majiayu000/claude-arsenal
xiaohongshu
xiaohongshu
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möbius-color-duality
Möbius inversion for Gay.jl color duality - closes sparsification spine gap
SKILL.md
Full skill instructions
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 spacesTriadicColorInverter: 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
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