Level 0 (Sigma):     Entire Claude Config (self-referential schema)
Level 1 (G_meta):    Skills + Routers + Agents (hyperedges connecting triads)
Level 2 (G):         Individual component files
Level 3 (V):         Sections within files (frontmatter, body, graph)
Level 4 (v):         Individual properties/concepts

                    SCALE INVARIANCE
    structure(Level_n) ≅ structure(Level_{n+1}) ≅ lambda.o.tau

Core Principles

1. Renormalization Group Optimization

Apply the same optimization at each scale level:

def renormalize(G_level):
    """Coarse-grain, identify universality class, extract relevant operators."""
    # Identify redundant patterns
    redundant = find_duplicates(G_level)
    # Merge into canonical forms
    canonical = merge_to_canonical(redundant)
    # Project to lower-dimension effective theory
    return project(canonical, dim=relevant_operators_only)

Scale	Renormalization Action	Universality Class
Config	Consolidate routers (15 -> 7)	Delegation pattern
Skill	Merge duplicate content	lambda.o.tau triad
Agent	Domain agents absorb specialists	Execution pattern
File	Deduplicate sections	Frontmatter schema

2. Power Law Resource Allocation (Pareto)

P(k) ~ k^{-alpha}  where alpha approx 2.5

Top 20% of agents handle 80% of tasks:
  oracle, sisyphus-junior, explore, engineer (4/20 = 20%)

Top 20% of skills deliver 80% of value:
  learn, ultrawork, git-master, lambda-skill, obsidian (5/23 approx 22%)

3. Homoiconic Self-Improvement

# Ultrawork can improve itself
assert ultrawork.can_process(ultrawork.schema) == True

def self_improve(skill):
    analysis = skill.analyze(skill.schema)
    improvements = skill.generate_improvements(analysis)
    validated = skill.validate(improvements)
    return skill.apply(validated) if validated else skill

Agent Routing Matrix (Power-Law Optimized)

Tier 1: High-Frequency (80% of delegations)

Agent	Model	Complexity	Use When
sisyphus-junior	Sonnet	0.4-0.7	Focused task execution, implementation
explore	Haiku	0.1-0.3	Pattern matching, file discovery, quick search
oracle	Opus	0.7-1.0	Root cause analysis, architecture decisions
engineer	Sonnet	0.5-0.8	Production-ready implementation

Tier 2: Medium-Frequency (15% of delegations)

Agent	Model	Complexity	Use When
librarian	Sonnet	0.3-0.5	Documentation research, codebase understanding
architect	Opus	0.8-1.0	System design, multi-domain decomposition
prometheus	Sonnet	0.5-0.7	Strategic planning, roadmap design
researcher	Sonnet	0.4-0.6	Deep research with MCP tools

Tier 3: Low-Frequency (5% of delegations)

Agent	Model	Complexity	Use When
document-writer	Haiku	0.2-0.4	README, API docs, technical writing
multimodal-looker	Sonnet	0.3-0.5	Screenshots, diagrams, visual analysis
momus	Haiku	0.3-0.5	Critical plan review, devil's advocate
metis	Sonnet	0.4-0.6	Pre-planning, hidden requirements
frontend-engineer	Sonnet	0.4-0.7	UI/UX, components, styling

External CLI Agents (Token Conservation)

Agent	Binary	Context Limit	Use When
gemini	/opt/homebrew/bin/gemini	2M tokens	Large context analysis (>100K)
codex	~/.local/bin/codex	128K tokens	GPT code generation preference
amp	~/.amp/bin/amp	200K tokens	Claude-specific delegation

Execution Patterns

Parallel Execution (Independent Tasks)

# Launch simultaneously in single message
spawn:
  - Task(explore, "find auth files", run_in_background: true)
  - Task(librarian, "search auth docs", run_in_background: true)
  - Task(researcher, "find auth best practices", run_in_background: true)
collect: TaskOutput for each
merge: Deduplicate by content hash

Sequential Execution (Dependent Tasks)

# Chain with explicit dependencies
pipeline:
  - result_1 = Task(explore, "find relevant code")
  - result_2 = Task(oracle, "analyze: ${result_1}")
  - result_3 = Task(engineer, "implement fix: ${result_2}")

Background Execution (Long-Running)

background_operations:
  - Package installation: npm install, pip install, cargo build
  - Build processes: npm run build, make, tsc
  - Test suites: npm test, pytest, cargo test
  - Docker operations: docker build, docker pull
  - Large file operations: >1000 files
  - Subagent delegations: complexity > 0.7

foreground_operations:
  - Quick status: git status, ls, pwd (<5s)
  - File reads/edits
  - Simple commands
  - Verification checks

Convergence Detection

Fixed-Point Termination

def at_fixed_point(state, epsilon=0.001):
    """Detect when further iteration yields no improvement."""
    new_state = iterate(state)
    return distance(state, new_state) < epsilon

def ultrawork_loop(task):
    state = initialize(task)
    while not at_fixed_point(state):
        state = parallel_execute(state)
        state = assess(state)
        if converged(state):
            break
    return finalize(state)

Convergence Thresholds

Pipeline	Threshold	Use When
R1 (Simple)	0.85	Single-domain, <10 files
R2 (Moderate)	0.92	Multi-domain, 10-50 files
R3 (Complex)	0.96	Architecture-level, >50 files

Verification Checklist (Invariants)

Before stopping, ALL must be true:

TODO LIST: Zero pending/in_progress tasks
FUNCTIONALITY: All requested features work
TESTS: All tests pass (if applicable)
ERRORS: Zero unaddressed errors
TOPOLOGY: eta >= 4 (if knowledge graph modified)
MONOTONICITY: len(K') >= len(K) (no knowledge lost)

If ANY checkbox is unchecked, CONTINUE WORKING.

Integration with Learn Skill

Ultrawork extends learn with:

-- Learn: Sequential knowledge compounding
lambda(o, K, Sigma).tau' = renormalize . compound . assess . execute . route . parse

-- Ultrawork: Parallel orchestration with scale invariance
G_meta(lambda, K, Sigma).tau' = parallelize . delegate . renormalize . compound

Post-Task Learning Loop

after_completion:
  - Extract learnings via learn skill
  - Crystallize patterns with vertex-sharing
  - Update knowledge graph (K -> K')
  - If schema improvement identified: propose to user

Invariants Preserved

Invariant	Expression	Enforcement
K-monotonicity	len(K') >= len(K)	Never delete knowledge
Topology	eta >= 4	Minimum connectivity maintained
Homoiconicity	Sigma.can_process(Sigma)	Self-referential capability
Scale Invariance	structure(L_n) cong structure(L_{n+1})	Same patterns at all levels
Power Law	P(k) ~ k^{-alpha}	80/20 resource allocation

Quick Reference

G_meta(lambda,K,Sigma).tau'   Parallelize -> Delegate -> Renormalize -> Compound
K grows                       Sigma evolves              eta>=4 preserved
Scale-invariant               Power-law optimized        Fixed-point convergent

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