Temporal Phase Encoding (TPE) (v1.2)

A Theoretical Framework for Phase-Coherent Artificial Intelligence

Status: Official Spec (v1.2)

DOI: 10.5281/zenodo.18064354

Abstract

Temporal Phase Encoding (TPE) is a minimal framework for injecting a cyclic phase coordinate into AI systems so they can reason and act in coordination with periodic processes and phase-window scheduling stacks. TPE represents time by a canonical normalized phase 𝜙(𝑡) ∈ [0, 1) ≅ S1 derived from protocol conventions (𝑡0, 𝑇cycle) and embeds this phase using Fourier features [sin(2𝜋𝑘𝜙), cos(2𝜋𝑘𝜙)]^K_k=1. We formalize the encoding, state basic approximation properties for continuous periodic functions, provide a wrap-safe latency-to-phase bound on the circle, and show how phase-aware AI can select cycle-anchored phase windows compatible with macro-window/micro-slot coordination protocols (Q-Address style). TPE is a representational and decision-layer component; coordination guarantees depend on external bootstrapping and execution layers (phase estimation error, oscillator stability, and latency), not on the encoding alone.

1. Canonical conventions and phase geometry

Definition 1.1 (Protocol conventions):

A deployment fixes:

a reference epoch t₀,
a cycle period T_cycle > 0.

These are protocol conventions for interoperability.

Definition 1.2 (Cycle index and canonical phase):

For an ideal reference time variable t ∈ R define

𝑛(𝑡) := ⌊(𝑡 − 𝑡₀) / 𝑇_cycle⌋ ∈ Z, 𝜙(𝑡) := ((𝑡 − 𝑡₀) / 𝑇_cycle) mod 1 ∈ [0, 1) ≅ S1.

The cycle index 𝑛(𝑡) prevents ambiguity because 𝜙 repeats every cycle.

Remark 1.1 (Conventions vs calibration / bootstrapping): Bootstrapping procedures (e.g. QBP-style initialization) estimate node-specific clock parameters (offset/drift) so that a node can compute a reliable local time variable t̂(t) and therefore a reliable phase estimate ϕ̂(t) = 𝜙(t̂(t)) relative to the already-chosen conventions (𝑡₀, 𝑇_cycle). Bootstrapping does not define the conventions themselves.

Definition 1.3 (Circular distance on S¹):

For ϕ, ψ ∈ [0, 1) define

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

2. Temporal Phase Encoding (TPE)

2.1 Fourier feature encoding

Definition 2.1 (TPE phase features):

For harmonic order K >= 1, define the Fourier feature map

z_K(𝜙) := [sin(2𝜋𝑘𝜙), cos(2𝜋𝑘𝜙)]^K_k=1 ∈ R^2K.

Given input features x ∈ R^d at phase 𝜙, define the TPE-augmented representation

x_TPE(𝜙) := [x, z_K(𝜙)] ∈ R^d+2K.

Lemma 2.1 (Circle embedding for K ≥ 1):

The map 𝜙 ↦ → (sin(2𝜋𝜙), cos(2𝜋𝜙)) is injective on the half-open interval representation [0, 1). Thus TPE contains a faithful representation of the phase point on S¹ under the chosen [0, 1) convention.

Proposition 2.1 (Periodicity): For any real 𝜙, z_K(𝜙) = z_K(𝜙 mod 1). Hence TPE is cycle-periodic by construction.

3. Approximation statements

Lemma 3.1 (Trigonometric approximation (density)):

Trigonometric polynomials are dense in the space of continuous 1-periodic functions on the circle (uniform norm), i.e. any continuous periodic function can be approximated arbitrarily well by finite Fourier sums [7].

Theorem 3.1 (TPE supports approximation of continuous periodic maps)

Let f : [0, 1) → R^m be continuous and 1-periodic. For any ε > 0, there exist a harmonic order K and a feedforward neural network N_θ (with a standard universal approximation activation) such that

sup_𝜙∈[0,1) || f(𝜙) − N_θ(z_K(𝜙))|| <= ε.

Remark 3.1 (What this does and does not say): This result is about representational capacity for periodic functions of a single phase variable. It does not guarantee generalization under nonperiodic drift, distribution shift, or adversarial timing; those require additional modeling and validation.

4. Latency-to-phase error (wrap-safe)

Lemma 4.1 (Latency bound on the circle):

Assume an inference/actuation latency bound L >= 0 seconds. Since canonical phase advances at rate 1/T_cycle, the phase drift over latency satisfies

d_S1(𝜙(𝑡), 𝜙(𝑡 − 𝐿)) <= 𝐿 / T_cycle.

Remark 4.1 (Engineering rule for window sizing): If an execution layer uses phase windows with half-width ΔΦ, a common engineering guideline is to choose ΔΦ >> L / T_cycle plus additional margins for phase-estimation error and short-horizon clock instability.

5. Phase-window scheduling and Q-Address interface

Definition 5.1 (Phase window (half-width convention)):

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

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

Protocol 5.1 (Phase-aware window selection (AI))

Given a task and constraints, an AI component selects a cycle-anchored macro window (𝑛, ϕ₀, ΔΦ). Execution is triggered when the local phase estimate enters W(ϕ₀, ΔΦ) within the intended cycle index 𝑛.

Definition 5.2 (Q-Address minimal view):

A coordination stack may encode an AI decision as a Q-Address-style instruction (minimal view):

(convention_id, scope, 𝑛, ϕ₀, ΔΦ, slot),

where slot specifies micro sequencing inside the window (index slots or local offsets). The full Q-Address schema and canonical encoding rules are specified in the dedicated Q-Address specification.

Remark 5.1 (Triadic stack positioning): TPE is the AI layer in a triadic Human–AI–Quantum stack: humans select/approve windows (HTIL-style), AI proposes/optimizes windows (TPE), and quantum/hardware executes inside windows with local micro-timing (TAQA/Q-Address/QBP-style layers).

6. Limitations and future work

Periodicity assumption: TPE targets cyclic phenomena; nonperiodic trends require additional features.
Resolution: finite K limits representation of high-frequency patterns.
Latency and drift: tight coordination depends on latency bounds and short-horizon stability.
Multi-scale cycles: deployments may require multiple phase variables (daily/weekly/seasonal), handled by multi-frequency encodings.

7. Conclusion

TPE injects a cyclic phase coordinate into AI models using Fourier features. It supports phase-aware reasoning and cycle-anchored window selection in coordination stacks where macro phase windows are combined with micro-slot local sequencing. The framework is minimal and interoperable: it uses a canonical phase variable 𝜙(𝑡) ∈ S¹ and explicit conventions, and it cleanly interfaces with Q-Address-style coordination metadata.

References

[1] Ouardi, T. (2025). Theorem of Temporal Resolution Limitation and the Phase-Coordination Principle. Zenodo. DOI: 10.5281/zenodo.17955430
[2] Ouardi, T. (2025). The Phase-Coordination Principle: From Fundamental Theorem to Quantum Network Implementation. Zenodo. DOI: 10.5281/zenodo.17969766
[3] Ouardi, T. (2025). Temporal-Angular Quantum Addressing (TAQA): A Deterministic Coordination Layer for Distributed Quantum Systems. Zenodo. DOI: 10.5281/zenodo.17956182
[4] Ouardi, T. (2025). Quantum Bootstrapping Protocol (QBP): Autonomous Initialization for Phase-Coordinated Quantum Networks. Zenodo. DOI: 10.5281/zenodo.17981366
[5] G. Cybenko, Approximation by superpositions of a sigmoidal function, Mathematics of Control, Signals and Systems 2(4), 303–314 (1989).
[6] K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural Networks 4(2), 251–257 (1991).
[7] Y. Katznelson, An Introduction to Harmonic Analysis, 3rd ed., Cambridge University Press (2004).