Curvature at Civilization Scale: Translating Cliodynamics to Coherence Geometry

Series: Cliodynamics | Part: 5 of 10

Turchin describes societies as integrating and disintegrating. He tracks rising and falling social cohesion. He measures when systems move cooperatively versus fragmenting into competing factions. He quantifies the transition from stability to crisis.

These aren't just metaphors. They're descriptions of geometric properties.

In AToM's coherence framework, M = C/T—meaning equals coherence over time (or tension). A society that integrates is a society achieving high coherence: its components (population, elites, state, norms) move through state-space along aligned trajectories. A society that disintegrates is experiencing coherence collapse: those same components pursuing incompatible paths, generating mutual interference, fragmenting the manifold.

The secular cycle isn't just a historical pattern. It's a description of coherence dynamics at macro scale—the same geometric principles operating on civilizations that operate on nervous systems, on cells, on any system that maintains organization over time while navigating constraint.

This isn't loose analogy. The mathematics of coherence geometry—curvature, dimensionality, coupling, attractor dynamics—map directly onto what cliodynamics measures. Integration is low-curvature flow. Disintegration is curvature spike and dimensional collapse. Elite overproduction is constraint accumulation. Crisis is the moment the manifold can no longer support coherent trajectories and the system fragments.

Understanding cliodynamics through coherence geometry reveals why the same patterns recur across scales, why instability follows predictable dynamics, and why meaning collapses along with social order.

Integration as Low-Curvature Coherent Flow

The integrative phase of the secular cycle—population growing sustainably, elites cooperating, state capacity building, norms holding—is a description of low-curvature flow through state-space.

What This Means Geometrically

In coherence terms, a society is a high-dimensional dynamical system. The state-space includes variables like population density, resource distribution, elite composition, state fiscal capacity, institutional norms, cultural narratives, and technological infrastructure. At any given moment, the society occupies a point in this space. Over time, it traces a trajectory.

During integration, this trajectory is smooth. The system isn't encountering sharp constraints that force rapid directional changes. Resources are sufficient relative to population. Elite positions are adequate relative to aspirants. State revenue covers expenditures. Norms align behavior without coercion.

This is low curvature: the path through state-space bends gently. Changes happen, but they're gradual and the system can adapt without destabilizing. Innovation occurs, institutions evolve, the economy grows—but these changes are integrable, meaning they can be incorporated into the system's trajectory without forcing sharp reversals.

Coherence in this phase is high because the system's components are aligned. Population growth supports economic expansion. Economic expansion funds state capacity. State capacity maintains order and infrastructure. Order enables further growth. The feedback loops are positive and mutually reinforcing. The society feels like it's working because geometrically, it is—the components are entrained, moving together.

Dimensionality and Freedom

Integration also means high effective dimensionality. The system has multiple degrees of freedom. There are many viable paths forward. Policy choices have real options. Elites can pursue different strategies without colliding. Regional variations can coexist. Cultural diversity enriches rather than fragments.

This isn't chaos—the system is still structured. But the structure is enabling rather than constraining. The manifold is vast enough that movement doesn't immediately encounter walls.

Stagflation as Rising Curvature

The stagflation phase—growth slowing, inequality rising, elite numbers expanding, state fiscal stress beginning—is curvature accumulation.

Constraint Tightening

The smooth manifold of the integrative phase starts developing sharp features. Population approaches carrying capacity, which introduces resource constraint. Elite positions fill up, which introduces competition constraint. State expenses rise faster than revenue, which introduces fiscal constraint. Norms that worked during expansion start breaking under compression.

These constraints don't immediately collapse the system. But they bend trajectories. What was possible in the integrative phase becomes difficult. What was easy becomes hard. The path forward narrows.

Geometrically, this is rising curvature. The trajectory through state-space is bending more sharply. The system is encountering walls, and navigating around them requires increasingly tight turns. This costs energy. It generates stress. It reduces the system's capacity to absorb shocks.

Brittleness

High curvature means brittleness. Small perturbations that would have been absorbed during integration now propagate. A financial crisis that would have been a correction becomes a panic. A political scandal that would have been forgotten destabilizes the government. A social movement that would have been integrated fractures into violent protest.

The system is still coherent—it hasn't collapsed—but coherence is increasingly forced rather than emergent. It requires active suppression of conflict, enforcement of norms, extraction of resources. This is expensive. And it's unstable.

Dimensional Collapse Begins

As curvature rises, effective dimensionality starts contracting. Options narrow. Policy choices become zero-sum. Elite competition intensifies because there's less room to maneuver. Cultural diversity becomes polarization because the system can't accommodate variation anymore.

This is the geometric signature of approaching crisis: the manifold that supported rich multidimensional exploration during integration is compressing into a narrow channel where every choice involves tradeoffs that generate conflict.

Crisis as Coherence Collapse and Curvature Spike

The crisis phase—civil war, revolution, state breakdown, mass violence—is coherence collapse.

The Manifold Fragments

When curvature exceeds critical threshold, the manifold breaks. The single coherent trajectory that integrated the society fragments into multiple incompatible trajectories. Elites split into factions pursuing conflicting goals. Regions pursue divergent policies. Cultural groups reject shared norms. The state loses its monopoly on violence and its capacity to coordinate.

This is dimensional collapse at macro scale. What was a high-dimensional system with many degrees of freedom becomes a low-dimensional system where only a few dimensions matter—usually violence, factional loyalty, and survival. The rich complexity of the integrative phase reduces to brutal simplicity: who controls the guns, who can mobilize crowds, who survives the purge.

Curvature Spikes

During crisis, not only does the manifold fragment, but trajectories through the fragments encounter extreme curvature. Rapid, violent changes of direction. What was unthinkable yesterday becomes policy today. Institutions that lasted centuries collapse overnight. Norms that held for generations evaporate.

This is the geometry of revolution and civil war: wild oscillation between extremes, radical instability, inability to predict or plan. The system is far from equilibrium, searching desperately for a new attractor but unable to find one while elite numbers remain high and state capacity remains broken.

Coupling Breakdown

The mechanism of coherence collapse is decoupling. During integration, the system's components were coupled—population dynamics affected elite dynamics affected state dynamics affected norm dynamics, all feeding back constructively. During crisis, these couplings break.

Elites stop coordinating with each other. The state stops coordinating with regions. Cultural narratives stop coordinating across groups. Each component pursues its own survival strategy independent of or in opposition to the others. The system that was a unified whole becomes a collection of warring subsystems.

This is coherence collapse in its purest form: the geometric property that made the system a functioning whole—aligned motion through shared state-space—disappears. What remains are fragments, each following its own logic, generating mutual interference that destabilizes all of them.

Depression as Attractor Search and Reset

The depression/intercycle phase—post-crisis exhaustion, population reduced, elite class culled, new political order emerging—is the system searching for a new attractor.

Low-Dimensional Stability

The crisis has simplified the system dramatically. Population is lower. Elite numbers are lower. State capacity is rebuilt on a smaller, simpler base. Many of the constraints that created rising curvature in the previous cycle have been eliminated—often violently.

The manifold is smaller now—lower dimensional, less complex—but it's also stable. The system has found a basin it can rest in. Trajectories aren't smooth yet, but they're not fracturing either. The extreme curvature of the crisis phase has relaxed.

This is low-dimensional coherence. The system isn't rich or complex, but it works at the level it's operating. The surviving components are aligned. Resources are adequate for the reduced population. Elite positions are adequate for the reduced elite class. The new political order (often radically different from what preceded it) has established functional coupling.

Conditions for New Integration

The depression sets initial conditions for the next integrative phase. Land is available (population is reduced). Capital is available (either through surviving wealth or new accumulation). Elite cooperation is possible (competition has relaxed through culling). State capacity can be rebuilt (fiscal stress has reset through default, inflation, or new taxation).

Geometrically, the manifold is expanding again. What was compressed to a narrow channel during stagflation and shattered during crisis is opening up. Dimensionality increases. Curvature decreases. The system can begin exploring state-space again without immediately encountering constraint.

This is the geometric reset that enables the next cycle to begin. The society hasn't learned how to avoid the cycle (that would require maintaining institutions that prevent elite overproduction and manage carrying capacity across centuries). But it has cleared the structural pressures that drove the previous disintegration.

Elite Dynamics as Constraint Propagation

Elite overproduction is the single most important variable in the secular cycle model. In coherence geometry, it's constraint propagation.

Constraint Accumulation

During the integrative phase, elite numbers are low relative to positions. This is low constraint—there's room to move, competition is manageable, cooperation dominates. As elite numbers grow, constraint accumulates. More people trying to navigate the same narrow space (political office, high status, significant wealth). The manifold of elite trajectories compresses.

This constraint doesn't stay localized to elites. It propagates through the system. Elites competing for power mobilize masses, which destabilizes politics. Elites competing for resources drive inequality, which immiserates the population. Elites competing for state positions undermine state capacity through factionalism.

Geometrically, constraint in one subsystem (elite dynamics) propagates to other subsystems (population wellbeing, state capacity, social norms), raising curvature across the entire manifold. This is coupling—the components aren't independent, so constraint in one generates constraint in others.

Curvature Forcing

Elite competition specifically generates curvature through incompatible objectives. When elite numbers are low, elites can cooperate because there's enough for everyone. When elite numbers are high, cooperation becomes defection—helping a competitor means losing yourself.

This incompatibility forces sharp curvature. Policies that would serve the collective good are rejected because they advantage a rival faction. Compromises that would work are impossible because any concession is framed as betrayal. The system can't maintain smooth trajectories because the competing factions keep forcing sharp turns.

This is why elite overproduction is the critical variable: it's the mechanism that transforms a smooth low-curvature manifold into a high-curvature fragmented one.

State Capacity as Coherence Maintenance Infrastructure

The state in structural-demographic theory is the institution that maintains order, enforces norms, provides public goods, and manages collective problems. In coherence terms, the state is the coherence maintenance infrastructure.

Coupling Coordination

A strong state coordinates the coupling between subsystems. It mediates elite competition (through laws, procedures, enforcement). It manages population-resource dynamics (through infrastructure, welfare, economic policy). It propagates norms (through education, media, law). It provides the shared framework that enables the system's components to move together.

This is active coherence maintenance. The state isn't neutral—it's the mechanism by which the society sustains alignment despite internal variation. When the state is strong, it can absorb perturbations, smooth conflicts, and maintain the low-curvature conditions that enable integration.

Fiscal Stress as Coherence Cost

State fiscal stress—when expenses exceed revenue—is the geometric cost of maintaining coherence under rising constraint. During stagflation, the state needs to spend more (managing elite competition, addressing inequality, maintaining legitimacy) while revenue becomes harder to extract (economic stagnation, elite resistance to taxation, political paralysis).

Geometrically, this is the cost of forcing coherence when the underlying dynamics are generating fragmentation. The state is spending energy to hold the manifold together. Eventually, the fiscal stress becomes unsustainable. The state can't pay its debts, can't fund its institutions, can't enforce its rules. Coherence maintenance infrastructure breaks down.

When the state collapses, coherence collapses with it. There's no longer a mechanism coordinating the subsystems. They decouple, pursue independent trajectories, and generate mutual interference. The crisis begins.

Social Norms as Attractor Geometry

Norms in structural-demographic theory are the unwritten rules that coordinate behavior, establish expectations, and enable cooperation without coercion. In coherence geometry, norms are attractor structure.

Shared Basins

Norms create shared basins of attraction in the behavioral manifold. When norms are strong, individual and group trajectories naturally flow toward norm-compliant states. You don't have to enforce honesty if everyone expects honesty. You don't have to enforce peaceful transfer of power if that's the stable attractor.

This is low-cost coherence. The system maintains alignment not through constant intervention but through shared attractor geometry. Trajectories converge naturally because that's the structure of the space they're moving through.

Norm Erosion as Attractor Decay

During disintegration, norms erode. What was unthinkable becomes thinkable, then debatable, then acceptable, then strategic. This is attractor decay. The basins that pulled trajectories toward norm-compliant behavior flatten out. Defection becomes as stable as cooperation. Violence becomes as stable as peace.

Geometrically, this transforms the manifold from one with deep, stable attractors (integration) to one with shallow, unstable attractors or no attractors at all (crisis). Behavior becomes unpredictable because there are no stable states to converge toward. Everything is contingent, contested, chaotic.

Why Norm Breakdown Is Irreversible (In the Short Term)

Once norms erode, they're extremely difficult to restore without going through crisis and reset. This is because norm erosion is a coordination failure at scale. Norms only work if everyone expects everyone else to follow them. Once that expectation breaks, individuals have no incentive to cooperate unilaterally.

Geometrically, rebuilding attractor structure requires reshaping the entire manifold, which requires either:

1. External imposition (a new state with capacity to enforce new norms)
2. Exhaustion (crisis reduces the system to such a simple state that new coordination is possible)
3. Generational replacement (new populations without memory of the broken norms)

This is why societies in crisis can't simply decide to cooperate again. The attractor geometry that enabled cooperation has been destroyed.

Meaning Collapse and Coherence Geometry

The secular cycle isn't just about politics and economics. It's about meaning. During integration, life makes sense. The system provides roles, narratives, futures. During disintegration, meaning collapses along with social coherence.

M = C/T at Macro Scale

In AToM's framework, meaning equals coherence over time. A society with high coherence—aligned components, smooth trajectories, stable attractors—generates meaning for its members. The roles you occupy make sense within the larger system. The narratives you inhabit are coherent. The future you plan for is imaginable.

During disintegration, coherence collapses, and meaning collapses with it. The roles that structured your identity become unstable. The narratives that explained the world fragment into competing, incommensurable stories. The future becomes unpredictable and terrifying.

This isn't metaphorical meaning-making. It's geometric. Your life trajectory depends on the manifold you're moving through. When that manifold is smooth, stable, and coherent, you can navigate it. When it fragments, you can't. Meaning is the lived experience of being able to maintain coherent trajectories. Meaninglessness is the lived experience of coherence collapse.

Personal Experience of Macro Curvature

When historians describe periods of crisis as times when "the center cannot hold" or "all that is solid melts into air," they're describing the phenomenology of navigating a high-curvature, fragmenting manifold.

You feel it as:

• Institutions you relied on becoming unreliable
• Plans you made becoming impossible
• Skills you developed becoming worthless
• Identities you built becoming contested
• Futures you expected disappearing

This is what it feels like, from the inside, to live through a period when the macro-social coherence you were embedded in is collapsing. The geometry of the civilization-scale system propagates down to the geometry of your personal life. You can't maintain smooth trajectories because the manifold won't support them.

Why the Same Patterns Recur: Coherence Dynamics Are Universal

The secular cycle recurs across civilizations not because of some mystical historical force but because coherence dynamics are universal. Any system that maintains organization over time while navigating constraint will exhibit similar patterns.

Cells, organisms, ecosystems, societies—all of them are dynamical systems moving through state-space under constraint. All of them exhibit:

• Integration when constraint is manageable and components are aligned
• Compression when constraint accumulates
• Crisis when constraint exceeds capacity and coherence collapses
• Reset when the system simplifies and finds a new stable configuration

The secular cycle at civilization scale is the same geometric process that occurs in biological development, in ecosystem succession, in individual psychological crisis. The timescales differ. The specific variables differ. But the underlying dynamics are identical.

This is why Turchin's predictions work. He's not predicting history. He's reading the geometry.

This is Part 5 of the Cliodynamics series, exploring Peter Turchin's mathematical history through AToM coherence geometry.

Previous: The Age of Discord
Next: Narrative Fragmentation

Further Reading

• Turchin, P., & Nefedov, S. A. (2009). Secular Cycles. Princeton University Press.
• Thurner, S., Klimek, P., & Hanel, R. (2018). Introduction to the Theory of Complex Systems. Oxford University Press.
• Scheffer, M., et al. (2009). "Early-Warning Signals for Critical Transitions." Nature 461(7260): 53-59.
• West, G. (2017). Scale: The Universal Laws of Life, Growth, and Death in Organisms, Cities, and Companies. Penguin Press.