The Phase-Unified Computing Stack (v2)

Geometric Foundations of Phase-Coordinated Systems

Status: Official Spec (v2.0)

DOI: 10.5281/zenodo.18057136

Abstract

This paper presents a geometric unification of quantum control, artificial intelligence scheduling, and human-facing coordination under a single phase-coordinated framework. All components are modeled as operations on phase manifolds, linked by structure-preserving mappings on relevant U(1) phase actions. We formalize a precise correspondence between qubit Z-axis phase rotations and temporal-circle phase shifts, define a stack architecture for triadic coordination using phase windows, and state coordination bounds under explicit assumptions on phase estimation and short-horizon oscillator stability.

Errata and Clarifications (v2)

This v2 clarifies mathematical scope and removes over-strong claims without changing the operational goal: phase-based coordination across domains.

Scope of the quantum–temporal mapping. “Isomorphism” statements involving the Bloch sphere are restricted to the U(1) phase-rotation subgroup generated by qubit Z-axis phase gates R_z(α). No claim is made about an isomorphism between the full rotation group of S² (or SU(2)) and the circle S¹.
Status of the epoch t₀. The parameter t₀ is an engineering convention used as a shared reference for interoperability (analogous to adopting a calendar epoch). It is not a universal physical constant.
Phase-window operators. Mappings from phase windows W ⊂ S¹ to qubit operators are interpreted as associations to positive operators (effects), not as a Boolean-algebra isomorphism. Boolean operations on windows do not generally correspond to algebraic operations on a single-qubit operator algebra.

1. Introduction

A recurring engineering problem across quantum systems, AI pipelines, and human-centered operations is temporal coordination under imperfect clocks, latency, and limited synchronization infrastructure. A phase-based approach models coordination as alignment on a shared cyclic coordinate φ ∈ S¹.

This paper presents a unification layer: different domains implement compatible U(1) phase actions (phase rotations), enabling a stack that coordinates via phase windows and structure-preserving mappings.

1.1 Unification Thesis

Principle 1.1 (Phase Unification): System components operate on phase manifolds:

Quantum: Bloch sphere S² with a distinguished U(1) phase subgroup.
Temporal: Circle S¹ (HS-phase coordination).
AI: Phase-embedded representation space P_AI.
Human: Cognitive phase circle C_cog.

These are connected by structure-preserving mappings on relevant phase actions, forming a phase-unified computing stack.

2. Phase Manifolds

2.1 Quantum Phase Manifold

Definition 2.1 (Bloch Sphere):

A single-qubit pure state can be represented by:

S² = {(θ, φ) ∈ [0, π] × [0, 2π)} ∼= S², |ψ⟩ = cos(θ/2)|0⟩ + e^iφ sin(θ/2)|1⟩.

Definition 2.2 (Qubit Z-Axis Phase Rotation):

A Z-axis phase gate implements a U(1) phase rotation:

R_z(α) = e^-iαZ/2, α ∈ [0, 2π).

2.2 Temporal Phase Manifold

Definition 2.3 (Public Reference Parameters (Engineering Convention)):

All phase calculations reference a fixed reference epoch:

t₀ = 2022-09-23 06:00:00 UTC (Unix timestamp 1663912800),

and a cycle duration:

T_cycle = 86,400 s (solar day).

These parameters are fixed by convention for interoperability within a given deployment.

Definition 2.4 (Temporal Circle):

The HS-phase coordinate resides on:

S¹ = [0, 1) ∼= S¹, φ_HS(t) = ((t - t₀) mod T_cycle) / T_cycle.

Definition 2.5 (Temporal Rotation):

Rot_S1(Δφ) : φ → (φ + Δφ) mod 1.

2.3 AI Phase Manifold

Definition 2.6 (Phase-Embedded AI Space):

A common embedding of φ ∈ S¹ into an AI feature space uses Fourier features:

P_AI = {x_phase(φ) : φ ∈ S1}} ⊂ R^d+2K, x_phase(φ) = [x, sin(2πkφ), cos(2πkφ)]^K_k=1.

2.4 Human Cognitive Phase Manifold

Definition 2.7 (Cognitive Circle):

Human circadian-like phase variables can be modeled on:

C_cog = [0, 1) ∼= S¹,

with a calibration mapping H : S¹ → C_cog defined operationally (e.g. via individual calibration).

3. Isomorphism and Embedding Statements

3.1 Quantum–Temporal Correspondence (U(1) Phase Actions)

Theorem 3.1 (U(1) Phase-Action Isomorphism)

Let {R_z(α)}α∈[0,2π) be the qubit Z-axis phase subgroup and let {Rot_S1(Δφ)}Δφ∈[0,1) be temporal circle rotations. Define

I_QT(R_z(α)) = Rot_S1(α / 2π).

Then I_QT is a group isomorphism between these two U(1) actions:

I_QT(R_z(α)R_z(β)) = I_QT(R_z(α)) ∘ I_QT(R_z(β)), I_QT(R_z(0)) = id, I_QT(R_z(2π)) = id.

Remark 3.1: This theorem concerns the U(1) phase-rotation subgroup of qubit control, not the full rotation group of the Bloch sphere.

3.2 Temporal–AI Embedding

Theorem 3.2 (Fourier Feature Embedding)

For K ≥ 1, the mapping

E_TA : S¹ → R^2K, φ → [sin(2πkφ), cos(2πkφ)]^K_k=1

is continuous and injective.

3.3 AI–Cognitive Learning

Theorem 3.3 (Circadian Mapping Learnability (informal))

A data-driven system can learn an approximation of the calibration map H : S¹ → C_cog given sufficient labeled data and a suitable function class. Formal learnability guarantees depend on modeling assumptions and are not specialized here.

3.4 Composite Correspondence

Corollary 3.3.1 (Triadic Phase Correspondence). There exists a composite correspondence acting on phase variables:

{R_z(α)} → S¹ → P_AI → C_cog,

enabling coordinated phase operations across domains at the level of their relevant U(1) phase actions.

4. Phase-Unified Computing Stack

4.1 Stack Architecture

Quantum Layer (S², U(1) phase gates R_z(α))

↓ I_QT

Coordination Layer (S¹, HS-phase φ_HS(t))

↓ E_TA

Intelligence Layer (P_AI via phase embedding)

↓ H

Interface Layer (C_cog via calibration)

Figure 1: Phase-Unified Computing Stack with structure-preserving mappings on relevant U(1) phase actions.

4.2 Layer Specifications

Layer	Phase Manifold	Role
Quantum	S² (with U(1) phase subgroup)	Phase-rotation control
Coordination	S¹	Shared phase variable for scheduling
Intelligence	P_AI	Phase-aware computation/planning
Interface	C_cog	Human-facing phase interpretation

Principle 4.1 (Stack Phase Coherence): All layers maintain coherence through a shared phase coordinate φ_HS and structure-preserving mappings on phase actions:

φ_quantum ↔ φ_HS ↔ φ_AI ↔ φ_cog.

5. Phase-Window Logic

Definition 5.1 (Phase Window):

A phase window is a measurable subset W ⊂ S¹.

Definition 5.2 (Phase-Window Gate):

A phase-window gate is a mapping G : P(S¹) → P(S¹), where P(S¹) denotes measurable windows on S¹.

Gate Operation	Definition
Identity	I(W) = W
Rotation	R_α(W) = {w + α mod 1 : w ∈ W}
Intersection	AND(W₁, W₂) = W₁ ∩ W₂
Union	OR(W₁, W₂) = W₁ ∪ W₂
Complement	NOT(W) = S¹ \ W

5.1 Window-to-Operator Association

Theorem 5.1 (Phase-window effect operator)

Let |w⟩ = 1/√2 (|0⟩ + e^2πiw|1⟩) for w ∈ [0, 1). For a measurable window W ⊂ S¹, define

E_W := ∫_w∈W |w⟩⟨w| dw.

Then E_W is positive semidefinite on C² and satisfies

0 ≤ E_W ≤ |W| I,

where |W| is the Lebesgue measure of W.

Remark 5.1: The association W → E_W can be interpreted as constructing effects (POVM elements up to normalization). No Boolean-algebra isomorphism is claimed.

6. Coordination Protocols

Protocol 6.1 (Phase-Unified Triadic Coordination).

Compute current phase φ = φ_HS(t) using the agreed convention.
Each agent declares an availability/validity window: W_Q, W_AI, W_H ⊂ S¹.
Compute intersection: W_align = W_Q ∩ W_AI ∩ W_H.
If W_align is not empty, pick a target φ_target ∈ W_align and execute when φ_HS(t) ∈ W_align.

6.1 Formal Bounds (under assumptions)

Theorem 6.1 (Phase-unified coordination bound (window form))

Assume agents implement phase-triggering with a target window of phase width Δφ and short-horizon relative timing uncertainty bounded by a stability term κ(Δφ) over the corresponding window duration. Then the maximum inter-agent execution skew satisfies the bound

max_i,j |t_i - t_j| ≤ Δφ · T_cycle + κ(Δφ),

up to the convention used to define Δφ (window width vs half-width).

Remark 6.1: This statement is an architectural restatement of window-based bounds: precise constants depend on whether Δφ denotes the full window width or the half-width, and on the implementation-specific stability model used to bound κ.

7. Mathematical Structures

7.1 Fiber-Bundle View (informal)

The stack can be viewed as a fibered structure over the temporal phase circle: E → S¹, with fibers representing quantum, AI, and cognitive states parameterized by phase.

7.2 Information Geometry (informal)

Each phase manifold admits natural metrics (e.g. Fubini–Study on S², arc-length on S¹). Approximate metric preservation depends on the chosen normalizations and calibration procedures.

8. Limitations

Periodicity assumptions are idealized; real cycles drift and must be modeled.
Bounds depend on explicit phase-estimation and short-horizon stability assumptions.
Human and AI layers require operational definitions and validation for specific deployments.

9. Conclusion

We presented a phase-unified stack that connects quantum phase control, temporal phase scheduling, phase-aware AI embeddings, and human-facing calibration through structure-preserving correspondences on relevant U(1) phase actions. The result is a coherent architecture for triadic coordination using phase windows, with coordination bounds stated under explicit assumptions.

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