Voice Channel UWD Skill

Status: ✅ Production Ready
Trit: 0 (ERGODIC - balanced flow between play/coplay)
Principle: Voice communication as undirected wiring diagram with GF(3) conservation
Source: plurigrid/VoiceChannelUWD.jl + UnwiringDiagrams.jl + Arena Protocol

---

Overview

Voice Channel UWD models real-time voice communication using the categorical framework of undirected wiring diagrams:

┌─────────────────────────────────────────────────────────────────────┐
│                    VOICE CHANNEL AS UWD                             │
├─────────────────────────────────────────────────────────────────────┤
│                                                                     │
│   ┌──────────┐     ┌──────────┐     ┌──────────┐                   │
│   │ 🔊 Alice │     │ 👂 Bob   │     │ 🔇 Carol │   ← Boxes         │
│   │ trit: +1 │     │ trit: 0  │     │ trit: -1 │     (Participants) │
│   │ #D82626  │     │ #26D826  │     │ #2626D8  │                    │
│   └────┬─────┘     └────┬─────┘     └────┬─────┘                   │
│        │                │                │                          │
│        └───────────┬────┴────┬───────────┘                          │
│                    │         │                                      │
│              ┌─────┴─────────┴─────┐                                │
│              │   JUNCTION          │  ← Audio Mix Point             │
│              │   oapply = colimit  │    (Shared State)              │
│              └─────────────────────┘                                │
│                       │                                             │
│               ┌───────┴───────┐                                     │
│               │ OUTER PORTS   │  ← External I/O                     │
│               │ 🎙️ Record     │    (WhiteHole/NATS)                  │
│               │ 📡 Stream     │                                     │
│               └───────────────┘                                     │
│                                                                     │
│   GF(3) Conservation: (+1) + (0) + (-1) = 0 ✓                      │
└─────────────────────────────────────────────────────────────────────┘

Categorical Mapping

Voice Concept	UWD Concept	Categorical Role
Participant	Box	ResourceSharer with audio state
Speaking (+1)	Port out	Generative contribution
Listening (0)	Port in/out	Neutral transport
Muted (-1)	No contribution	Observation only
Audio Mix	Junction	Pushout gluing shared state
oapply	Composition	Colimit of audio streams
Play	Forward pass	Arena: Ω × X → Y
Coplay	Backward pass	Arena: Ω × X × R → ℧ × S

Core Components

1. VoiceChannelUWD.jl

Core undirected wiring diagram model:

using VoiceChannelUWD

# Create channel with 3 participants (GF(3) balanced)
channel = create_voice_uwd("room-1", ["alice", "bob", "charlie"])

# Compose audio via oapply (colimit)
mixed_audio = oapply_voice(channel)

# Arena play/coplay
play_result = voice_play(channel, "alice")
coplay_result = voice_coplay(channel, "alice", mixed_audio, feedback)

2. VoiceChannelNATS.jl

NATS message broker integration:

using VoiceChannelNATS

# Connect to NATS
client = NATSVoiceClient(channel, "alice")
connect!(client)

# Publish audio (play)
publish_audio(client, audio_frame)

# Subscribe to mixed audio (coplay)
subscribe_coplay!(client) do msg, feedback
    # Deliver to speakers
end

NATS Topics:

plurigrid.voice.{channel}.play.{participant}     # Outbound audio
plurigrid.voice.{channel}.coplay.{participant}   # Inbound mixed
plurigrid.voice.{channel}.peers                  # Peer discovery
plurigrid.voice.{channel}.meta                   # Channel metadata

3. VoiceChannelWhiteHole.jl

macOS virtual audio driver binding:

using VoiceChannelWhiteHole

# Create WhiteHole binding
binding = create_whitehole_binding(channel, "alice")

# Start audio I/O
start_input!(binding)   # Microphone → Channel
start_output!(binding)  # Channel → Speakers

# Color-tagged stream mixing
result = mix_colored_streams(streams)

Stream Colors:

• Hue 0-120°: PLUS (+1) speaking streams
• Hue 120-240°: ERGODIC (0) transport streams
• Hue 240-360°: MINUS (-1) monitoring streams

GF(3) Trit Assignment

State	Trit	Color	Meaning
SPEAKING	+1	#D82626 (Red)	Generative - producing audio
LISTENING	0	#26D826 (Green)	Ergodic - processing/routing
MUTED	-1	#2626D8 (Blue)	Observation - silent consume

Conservation Law:

∀ channel: Σ participant.trit ≡ 0 (mod 3)

Arena Protocol

From Capucci et al. "Parametrised Lenses":

Arena A_G : Lens_{(Ω,℧)}(X,S)(Y,R)

where:
  Ω = Π_{p∈P} {speaking, listening, muted}  (strategy profiles)
  ℧ = Π_{p∈P} ℝ                              (latency/quality)
  X = audio buffer states
  S = listening buffer costates
  Y = mixed audio output
  R = quality feedback

Play:   play_A : Ω × X → Y      (participant → channel)
Coplay: coplay_A : Ω × X × R → ℧ × S  (channel → participant)

Commands

# Run voice session demo
julia plurigrid/VoiceChannelUWD.jl

# Run NATS-integrated session
julia plurigrid/VoiceChannelNATS.jl

# Run WhiteHole audio session
julia plurigrid/VoiceChannelWhiteHole.jl

# Create voice channel
just voice-channel create room-1 alice bob charlie

# Join voice channel
just voice-channel join room-1 --as alice

# Check GF(3) balance
just voice-channel gf3-check room-1

Integration with Existing Skills

unwiring-arena

# Voice channel uses same Play/Coplay protocol
using UnwiringArena

arena = VoiceArena(channel)
arena_step!(arena)  # Full play/coplay cycle

oapply-colimit

# Audio mixing IS oapply
using AlgebraicDynamics

# Each participant is a ResourceSharer
participants = [ResourceSharer([:audio], produce_fn) for p in channel]

# Mix via colimit
mixed = oapply(voice_uwd, participants)

gay-mcp

# Deterministic color assignment
using Gay

color = gay_color(participant.seed)
trit = color_to_trit(color)

Mathematical Foundation

Undirected Wiring Diagram

UWD = (B, J, π: P → J, ρ: P → B, O, o: O → J)

where:
  B = set of boxes (participants)
  J = set of junctions (mix points)
  P = set of ports (audio in/out)
  O = outer ports (external I/O)

oapply as Colimit

function oapply(d::UndirectedWiringDiagram, xs::Vector{ResourceSharer})
    # 1. Coproduct of state spaces (all audio buffers)
    S = coproduct((FinSet ∘ nstates).(xs))
    
    # 2. Pushout identifies shared at junctions (audio mixing)
    S′ = pushout(portmap, junctions)
    
    # 3. Induced dynamics sum at junctions
    return ResourceSharer(induced_interface, sum_dynamics)
end

GF(3) Conservation

For any voice channel with n participants:
  
  1. Assign trits: trit(pᵢ) ∈ {-1, 0, +1}
  2. Conservation: Σᵢ trit(pᵢ) ≡ 0 (mod 3)
  3. If n = 3k: assign k each of {-1, 0, +1}
  4. If n ≠ 3k: some participants must share trit

GF(3) Triads

voice-channel-uwd (0) ⊗ gay-mcp (+1) ⊗ temporal-coalgebra (-1) = 0 ✓
voice-channel-uwd (0) ⊗ oapply-colimit (+1) ⊗ sheaf-cohomology (-1) = 0 ✓
unwiring-arena (0) ⊗ voice-channel-uwd (0) ⊗ bisimulation-game (0) = 0 ✓

Example Session

═══════════════════════════════════════════════════════════════
  Voice Session via NATS + UWD
═══════════════════════════════════════════════════════════════

[1] Created voice channel: demo-room
    Participants: alice, bob, charlie

[2] Created NATS client for: alice

[3] Connected to NATS: nats://nonlocal.info:4222

[4] Announced presence

[5] Registered coplay callback

[6] Running play/coplay loop (9 frames):
    Frame 1: play=1, GF(3)=0 ✓
    Frame 2: play=1, GF(3)=0 ✓
    Frame 3: play=1, GF(3)=0 ✓
    ...

[7] Final GF(3) conservation check:
    Accumulator: 0
    Mod 3: 0
    Balanced: true

[8] Left channel

✓ Voice session complete

References

• Libkind "An Algebra of Resource Sharers"
• Capucci et al. "Parametrised Lenses"
• AlgebraicJulia/AlgebraicDynamics.jl
• plurigrid/UnwiringDiagrams.jl
• WhiteHole Audio Driver

---

Skill Name: voice-channel-uwd
Type: Real-time Communication / Compositional Audio
Trit: 0 (ERGODIC)
Color: #26D826 (Green)
Principle: Voice as categorical composition with GF(3) conservation

Scientific Skill Interleaving

This skill connects to the K-Dense-AI/claude-scientific-skills ecosystem:

Graph Theory

• networkx [○] via bicomodule
  • Universal graph hub

Bibliography References

• general: 734 citations in bib.duckdb

Cat# Integration

This skill maps to Cat# = Comod(P) as a bicomodule in the equipment structure:

Trit: 0 (ERGODIC)
Home: Prof
Poly Op: ⊗
Kan Role: Adj
Color: #26D826

GF(3) Naturality

The skill participates in triads satisfying:

(-1) + (0) + (+1) ≡ 0 (mod 3)

This ensures compositional coherence in the Cat# equipment structure.