Quick Reference for Core Concepts
Start here if you're new to Quannex mathematics
This is your entry point to understanding Quannex's mathematical framework. Think of it as the "map before the territory" - a high-level overview before diving into specific formulas.
After reading this, explore:
- PENTAGRAM_ANALYSIS.md - How we calculate harmonic resonance
- BREATH_DYNAMICS.md - The 6 breath axes and flow balance
- OCTAVE_FRAMEWORK.md - 7 developmental stages
- FOUNDATION_PRINCIPLE.md - Why coherence ≠ promotion (critical insight!)
- SPECTRAL_SHADOW.md - Advanced analysis techniques
Traditional dashboards ask: "How much?"
- How much revenue?
- How much growth?
- How much profit?
Quannex asks: "How well do parts work together?"
- Is revenue growth sustainable given team capacity?
- Is innovation balanced with operational stability?
- Are stated values aligned with actual behavior?
This is the shift from magnitude to coherence.
Global Coherence = f(Individual Health, Relational Harmony, Structural Integrity)
↓ ↓ ↓
Face Energy Edge Tension Vertex Vortex
Question: Is each organizational domain healthy on its own?
Measures:
- Each of 12 faces has 5 elemental KPIs
- Pentagram analysis finds harmony among elements
- Face energy = base health × harmonic boost
Question: Do domains work well together?
Measures:
- 30 edges connect the 12 faces
- Each edge has a relationship KPI
- Breath analysis checks for sustainable flow (6 axes)
See: BREATH_DYNAMICS.md
Question: Where are the transformation leverage points?
Measures:
- 20 vertices where 3 faces meet
- Vortex potential based on face energy variance
- High variance = transformation opportunity
See: OCTAVE_FRAMEWORK.md
The dodecahedron is the most complex Platonic solid:
- 12 pentagonal faces → 12 organizational domains
- 30 edges → 30 relationships
- 20 vertices → 20 transformation points
Each face is a pentagon (5 sides), which creates natural pentagram geometry:
- 5 elements: Earth, Water, Fire, Air, Ether
- 5 KPIs per face, one per element
- Pentagram star reveals harmonic relationships
φ = (1 + √5) / 2 ≈ 1.618033988749895
The golden ratio appears in:
- Natural growth patterns (spirals, flowers, DNA)
- Harmonic proportions (music, art)
- Quannex tuning constants (harmony rewards)
All KPIs must be normalized to 0-1 scale before calculations.
Examples: Revenue, customer satisfaction, team morale
Formula:
value - target_min
N(v) = ──────────────────────
target_ideal - target_min
Capped: N(v) ∈ [0, 1]
Quick Check:
- At
target_min→ Score = 0.0 - At
target_ideal→ Score = 1.0 - Above
target_ideal→ Score = 1.0 (capped)
Examples: Turnover rate, cost per acquisition, error rate
Formula:
value - target_min
N(v) = 1 - ──────────────────────
absolute_max - target_min
Capped: N(v) ∈ [0, 1]
Quick Check:
- At
target_min→ Score = 1.0 (ideal - as low as possible) - At
absolute_max→ Score = 0.0 (worst case)
Examples: Meeting frequency, work-life balance, inventory levels
Formula:
Inside plateau [healthy_min, healthy_max]:
N(v) = 1.0
Below plateau [target_min, healthy_min):
value - target_min
N(v) = ──────────────────────
healthy_min - target_min
Above plateau (healthy_max, absolute_max]:
value - healthy_max
N(v) = 1 - ──────────────────────
absolute_max - healthy_max
Capped: N(v) ∈ [0, 1]
Quick Check:
- Too low → Score increases toward 1.0
- In sweet spot → Score = 1.0
- Too high → Score decreases toward 0.0
Example: Meeting Hours Per Week
target_min = 0 hours (no meetings at all)
healthy_min = 4 hours (minimum needed)
healthy_max = 8 hours (maximum before overhead)
absolute_max = 20 hours (all meetings, no work!)
Value = 6 hours → N(6) = 1.0 (perfect, in plateau)
Value = 2 hours → N(2) = 0.5 (below minimum, only halfway)
Value = 15 hours → N(15) = 0.58 (too many, performance drops)
Here's how data becomes coherence:
1. Load Raw KPIs (60 total: 5 per face × 12 faces)
↓
2. Normalize Each KPI → Score ∈ [0, 1]
↓
3. Group by Face (12 groups of 5)
↓
4. Pentagram Analysis → Find Harmonic Resonance
↓
5. Face Energy = Weighted Average × (1 + Harmonic Boost)
↓
6. Breath Analysis → Check 6 Axes for Balance
↓
7. Edge Tension = Relationship Health
↓
8. Vertex Vortex = Transformation Potential
↓
9. Global Coherence = Weighted(Faces, Edges, Vertices)
See PENTAGRAM_ANALYSIS.md for step-by-step walkthrough
E_face = E_base × (1 + α × R_harmonic)
Where:
- E_base = Weighted average of 5 KPIs
- R_harmonic = Pentagram resonance score ∈ [0, 1]
- α = 0.3 (30% boost for perfect harmony)
R_harmonic = 1 - (variance / max_variance)
Where variance is calculated from pentagram star pair distances
C_global = 0.4 × AvgFaceEnergy +
0.3 × AvgEdgeHealth +
0.3 × AvgVertexCoherence
Ratio = Energy(Projection) / Energy(Reception)
Balanced: 0.8 ≤ Ratio ≤ 1.2
Over-exhaling: Ratio > 1.2 (burnout risk)
Over-inhaling: Ratio < 0.8 (stagnation risk)
coherence < 0.382 (φ⁻²) → O1 Survival
0.382 ≤ c < 0.500 → O2 Structure
0.500 ≤ c < 0.618 (φ⁻¹) → O3 Relationships
0.618 ≤ c < 0.764 → O4 Creativity
0.764 ≤ c < 0.854 → O5 Expression
0.854 ≤ c < 0.950 → O6 Vision
coherence ≥ 0.950 → O7 Radiance
Where: φ = 1.618033988749895 (Golden Ratio)
Org_Octave = floor(GeometricMean(face_octaves) - SpreadPenalty)
GeometricMean = (∏ face_octaves)^(1/12)
SpreadPenalty:
spread ≤ 2: 0
spread = 3: 0.5
spread = 4: 1.0
spread ≥ 5: 1.5 + (spread - 5) × 0.5
Clamped: 1 ≤ Org_Octave ≤ LifecycleMax
Key Insight: High coherence at O1 = excellent survival, NOT promotion to O2. See FOUNDATION_PRINCIPLE.md for the full treatment.
- Linear thinking (more is better)
- Isolated metrics (revenue vs. team health are separate)
- No relational awareness (doesn't detect harmony vs. discord)
- Non-linear rewards (extreme variance penalized via pentagram)
- Relational intelligence (edges measure how well domains connect)
- Harmonic thinking (5 elements must thrive together)
- Sustainable flow (breath analysis prevents burnout)
Every formula carries philosophical meaning:
Perfect pentagram (all 5 elements equal) → Maximum resonance boost
Unbalanced pentagram (one dominant) → Resonance penalty
Why? Sustainable organizations grow all parts together, not just one function.
Balanced inhale/exhale → Healthy flow
Over-exhaling → Burnout, depletion
Over-inhaling → Stagnation, accumulation
Why? Organizations are living systems that need rest, integration, and cyclical flow.
α (harmonic boost) = 0.3 ≈ φ/5
γ (blending constant) = 0.7 ≈ 1 - φ/5
Why? Nature uses φ for sustainable, beautiful growth. We mirror that.
A: Pentagon geometry. Each face is a 5-sided pentagon, creating natural pentagram star patterns. Also maps to classical elements (Earth, Water, Fire, Air, Ether) from wisdom traditions.
A: Real organizational health requires balance. A company with amazing marketing but failing operations will collapse. Pentagram math encodes "all elements must thrive together."
A:
- Quick mode: 12 KPIs (1 per face, no pentagram analysis)
- Full mode: 60 KPIs (5 per face, full harmonic analysis)
A: Yes! Map your KPIs to the 12 faces and 5 elements. The demo-orchestrator.html wizard helps with this.
To understand the core math:
- Read this overview (you just did! ✅)
- Read PENTAGRAM_ANALYSIS.md - The heart of harmonic calculation
- Read BREATH_DYNAMICS.md - Flow and balance
- Read OCTAVE_FRAMEWORK.md for developmental progression
- Essential: FOUNDATION_PRINCIPLE.md - Why coherence ≠ promotion
- Advanced: SPECTRAL_SHADOW.md for leverage points and ethics
To see it in action:
- Open ../DEMO.html
- Load "Quannex" company
- Look at calculations while reading formulas
- It will click! 💡
Questions? See ../DEMO_GUIDE.md for walkthroughs
Ready for deep dive? Continue to PENTAGRAM_ANALYSIS.md
Created: 2025-01-16 | Updated: 2025-12-09 | Part of Quannex Mathematical Framework