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Phase-Coordination Series Conventions

Epoch, Cycle Period, Cycle Index, Wrap-Safe Phase Windows, HS Notation, and Fixed-Point Encoding

Status: Official Spec
DOI: 10.5281/zenodo.18068999

Abstract

This note fixes canonical conventions used across a phase-coordination document series: (i) protocol conventions (t0, Tcycle) together with an explicit computation time-scale requirement, (ii) canonical cyclic phase ϕ(t) ∈ [0, 1) ≅ S1 and cycle index n(t), (iii) circular distance and wrap-around-safe phase windows with half-width tolerance ΔΦ, (iv) derived HS◦ degrees and HS index displays, (v) mapping between a U(1) angle α and normalized phase ϕ, and (vi) a fixed-point (ticks) encoding rule to avoid float boundary ambiguity. These are interoperability conventions, not claims about universal physical time.

1. Purpose and scope

Principle 1.1 (One canonical phase, multiple representations): Coordination is performed on a single cyclic phase variable ϕ(t) ∈ S1. Different layers may represent the same phase point as a normalized number in [0, 1), an angle in degrees, or a discrete segment index. These are representational choices only.

This conventions note standardizes:

  • the meaning of t0 and Tcycle,
  • the canonical phase map ϕ(t) and cycle index n(t),
  • phase distance and wrap-safe window definitions on S1,
  • the tolerance convention (half-width) and window-duration mapping,
  • HS◦ (degrees) and HS index (1–12) as human-friendly views of ϕ,
  • the mapping α ↔ ϕ for U(1) parameterizations,
  • fixed-point encoding for robust implementations and verification.

2. Time scale requirement

Definition 2.1 (Arithmetic time variable (autonomous monotone seconds)): Implementations MUST perform arithmetic on a linear/monotone time variable t ∈ ℝ (or fixed-point integer), not directly on UTC wall-clock timestamps. The arithmetic time variable MUST be autonomous (no leap seconds). UTC/RFC3339 timestamps are permitted for human-facing I/O and logging, but MUST be mapped into the chosen arithmetic time variable according to an explicit policy carried by convention_id.

Remark 2.1 (Why not compute directly in UTC?): UTC includes leap-second adjustments (and operationally also “leap smear” policies), which can introduce ambiguity for arithmetic unless the mapping policy is explicit [1, 2]. The key requirement is: all parties must share the same arithmetic time definition [3].

Definition 2.2 (Recommended autonomy policy tag): Deployments SHOULD declare an explicit timescale tag in convention_id, e.g. timescale=TV_MONO_LEAP, meaning: a monotone second-count used for protocol arithmetic, with no leap seconds.

3. Reference parameters

Definition 3.1 (Reference epoch and cycle period (protocol conventions)): A deployment fixes:

  • a reference epoch t0,
  • a cycle period Tcycle > 0.
These are protocol-level conventions chosen for interoperability.

4. Canonical phase variable and cycle index

Definition 4.1 (Normalized phase on the circle): For an arithmetic time variable t ∈ ℝ, define the normalized phase

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

Definition 4.2 (Modulo convention (negative-safe)): Throughout the series, “x mod 1” denotes the unique representative in [0, 1):

x mod 1 := x − ⌊x⌋.

This convention is required for consistent behavior when (t − t0)/Tcycle < 0.

Definition 4.3 (Cycle index): Define the cycle index

n(t) := ⌊(t - t0) / Tcycle⌋ ∈ ℤ.

A coordination instruction SHOULD include the intended cycle index n (or an equivalent cycle label) to avoid phase-cycle ambiguity.

Remark 4.1 (Local phase estimates): A node i computes a local phase estimate by applying the same mapping to a locally available arithmetic time variable t̂i(t) (e.g. a local clock corrected by an estimated offset/drift):

ϕ̂i(t) = ((t̂i(t) - t0) / Tcycle) mod 1,

n̂i(t) = ⌊(t̂i(t) - t0) / Tcycle⌋.

How t̂i(t) is obtained is implementation-dependent (bootstrapping/calibration layers).

5. Circular distance and phase windows

Definition 5.1 (Circular distance on S¹): For ϕ, ψ ∈ [0, 1) define the circular distance

dS¹(ϕ, ψ) := min{|ϕ − ψ|, 1 − |ϕ − ψ|}.

Definition 5.2 (Phase acceptance window (half-width convention)): Given a target phase ϕ0 ∈ [0, 1) and tolerance ΔΦ ∈ (0, 1/2), define

W(ϕ0, ΔΦ) := {ϕ ∈ [0, 1) : d<sub>S¹</sub>(ϕ, ϕ<sub>0</sub>) ≤ ΔΦ}.

Definition 5.3 (Physical duration of a phase window): The physical duration corresponding to half-width ΔΦ is

Δtw = 2ΔΦ · Tcycle.

Remark 5.1 (Wrap-around is intrinsic): Windows are subsets of the circle. When ϕ0 is near 0, the window may include phases near 1. Implementations must use dS¹ (not naive absolute difference) to avoid edge-case errors.

6. HS notation: degrees and index (derived displays)

Definition 6.1 (HS◦ degrees display): The HS◦ angular representation (degrees) is

HS◦deg(t) := 360◦ · ϕ(t) ∈ [0◦, 360◦).

Definition 6.2 (HS index (12-segment display)): For a human-facing discretization into 12 equal segments:

HSidx(t) := ⌊12 · ϕ(t)⌋ + 1 ∈ {1, 2, . . . , 12}.

Remark 6.1 (One reference, three representations):

ϕ(t) ∈ [0, 1) ⇔ HS◦deg(t) ∈ [0◦, 360◦) ⇔ HSidx(t) ∈ {1, . . . , 12}.

ϕ is canonical for computation, HS◦deg for visualization, and HSidx for simple human indexing.

7. Mapping to U(1) parameterizations

Definition 7.1 (U(1) angle α and normalized phase ϕ): The U(1) angle α ∈ [0, 2π) and normalized phase ϕ ∈ [0, 1) are related by

α = 2πϕ, ϕ = α/2π.

8. Convention identifier

Definition 8.1 (convention_id requirements (normative)): A convention_id MUST uniquely identify the conventions required to interpret (n, ϕ) and window membership. At minimum it MUST bind (t0, Tcycle) and MUST declare:

  • the arithmetic timescale policy (recommended: TV_MONO_LEAP),
  • the encoding policy (float vs fixed-point ticks),
  • boundary conventions (e.g. window membership uses inclusive edge “≤”).
If ticks are used, it MUST bind the resolution R (ticks per cycle).

9. Fixed-point (ticks) encoding

Principle 9.1 (No float boundary ambiguity for verification): Implementations MUST NOT use floating-point comparisons for window membership in verification/gating logic or in signed/verified semantics. Fixed-point ticks MUST be used for boundary checks. Float displays are UI-only.

Definition 9.1 (Tick fields): Fix a resolution R ∈ ℕ (ticks per cycle). A phase window is encoded by integers:

ϕticks ∈ {0, . . . , R − 1}, ΔΦticks ∈ {1, . . . , ⌊R/2⌋ − 1}.

The corresponding reals are ϕ0 = ϕticks/R and ΔΦ = ΔΦticks/R (derived only).

Definition 9.2 (Time-to-ticks conversion (recommended)): Let t be the arithmetic time variable and assume Tcycle and t0 are expressed in the same units. Define the within-cycle offset

δ(t) := (t - t0) mod Tcycle ∈ [0, Tcycle),

where x mod Tcycle := x − Tcycle⌊x/Tcycle⌋. A robust tick encoding is:

ϕticks(t) := ⌊(R/Tcycle)δ(t)⌋ ∈ {0, . . . , R − 1}.

This uses floor (not rounding), giving an unambiguous half-open binning of the cycle.

Definition 9.3 (Circular distance and window membership in ticks): Let a0 := ϕticks denote the center tick. For ticks a, b ∈ {0, . . . , 𝑅 − 1} define

dticks(a, b) := min{|a − b|, R − |a − b|}.

Window membership is computed as:

a ∈ W ⇔ dticks(a, a0) ≤ ΔΦticks.

10. Conclusion

This note fixes canonical definitions of ϕ(t) on S1, cycle index n(t), circular distance and wrap-safe windows, the half-width tolerance convention, HS◦ degrees and HS index displays, the mapping between U(1) and S1 parameterizations, and a tick-canonical encoding suitable for verification. Using a single conventions reference reduces ambiguity and implementation errors across documents and deployments.

References

Timeverse Normative References

  • [4] T. Ouardi, Theorem of Temporal Resolution Limitation and the Phase-Coordination Principle (v1.1), Zenodo (2025). DOI: 10.5281/zenodo.17955430.
  • [5] T. Ouardi, The Phase-Coordination Principle: From Fundamental Theorem to Quantum Network Implementation (v1.0), Zenodo (2025). DOI: 10.5281/zenodo.17969766.
  • [6] T. Ouardi, Temporal-Angular Quantum Addressing (TAQA): A Deterministic Coordination Layer for Distributed Quantum Systems (v1.0), Zenodo (2025). DOI: 10.5281/zenodo.17956182.
  • [7] T. Ouardi, Quantum Bootstrapping Protocol (QBP): Autonomous Initialization for Phase-Coordinated Quantum Networks (v1.2), Zenodo (2025). DOI: 10.5281/zenodo.18064435.
  • [8] T. Ouardi, Temporal Phase Encoding (TPE): A Theoretical Framework for Phase-Coherent Artificial Intelligence (v1.2), Zenodo (2025). DOI: 10.5281/zenodo.18064354.
  • [9] T. Ouardi, Quantum HS◦: A Resource Theory of Temporal Alignment (v1.1), Zenodo (2025). Version DOI: 10.5281/zenodo.18035886. Concept DOI: 10.5281/zenodo.18011999.
  • [10] T. Ouardi, Experimentally Measurable Quantification of Quantum HS◦ Resource Consumption (v1.0), Zenodo (2025). DOI: 10.5281/zenodo.18056884.