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

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

#!/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

@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:

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

-- 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

# 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

discohy-streams