HyJAX Relational Thinking Skill

Apply relational thinking (ACSets/C-Sets) to Amp thread analysis using HyJAX patterns.

When to Use

• Analyzing thread relationships and concept networks
• Extracting patterns from conversation history
• Building relational databases from unstructured thread data
• Generating Colored S-expressions for visualization

Core Concepts

ACSet Schema for Threads

Objects: Thread, Message, Concept, File
Morphisms: thread_msg, mentions, discusses, related
Attributes: content, timestamp, info_gain

Colored S-expressions

(acset-gold
  (threads-red (thread T-001 "Title" 42))
  (concepts-green (concept skill 5) (concept MCP 3))
  (relations-purple (edge skill co-occurs subagent)))

Key Files

Quick Start

1. Query the Thread Lake

duckdb /Users/bob/ies/music-topos/lib/unified_thread_lake.duckdb -c "
  SELECT name, hub_score FROM concepts ORDER BY hub_score DESC LIMIT 10
"

2. Find 2-Hop Concept Paths

duckdb /Users/bob/ies/music-topos/lib/unified_thread_lake.duckdb -c "
  SELECT r1.from_concept || ' → ' || r1.to_concept || ' → ' || r2.to_concept as path
  FROM concept_relations r1
  JOIN concept_relations r2 ON r1.to_concept = r2.from_concept
  WHERE r1.from_concept = 'skill'
"

3. Run Full Analysis

cd /Users/bob/ies && source .venv/bin/activate
python3 music-topos/lib/full_thread_analysis.py

Relational Patterns

Hub Concepts (Most Connected)

Strongest Relations

• skill ↔ subagent (weight 2)
• skill → MCP → alife
• skill → ACSet → discohy
• HyJAX ↔ relational

Integration with Other Skills

With acsets-algebraic-databases

@present SchThread(FreeSchema) begin
  Thread::Ob; Message::Ob; Concept::Ob
  thread_msg::Hom(Message, Thread)
  discusses::Hom(Message, Concept)
  related::Hom(Concept, Concept)
end

With gay-mcp

Each concept gets a deterministic color via Gay.jl seed:

using Gay
concept_color = gay_color(hash("skill"))  # Reproducible color

With entropy patterns

H(concepts) = 4.55 bits  # Shannon entropy of concept distribution
efficiency = 95.6%        # vs max entropy

DuckDB Schema

CREATE TABLE threads (thread_id VARCHAR PRIMARY KEY, title VARCHAR, message_count INT);
CREATE TABLE concepts (concept_id VARCHAR PRIMARY KEY, name VARCHAR, frequency INT, hub_score INT);
CREATE TABLE concept_relations (from_concept VARCHAR, to_concept VARCHAR, weight INT);
CREATE TABLE colored_sexprs (sexpr_id VARCHAR PRIMARY KEY, root_color VARCHAR, tree_json JSON);

Workflow

1. Ingest: Use find_thread to get thread data
2. Extract: Apply concept patterns to titles/content
3. Build: Create ACSet with objects and morphisms
4. Query: Run relational queries (pullbacks, 2-hop paths)
5. Output: Generate Colored S-expressions

Example Output

THREAD RELATIONAL ANALYSIS - 30 THREADS
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
Threads:    30
Messages:   2,951
Concepts:   27
Relations:  48
Entropy:    4.55 bits (95.6% efficiency)

TOP CONCEPTS:
  skill           5 █████
  subagent        3 ███
  MCP             3 ███
  GF3             3 ███

COLORED S-EXPRESSION:
(acset-gold 
  (threads-red ...) 
  (concepts-green ...) 
  (relations-purple ...))

Scientific Skill Interleaving

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

Autodiff

• jax [○] via bicomodule

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: Span
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.