Attractor Basins: The Tristable Geography of Freedom
Political systems behave like a ball on a landscape with three gravitational wells. The tyranny well (L≈7) is the deepest. The hybrid trap (L≈57) is shallow but genuine — countries dwell here for decades. The democratic plateau (L≈90) is elevated but fenced by institutional guardrails. The Event Horizon at L≈52-55 marks the ridgeline of maximum volatility.
31
Democratic plateau
32
Hybrid trap
28
Tyranny well
8%
Recovery rate below L≈52-55
Distribution of Liberty scores — the trimodal fingerprint of tristability
Political stability landscape — countries sink into basins, must climb to escape
Democratic Plateau (L ≥ 70)
31
countries · self-reinforcing institutions
Norway 97 · NZ 96 · Finland 96 · Denmark 95 · Switzerland 95 · Sweden 93 · Netherlands 93 · Estonia 93 · Ireland 93 · Canada 92 · Australia 92 · Germany 91 · Taiwan 91 · Uruguay 91 · Portugal 91 · Belgium 90 · Japan 89 · Austria 88 · UK 87 · Costa Rica 86 · Spain 85 · Czech Rep 85 · France 83 · S. Korea 83 · Italy 82 · Chile 82 · Poland 82 · Greece 79 · Romania 74 · Brazil 72 · Botswana 71
Hybrid Trap (20 ≤ L < 70)
32
countries · shallow basin, decades of stagnation
Ghana 68 · Bulgaria 66 · Argentina 65 · Hungary 63 · India 62 · S. Africa 62 · Israel 60 · Senegal 60 · Moldova 55 · Colombia 53 · Indonesia 50 · US 48 · Mexico 48 · Serbia 48 · Singapore 47 · Malaysia 45 · Armenia 42 · Philippines 42 · Kenya 40 · Georgia 38 · Nigeria 38 · Guatemala 35 · Ukraine 35 · Sri Lanka 35 · Morocco 30 · Pakistan 25 · Bangladesh 25 · Tunisia 22 · UAE 22 · Thailand 20 · Iraq 20 · Jordan 20
THEORETICAL BASIS: The ternary constraint models political power as a zero-sum allocation: Liberty (distributed with institutional constraints), Tyranny (concentrated), Chaos (fragmented/contested). T is computed as the residual (T = 100 − L − C), which the author acknowledges as a measurement limitation — the constraint holds definitionally, not as an independent empirical finding. L is measured via Freedom House; C via the Fragile States Index. Future work should develop independent Tyranny measures (e.g., executive concentration indices) to test the constraint empirically rather than impose it.
⚠️ CLASSIFICATION NOTE: Zone velocities use ending-zone assignment (countries classified by period-end score). Starting-zone assignment yields materially different results (e.g., Tyranny Basin: +0.72/yr starting-zone vs −0.64/yr ending-zone). This sensitivity means zone velocity claims should be interpreted with caution. The "gravitational pull" narrative depends on the classification method chosen.
Mathematical Framework Reconciliation
This visualization employs three mathematical frameworks that each capture a different facet of political tristability. (a) The Gaussian Mixture Model (GMM) underlies the trimodal histogram above: fitting a three-component mixture to the cross-sectional distribution of liberty scores reveals where the basins are — peaks near L≈7 (tyranny), L≈48 (hybrid), and L≈90 (democracy). The GMM is a static, cross-sectional tool: it identifies equilibrium structure from a snapshot of all countries at a single point in time. (b) The discrete Markov chain transition matrix (presented in the Complete Model) estimates how often countries actually move between basins over observed time intervals, providing empirical transition dynamics from the longitudinal panel. (c) The continuous SDE/Langevin formulation (dL = f(L)dt + σdW) generates the potential energy landscape shown in the second chart, where basin depth corresponds to attractor strength and saddle heights determine escape difficulty. In short: the GMM identifies WHERE the basins are; the Markov chain quantifies HOW often countries move between them; the SDE provides an analytical framework for deriving properties like escape times and basin stability.
The complementarity of these tools should not obscure the tension between them. Strictly, the SDE and Markov models make different assumptions about the data-generating process. The SDE assumes continuous paths; the Markov model assumes discrete jumps. With irregular time spacing (5–20 year gaps pre-1972), neither is exact. We treat both as useful approximations and note where they diverge. The potential well diagram (SDE-derived) is best understood as a qualitative metaphor grounded in the GMM’s empirical peak locations: the well depths and saddle heights are calibrated to match observed basin persistence and transition rates, but the smooth curve imposes a continuity assumption the data do not strictly support. The Markov chain, by contrast, makes no continuity assumption but discretizes what may be a continuous underlying process. Where both frameworks agree — tyranny is the deepest well, the hybrid trap is shallow but genuine, recovery below L≈52–55 is rare — we have convergent evidence. Where they diverge, the divergence flags modeling assumptions rather than empirical facts.
Recommended future work: A continuous-time Markov chain (CTMC) would bridge both frameworks, with transition rates estimated via maximum likelihood from the irregularly-spaced panel. The CTMC handles variable observation gaps naturally (via matrix exponentials eQt) and converges to the Langevin SDE in the fine-discretization limit, providing a single unified model that respects both the empirical transition data and the analytical tractability of the continuous formulation.
The physics of political tristability. Like a marble in a triple well, countries settle into one of three stable basins. Consolidated democracies (L≥80) have self-reinforcing institutions that resist erosion. The hybrid trap (20≤L<70) is shallow but genuine — countries can dwell here for decades, neither consolidating democracy nor collapsing into tyranny. Consolidated tyrannies (L<20) have self-reinforcing repression that resists liberalisation. The hybrid trap is the key insight: it is not merely a volatile transition zone, but a distinct attractor with its own gravitational pull.
The hybrid trap is shallow but real. Countries like Singapore (47), Indonesia (50), and Mexico (48) have occupied the hybrid zone for decades. They are neither transitioning toward democracy nor sliding into tyranny. The hybrid trap captures patronage networks, managed elections, and partial press freedom — a stable equilibrium of controlled openness.
Historical recovery rate below the Event Horizon (L≈52-55): just 8%. Of the 50 countries that dropped below this threshold since 1989, only 4 climbed back. Escaping the hybrid trap requires crossing the volatility ridgeline at L≈52-55. Escaping the tyranny well is even harder — countries must first climb into the hybrid trap before attempting the second ascent to democracy.