Phase Harmonic Encoding (PHE) (v1.0)

Deterministic Frequency Mapping from Canonical Phase to Auditory, AI, and Phononic Representations

Status: Official Spec (v1.0)

DOI: 10.5281/zenodo.XXXXXXXX

Abstract

Phase Harmonic Encoding (PHE) specifies a deterministic mapping from the canonical phase variable φ(t) and its discrete components (hour-sol, binute, φ-tick) to frequency-domain representations suitable for three target domains: (i) auditory (human-perceptible pitch and rhythm), (ii) feature (AI-consumable spectral vectors extending TPE), and (iii) phononic (frequency parameters for quantum acoustic transduction at cryogenic microwave frequencies).

The encoding is bijective within each day, periodic across days, and deterministic from the protocol conventions (t₀, Tcycle) alone. For the auditory target, PHE maps the 12 hour-sols to the 12 chromatic pitch classes and the intra-HS position to duration and fine frequency.

1. Motivation and scope

Frequency is a natural target for three reasons:

Human perception: A deterministic phase-to-pitch mapping creates an auditory channel for temporal information, enabling verification by ear and accessibility.
AI features: Spectral representations provide an alternative feature space to TPE's trigonometric encoding.
Phononic quantum interfaces: Mapping temporal coordinates to resonator modes for quantized mechanical vibrations (phonons).

Honest scope for phononic target: The phononic target defines a parameterization interface, not a complete control protocol. Transduction operates at GHz frequencies (~15mK), seven orders of magnitude above audible sound.

2. Phase decomposition

Definition 2.1 (Temporal triple):

From the canonical phase φ(t) ∈ [0,1) with resolution R=86,400, define:

HS(t) ∈ [1,12] (hour-sol index)
bin(t) ∈ [0,59] (binute within HS)
φ-tick(t) ∈ [0,R) (tick index)

Property 2.1 (Intra-day injectivity): For fixed day index d, the map t ↦ (HS(t), bin(t), φ-tick(t)) is injective on [0, Tday).

3. Auditory target: phase-to-pitch encoding

3.1 Pitch mapping

The day divides into 12 hour-sols. The chromatic scale divides the octave into 12 pitch classes.

Definition 3.1 (HS-to-pitch map π):

π: [1,12] → {C, C#, D, D#, E, F, F#, G, G#, A, A#, B}

Maps HS 1 → C, HS 2 → C#, ..., HS 12 → B.

3.2 Duration and Fine Frequency

Duration mapping (δ)

Maps binute to duration in [100, 690] ms.

δ(b) = 100ms + b * 10ms

Fine Frequency mapping (ν)

Maps φ-tick to fine frequency in [220Hz, 440Hz].

ν(φ) = 220 * 2^(φ/R)

Theorem 3.1 (Intra-day injectivity of auditory encoding)

For fixed day d, the map t ↦ a(t) = (π(HS), δ(bin), ν(φ-tick)) is injective on [0, Tday).

4. AI target: spectral feature extension

PHE-AI extends Temporal Phase Encoding (TPE) by appending frequency-domain components.

Definition 4.1 (PHE-AI feature vector):

x_PHE(φ) = [zK(φ), p(HS)/12, δ(bin)/max, log2(ν/lo)] ∈ R^(2K+3)

Compatibility: PHE-AI is a strict extension; systems consuming only TPE can ignore the appended PHE components.

5. Phononic target: parameterization interface

PHE defines parameters to address phononic modes in piezoelectric resonators (GHz frequencies).

mode_index(t) = (HS(t)-1) mod N_modes

theta_phase(t) = 2π * φ-tick(t) / R

tau_duration(t) = δ(bin(t))

6. D-Calendar harmonic structure

Definition 6.1 (D-Calendar chord):

For D-Calendar coordinates (C,Y,M,W,D), the protocol defines a five-note chord representing the hierarchical temporal position.

Week (W): Maps to 5 pitch classes (Pentatonic).
Day (D): Maps to 6 pitch classes (Hexatonic).

7. Applications

Auditory Monitoring

Monitoring Clockchain integrity by ear. Forks or missing blocks produce audible dissonance.

Phononic Memory

Encoding φ-ticks into the phase of a mechanical oscillator for physics-layer attestation.

8. Conclusion

Frequency-Domain Projection: PHE adds a frequency-domain projection to the phase-coordination stack without modifying existing layers, providing a universal language for hearing, training, and vibrating time.

References

[1] Ouardi, T. (2025). Phase-Coordination Series Conventions. DOI: 10.5281/zenodo.18068999
[2] Ouardi, T. (2025). Temporal Phase Encoding (TPE) v1.2. DOI: 10.5281/zenodo.18064354
[3] Chu et al. (2017). Quantum acoustics with superconducting qubits. Science 358(6360).