state the Phase-Coordination Principle: replace absolute-time targets with a shared cyclic phase convention and trigger actions within wrap-around-safe acceptance windows on S¹, anchored to an intended cycle index. Under explicit short-horizon phase estimation error bounds, we give an operational bounded-skew statement: the inter-node execution skew is bounded by a chosen window duration plus short-horizon estimation terms, and does not scale with time since initialization. Finally, we provide an implementation-facing view compatible with tick-canonical macro-window/micro-slot addressing (Q-Address style) and triadic Human–AI–Quantum coordination stacks.

1. Fundamental limitation motivating phase-window coordination

This paper follows the limitation formalized in the companion theorem note [4]: without persistent coupling, offset uncertainty grows and long-horizon absolute timestamp guarantees are not time-uniform.

Definition 1.1 (Integrated frequency deviation model):

Let a local clock be modeled as

Cᵢ(t) = t + ∫₀ᵗ εᵢ(τ) dτ,

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

Theorem 1.1 (Asynchronous coordination limitation (variance identity))

Let Δᵢⱼ(t) = Cᵢ(t) − Cⱼ(t) = ∫₀ᵗ (εᵢ(τ) − εⱼ(τ)) dτ. Assume εᵢ, εⱼ are zero-mean, uncorrelated, wide-sense stationary, with autocorrelations Rᵢ(u), Rⱼ(u) satisfying ∫₀^∞ |Rᵢ(u)|du < ∞ and ∫₀^∞ |Rⱼ(u)|du < ∞. Then

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

which grows without bound as t → ∞ whenever Rᵢ + Rⱼ ≠ 0. Consequently, fixed-tolerance absolute timestamp targeting cannot be guaranteed over arbitrarily long horizons without continuous coupling or periodic recalibration.

Remark 1.1:

This theorem motivates phase-window coordination. The remainder of the paper is an operational coordination construction whose bound depends on explicit short-horizon estimation/stability assumptions.

2. Canonical conventions and wrap-safe windows

Remark 2.1 (Canonical conventions are defined externally):

The canonical meanings of (t₀, T_cycle, n(t), ϕ(t)), the negative-safe modulo convention, wrap-safe windows on S¹, and tick encodings are normatively defined by the Conventions document [1]. This paper uses those conventions and does not redefine them.

Definition 2.1 (Phase acceptance window (half-width)):

Given a target phase ϕ₀ ∈ [0, 1) and tolerance ΔΦ ∈ (0, 1/2) define

W(ϕ₀, ΔΦ) := {ϕ ∈ [0, 1) : d_S¹(ϕ, ϕ₀) ≤ ΔΦ}.

The corresponding physical duration is

Δt_w = 2ΔΦ · T_cycle.

3. Phase-Coordination Principle

Principle 3.1 (Phase-Coordination Principle): Replace absolute-time targets by a shared cyclic phase convention. Nodes trigger actions when their local phase estimate enters a wrap-around-safe acceptance window around a target phase, within an explicitly specified cycle index (cycle anchoring).

Definition 3.1 (Local phase estimate and phase error bound):

Node 𝑖 computes a local phase estimate ϕ̂ᵢ(t) = ϕ(t̂ᵢ(t)) from a locally available time variable t̂ᵢ(t) (e.g. corrected monotone clock as specified by the deployment). We say node 𝑖 has a short-horizon phase error bound εᵢ ∈ [0, 1/2) over an operational interval if

d_S¹(ϕ̂ᵢ(t), ϕ(t)) ≤ εᵢ

throughout that interval.

Remark 3.1 (Why cycle anchoring is mandatory):

Because ϕ(t) is cyclic, the same phase recurs every cycle. A coordination instruction must include an intended cycle index; otherwise “same phase” can refer to the wrong cycle.

4. Bounded coordination statement

Theorem 4.1 (Bounded coordination via cycle-anchored phase windows)

Fix a target cycle index n^★ ∈ Z, target phase ϕ₀ ∈ [0, 1) and half-width tolerance ΔΦ ∈ (0, 1/2). Each node triggers at the first time after an arming time t_arm when it detects both:

n(t̂ᵢ(t)) = n^★ and ϕ̂ᵢ(t) ∈ W(ϕ₀, ΔΦ).

Assume nodes 𝑖 and 𝑗 satisfy phase error bounds εᵢ, εⱼ over the triggering interval and that

εᵢ < 1/2 − ΔΦ, εⱼ < 1/2 − ΔΦ,

so that a window expanded by estimation error cannot straddle the cycle boundary. Then their trigger times tᵢ, tⱼ satisfy

|tᵢ − tⱼ| ≤ (2ΔΦ + εᵢ + εⱼ)T_cycle = Δt_w + (εᵢ + εⱼ)T_cycle.

In particular, the bound depends on the chosen window size and short-horizon estimation/stability terms and does not grow with total elapsed time since initialization.

5. Implementation-facing view

Definition 5.1 (Tick-canonical Q-Address minimal view):

An implementation may encode an action as a macro window plus micro slot using Q-Address. The tick-canonical minimal view is:

(qaddr_schema, convention_id, scope, cycle_index, resolution, phi_ticks, deltaPhi_ticks, slot).

This paper recommends qaddr_schema=TV-QADDR-2025-12 as a machine-facing compatibility tag. Full schema, canonical encoding rules, and slot semantics are normatively defined in the Q-Address specification [2].

Remark 5.1 (Derived displays are non-normative):

HS degrees, HS index, SWT labels, and other UI projections are derived-only and must not drive boundary checks or verification. For verification, use tick fields and the Conventions-defined tick-distance membership rule.

6. Security scope

Remark 6.1 (Security scope):

Phase-window coordination does not provide security by itself. Security requires authenticated messaging, nonces, anti-replay policies, and explicit adversary models. Timeverse/Q-Address/TSAE fields are public context (not secrets) and may be bound as associated data; cryptographic suites and policies are defined by Security Profiles [3].

7. Conclusion

The Phase-Coordination Principle replaces long-horizon absolute timestamp targeting by cycle-anchored phase-window triggering. Under explicit short-horizon phase estimation bounds, inter-node execution skew is bounded by the chosen window tolerance plus estimation terms and does not scale with time since initialization. Tick-canonical Q-Address representations provide an implementation-facing encoding that composes naturally with triadic Human–AI–Quantum coordination stacks.

References

[1] T. Ouardi, Phase-Coordination Series Conventions, Zenodo (2025). DOI: 10.5281/zenodo.18068999.
[2] T. Ouardi, Q-Address: Macro Phase + Micro Slot, Zenodo (2025). DOI: 10.5281/zenodo.18068997.
[3] T. Ouardi, Timeverse Security Profile, Zenodo (2025). DOI: 10.5281/zenodo.18069423.
[4] T. Ouardi, Theorem of Temporal Resolution Limitation and the Phase-Coordination Principle (v1.1), Zenodo (2025). DOI: 10.5281/zenodo.17955430.
[5] D. W. Allan, Statistics of atomic frequency standards, Proc. IEEE 54(2), 221–230 (1966). DOI: 10.1109/PROC.1966.4634.
[6] S. Wehner, D. Elkouss, and R. Hanson, Quantum internet: A vision for the road ahead, Science 362(6412), eaam9288 (2018). DOI: 10.1126/science.aam9288.