Möbius Color Duality

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

SDF Interleaving

This skill connects to Software Design for Flexibility (Hanson & Sussman, 2021):

Primary Chapter: 3. Variations on an Arithmetic Theme

Concepts: generic arithmetic, coercion, symbolic, numeric

GF(3) Balanced Triad

möbius-color-duality (−) + SDF.Ch3 (○) + [balancer] (+) = 0

Skill Trit: -1 (MINUS - verification)

Secondary Chapters

• Ch6: Layering

Connection Pattern

Generic arithmetic crosses type boundaries. This skill handles heterogeneous data.