# Möbius Color Duality

> Möbius inversion for Gay.jl color duality - closes sparsification spine gap

- Skill: `majiayu000/m-bius-color-duality` (Agent Skill, multi-file: 2 files)
- Install (CLI): `npx skillmds add majiayu000/m-bius-color-duality`
- Raw SKILL.md: https://api.skillmd.com/api/skills/majiayu000/m-bius-color-duality/raw
- Safety review: pending
- Works with: Claude Code, Claude.ai, OpenAI Codex
- Category: Coding & Dev Tools
- Author: majiayu000 (https://skillmd.com/u/majiayu000)
- Updated: 2026-09-09
- Page: https://skillmd.com/skills/majiayu000/m-bius-color-duality

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# Möbius Color Duality Skill

> *"Möbius inversion recovers local structure from global aggregates."*

## The Gap

The Amp thread corpus (1,807 threads) is **85.6% mapped to sparsification spine** but has a critical gap:

```
Generation (Synthesis)     ████████░  49.2% ⭐ WELL-DEVELOPED
Validation (Verification)  ████░░░░░  21.6% ✓ SOLID
Execution (Control)        ███░░░░░░  15.2% ✓ ADEQUATE
Ontology (ACSet)           ██░░░░░░░   6.8% ⚠️ THIN
Decomposition (Hierarchy)  █░░░░░░░░   5.6% ⚠️ WEAK
Inversion (Duality)        ░░░░░░░░░   1.6% ❌ CRITICAL GAP
```

The system can **generate** but not **invert**. This skill closes that gap.

## Core Insight

**Forward (Generation)**: seed → color indices → structures  
**Backward (Inversion)**: structures → color distributions → recover seed

## Implementation

Module: `lib/gay_möbius_inversion.py` (490 lines)

Key classes:
- `ColorMöbiusInverter`: Numerical inversion for color spaces
- `TriadicColorInverter`: GF(3) ternary extension

## Status

✓ **Core implementation**: Möbius function, forward/backward inversion  
✓ **Duality graph generation**: Contravariant functor structures  
⊘ **GF(3) integration**: Ternary state extension  
⊘ **Amp corpus application**: Test on actual thread colorization  

## Sparsification Spine Integration

**Tier**: Layer 5 - INVERSION (Duality/Reversal)  
**Trit**: +1 (PLUS/Generator)  
**Coverage**: Begins to fill 1.6% → expand to 10%+ target