Discohy Streams

---
name: discohy-streams
description: DisCoPy categorical color streams via Hy with 3 parallel TAP streams
  and 7 operad variants
metadata:
  trit: 0
---

# discohy-streams - DiscoHy Operadic Color Streams

## Overview

Provides personalized color streams using DisCoPy categorical diagrams via Hy (discohy). Each human gets a self-learning color embedding with 3 parallel streams, now extended with **7 operad variants** for compositional structure.

**Trit**: 0 (ERGODIC - Coordinator)
**GF(3) Triad**: `three-match (-1) ⊗ discohy-streams (0) ⊗ gay-mcp (+1) = 0 ✓`

## Core Concepts

### 3 Parallel Streams (Balanced Ternary)

```
color://human-id/LIVE      → +1 (forward, real-time)
color://human-id/VERIFY    →  0 (verification, BEAVER)
color://human-id/BACKFILL  → -1 (historical, archived)
```

### 7 DiscoHy Operad Variants

| Operad | File | Trit | Structure |
|--------|------|------|-----------|
| **Little Disks** (E₂) | `discohy_operad_1_little_disks.py` | ⊕ +1 | Configuration of non-overlapping disks |
| **Cubes** (E_∞) | `discohy_operad_2_cubes.py` | ⊖ -1 | Hypercube parallel structures |
| **Cactus** | `discohy_operad_3_cactus.py` | ⊖ -1 | Trees with cycles (self-modification) |
| **Thread** | `discohy_operad_4_thread.py` | ⊙ 0 | Thread continuations + DuckDB |
| **Gravity** | `discohy_operad_5_gravity.lisp` | ⊖ -1 | Moduli M_{0,n} with involutions |
| **Modular** | `discohy_operad_6_modular.bb` | ⊕ +1 | Genus-labeled runtime polymorphism |
| **Swiss-Cheese** | `discohy_operad_7_swiss_cheese.py` | ⊕ +1 | Open/closed for forward-only learning |

**GF(3) Total**: (+1) + (-1) + (-1) + (0) + (-1) + (+1) + (+1) = 0 ✓

### Libkind-Spivak Operads (AlgebraicDynamics)

From Sophie Libkind's thesis and AlgebraicJulia:

| Operad | Trit | Description |
|--------|------|-------------|
| **Directed** (⊳) | +1 | Output→Input wiring diagrams |
| **Undirected** (○) | -1 | Interface matching via pullback |
| **Machines** | 0 | State machines with dynamics |
| **Dynamical** | +1 | Open ODEs: dx/dt = f(x,u) |

### ∞-Operads

| Model | Description |
|-------|-------------|
| **Dendroidal** | Trees as colored operads (Cisinski-Moerdijk) |
| **Lurie** | coCartesian fibrations over Fin_* |
| **Segal** | Quillen equivalent to simplicial operads |

## DisCoPy Integration

```python
from discopy.monoidal import Ty, Box, Id
from discopy.drawing import draw

# Types for operad network
LittleDisks = Ty('E₂')
Cubes = Ty('E_∞')
Thread = Ty('Thread')

# Morphisms (operad maps)
stabilize = Box('stabilization', LittleDisks, Cubes)
linearize = Box('linearization', LittleDisks, Thread)

# Compose diagram
diagram = stabilize >> linearize.dom @ linearize
```

## Hy Usage

```hy
#!/usr/bin/env hy
(import [discohy_thread_operad [RootedColorOperad build-operad-from-threads]])

;; Build operad from thread tree
(setv threads [
  {:id "T-001" :title "Root" :parent nil}
  {:id "T-002" :title "Child1" :parent "T-001"}
  {:id "T-003" :title "Child2" :parent "T-001"}])

(setv operad (build-operad-from-threads threads 0x42D))

;; Get operad variant
(.set-variant operad "dendroidal")

;; Compose operations
(setv composed (.compose operad "T-001" ["T-002" "T-003"]))
```

## ACSet Schema for Operads

```julia
@present SchOperadNetwork(FreeSchema) begin
  Operad::Ob
  Morphism::Ob
  
  src::Hom(Morphism, Operad)
  tgt::Hom(Morphism, Operad)
  
  Name::AttrType
  Trit::AttrType
  
  name::Attr(Operad, Name)
  trit::Attr(Operad, Trit)
  morph_type::Attr(Morphism, Name)
end

@acset_type OperadNetwork(SchOperadNetwork)
```

## Relational Interleaving

The 7 operads form a relational network (ACSet):

```
                    ┌─────────────┐
                    │   Modular   │⊕
                    └──────┬──────┘
                    ┌──────┴──────┐
               ┌────┴────┐   ┌────┴────┐
               │ Cactus  │⊖  │  Swiss  │⊕
               └────┬────┘   │ Cheese  │
                    │        └────┬────┘
                    ▼             │
               ┌─────────┐       │
               │ Thread  │⊙◄─────┘
               └────┬────┘
          ┌────────┼────────┐
          ▼        ▼        ▼
     ┌────────┐ ┌──────┐ ┌─────────┐
     │ Cubes  │⊖│Gravity│⊖│ Little  │⊕
     │  E_∞   │ │ M_0,n │ │ Disks   │
     └────────┘ └──────┘ └─────────┘
```

## Triad Interleaving Schedule

Build balanced schedules with GF(3) = 0 per triplet:

```python
from operads.relational_operad_interleave import build_triad_from_operads

triad = build_triad_from_operads()
schedule = triad.build_round_robin(7)

# Output:
#   0: cubes ⊗ thread ⊗ little_disks
#   1: cactus ⊗ thread ⊗ modular
#   2: gravity ⊗ thread ⊗ swiss_cheese
#   ...
```

## DuckDB Integration

```sql
-- Query operad compositions
SELECT 
  src.name as source,
  tgt.name as target,
  m.morph_type,
  (src.trit + tgt.trit) % 3 as combined_trit
FROM operad_morphisms m
JOIN operads src ON m.src_id = src.id
JOIN operads tgt ON m.tgt_id = tgt.id;

-- Find GF(3)-conserving triads
SELECT o1.name, o2.name, o3.name
FROM operads o1, operads o2, operads o3
WHERE (o1.trit + o2.trit + o3.trit) % 3 = 0
  AND o1.id < o2.id AND o2.id < o3.id;
```

## File Locations

```
src/operads/
├── __init__.py                      # Registry
├── relational_operad_interleave.py  # ACSet + Triad
├── libkind_spivak_dynamics.py       # Directed/Undirected/Machines
└── infinity_operads.py              # Dendroidal + Lurie

scripts/
├── discohy_operad_1_little_disks.py
├── discohy_operad_2_cubes.py
├── discohy_operad_3_cactus.py
├── discohy_operad_4_thread.py
├── discohy_operad_5_gravity.lisp
├── discohy_operad_6_modular.bb
└── discohy_operad_7_swiss_cheese.py
```

## Commands

```bash
# Run relational interleaving demo
python3 src/operads/relational_operad_interleave.py

# Run Libkind-Spivak operads
python3 src/operads/libkind_spivak_dynamics.py

# Test individual operad
python3 scripts/discohy_operad_4_thread.py
```

## See Also

- `acsets` - Algebraic databases (schema category)
- `triad-interleave` - GF(3) balanced scheduling
- `gay-mcp` - Deterministic color generation
- `three-match` - 3-SAT via colored subgraph isomorphism
- `world-hopping` - Badiou triangle navigation



## 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.