# Canonical Parameters: Governance Topology Thesis

**Date:** February 2026
**Status:** Definitive — supersedes all prior parameter specifications

---

## 1. Stage Definitions

| Stage | Liberty Range | Name | Description |
|-------|-------------|------|-------------|
| 1 | 85-100 | Consolidated Democracy | Full democratic institutions |
| 2 | 80-84 | Early Warning | Initial stress signals |
| 3 | 70-79 | Democratic Erosion | Institutional degradation begins |
| 4 | 60-69 | Competitive Authoritarian | Electoral competition constrained |
| 5 | 50-59 | Electoral Autocracy | Elections occur but lack integrity |
| 6 | 40-49 | Soft Dictatorship | Opposition marginalized |
| 7 | 25-39 | Consolidated Autocracy | Single-party or strongman rule |
| 8 | 0-24 | Totalitarianism | Total state control |

## 2. Ternary Constraint

L + T + C = 100

Where L = Liberty, T = Tyranny, C = Chaos. All values 0-100.

## 3. Event Horizon

- **Revised threshold:** L approximately 52-55 (three-method convergence: survival analysis, Markov, potential function)
- **Recovery rate below threshold:** 3.0% (95% CI: 0.7-6.0%)
- **Original thesis threshold:** L=60 (SUPERSEDED)

## 4. AR(1) Model Coefficients

The AR(1) persistence model outperforms all stage-based transition models (delta-AIC > 300).

| Parameter | Symbol | Estimate | 95% CI | Method |
|-----------|--------|----------|--------|--------|
| Intercept | alpha | 3.56 | [2.80, 4.32] | Clustered SE |
| Persistence | beta | 0.956 | [0.941, 0.971] | Clustered SE |
| R-squared | R2 | 0.872 | [0.849, 0.893] | Bootstrap |

**Equilibrium:** L* = alpha / (1 - beta) = 80.9

## 5. Yield-Liberty Regression

Y = a + b*L (cross-sectional, N=91)

| Parameter | Estimate | SE | Interpretation |
|-----------|----------|-----|----------------|
| Intercept (a) | ~15 | - | Yield at L=0 |
| Liberty (b) | -0.35 | - | 35bp per 1pt Liberty decline |
| R-squared | 0.37 | - | - |

Four-factor model (Liberty + Debt/GDP + Reserve + Velocity): R2 = 0.79
Reserve currency premium: approximately 2,080bp

## 6. Stage 5 Structural Break

- Pre-2006 momentum: +38% (improving)
- Post-2006 momentum: -23.3% (declining)
- Global structural break: approximately 2000 (F=21.2)
- Stage-specific break: 2006

## 7. Markov Property Test

The Markov assumption is REJECTED at Stages 2, 5, and 6. Transition probabilities are path-dependent:
- Stage 6 (Soft Dictatorship): countries arriving via decline show -77.8% recovery vs +25.5% for those arriving via improvement
- Direction of travel carries more predictive power than current state

## 8. US Liberty Score Range

| Source | Score | Notes |
|--------|-------|-------|
| Author PTI (2026) | 48 | Real-time institutional assessment |
| TCF Democracy Meter | 57 | Newer index |
| V-Dem LDI (scaled) | 65-72 | Electoral autocracy reclassification (Sep 2025) |
| Freedom House (2024) | 83 | Annual report, lagged |
| Cross-validation mean | 76.6 | Mean of 7 external indices |

**Credible range:** 57-84. The PTI score (48) is an outlier.

## 9. Decoupling Statistics

- Pre-1900 correlation (Liberty vs HCI): r = 0.79
- Post-1990 correlation: r = 0.57
- Capable Autocracies (HCI >= 70, Liberty < 60): 39 countries (2023)
- Free & Capable (HCI >= 70, Liberty >= 60): 39 countries (2023)

---

## 10. Canonical Monte Carlo Parameters

**Model:** AR(1) with mean reversion and data-driven stage-specific volatility.

```
L(t+1) = alpha + beta * L(t) + sigma_stage * epsilon
where epsilon ~ N(0, 1), L clamped to [0, 100]
```

### AR(1) Coefficients

| Parameter | Value |
|-----------|-------|
| alpha (intercept) | 3.56 |
| beta (persistence) | 0.956 |
| L* (equilibrium) | 80.9 |

### Data-Driven Sigma by Stage

| Stage | Name | sigma (data-driven) | sigma (thesis, SUPERSEDED) | Ratio |
|-------|------|--------------------:|---------------------------:|------:|
| 1 | Consolidated Democracy | 0.45 | 3 | 6.7x |
| 2 | Early Warning | 3.27 | 5 | 1.5x |
| 3 | Democratic Erosion | 2.10 | 5 | 2.4x |
| 4 | Competitive Authoritarian | 1.82 | 6 | 3.3x |
| 5 | Electoral Autocracy | 2.45 | 7 | 2.9x |
| 6 | Soft Dictatorship | 2.97 | 7 | 2.4x |
| 7 | Consolidated Autocracy | 4.45 | 6 | 1.3x |
| 8 | Totalitarianism | 3.11 | 4 | 1.3x |

**Note:** All thesis stipulated sigma values are SUPERSEDED. Data-driven values are computed from annualized velocity residuals across 1,656 observations (91 countries, 1800-2025).

### Simulation Parameters

| Parameter | Value |
|-----------|-------|
| Number of paths | N = 10,000 |
| Random seed | 42 |
| Horizons tested | 5, 10, 15 years |
| Starting values tested | 48, 57, 65, 70, 75, 84 |

### Key Monte Carlo Results (L=48, 15yr)

| Metric | Data-driven sigma | Thesis sigma (SUPERSEDED) |
|--------|------------------:|-------------------------:|
| Median L (2040) | 64.7 | ~8 |
| 5th percentile | 52.2 | ~2 |
| 95th percentile | 74.5 | ~82 |
| P(L<25) | ~0% | ~62% |
| P(L>70) | ~15% | ~5% |

### Retracted Claims

- **P(tyranny by 2040) = 62%** — RETRACTED. Data-driven = ~0%.
- **Median trajectory to L=8** — RETRACTED. Data-driven median = L=65.
- **Thesis sigma values** — SUPERSEDED by data-driven estimates (0.45-4.45 range).

---

## Source Scripts

| Script | Phase | Focus |
|--------|-------|-------|
| `phase1-foundation-audit.py` | 1 | Crosswalk, Event Horizon, velocity CIs |
| `phase2-model-hardening.py` | 2 | Sigma estimation, Markov test, AR(1) |
| `phase3-us-case-hardening.py` | 3 | US cross-validation, institutional resilience |
| `phase4-missing-evidence.py` | 4 | Recalibration, MC sensitivity, counter-arguments |
| `phase5-recalibrated-monte-carlo.py` | 5 | Recalibrated MC engine (Tasks 2, 3, 4, 8) |
| `phase5-gdp-covariate.py` | 5 | GDP as covariate (Task 7) |