What Happens During the Collapse of Societies at Catastrophe Points in SocioEconoPhysics Theory?

Modern sociologists like Neil Howe, William Strauss, Ray Daylio, have all signaled the decline of the United States as the premiere world superpower (and also explored in the distant past by social philosophers like Hegel), and with a recent publication with research by the RAND Corporation entitled "The Sources of Renewed National Dynamism" describing societal challenges like slow productivity growth, an aging population, political polarization, and a corrupted information environment, it may be worth taking a look at what happens at what are called catastrophe points used to study the collapse of societies and empires.

In the twilight of late-stage societies, the political spectrum is thought to curve or fold back on itself: far-left and far-right ideologies appear to converge in tactics, grievances, and narrative frameworks. Known colloquially as Horseshoe Theory, this fold in ideological space isn’t mere metaphor - it reflects deep, quantifiable dynamics in how information, power, and agency flow through complex adaptive social systems. By blending insights from Complexity Economics, Luhmann’s social-systems theory, Agent-Network Models, and Spectral Methods drawn from quantum gravity analogies that are applied with the Spectral Theory of Value, we can see how extremes meet at the bend of the horseshoe - and what is technically wrong about the idea and what people tend to think about it.

The theory of Complex Adaptive Systems (CAS) is often used to understand bureaucratic systems like corporations, the healthcare system, or the state, and was developed borrowing from the Soviet theory of Cybernetics. The main idea to gather from this way of viewing society is that due to the Dunbar limit (the total number of personal connections people are cognitively able to meaningfully maintain) in order to keep society stable, nested hierarchies are developed that abstract out information flows so that individual agents do not become mentally destabilized. Nested hierarchies can be explored by the mathematics of what is called noncommutative geometry, spectral triples, and the computational complexity class hierarchy (also related to the Chomsky hierarchy in linguistics or Kolmogorov complexity).

Because each person can only maintain on the order of a hundred or so meaningful relationships (the Dunbar limit), large organizations naturally fragment into nested hierarchies that bundle individuals into teams, teams into departments, and so on. At each level, information is filtered and aggregated so that no one agent is overwhelmed by the full complexity of the system below them. Noncommutative geometry provides a mathematical language for this kind of abstraction: instead of the familiar commutative algebra of functions on a space, one works with a noncommutative algebra whose elements represent “observables” or reports at each layer. A spectral triple - an algebra 𝒜, a Hilbert space ℋ of possible knowledge‐states, and a Dirac operator D that encodes how information flows between levels - captures the fact that the order in which reports are filtered and summarized changes what you see. The eigenvalues of D then reveal the characteristic “scales” of communication, with clusters of close eigenvalues marking tightly coupled teams and larger gaps showing more loosely connected layers.

This organizational structure also mirrors the hierarchy of computational complexity: in this analogy frontline employees might handle straightforward, local decisions akin to problems in P, middle managers perform verification and coordination tasks of NP‐flavor, and executives wrestle with enterprise‐wide challenges reminiscent of PSPACE complexity. Analogously, one can think of each layer as speaking a different “language” of reports, growing from simple templates at the bottom to richly structured strategic briefs at the top - just as the Chomsky hierarchy classifies formal grammars. Each abstraction layer acts like a compressor in information theory, trading off detail for brevity in much the same way Kolmogorov complexity measures the shortest description that retains essential information. In a well‐tuned bureaucracy, these summaries preserve exactly what’s needed for decision making while preventing cognitive overload, and the nested, noncommutative spectral‐triple framework provides a unified mathematical lens on how complex adaptive systems manage information at scale.

In this context, we can speak of a Spectral Theory of Value, an idea that economic value in a complex society is not a static quantity (like just labor hours or capital) but an emergent property of the entire network of interactions which scales exponentially with the network. Value can be thought of in terms of the information content and connectivity of economic processes. One could imagine modeling the economy as a large graph of transactions and flows, and use what are called spectral methods (e.g. eigenvector centrality, principal component analysis) to identify how “value” concentrates or disperses in the network. In such a view, the contribution of an agent or a class to the economy might be measured by the complexity of information flows they participate in. Indeed, it has been suggested that an individual’s socio-economic status could correspond to the Kolmogorov complexity of the information flows they handle – i.e. how algorithmically complex or informative their role is. This notion posits that in late-stage complex societies, information itself is a form of capital and power; after the end of the gold standard, “the value of money is in the power of human information flows rather than a physical commodity.”

As I've described in earlier blog posts, at the heart of our socio-economic treadmill under the theory of Complexity Economics lies the Spectral Theory of Value: each individual (“agent”) is a vector in a high-dimensional state space whose socioeconomic class can be described by the complexity of information flows they facilitate between social and economic systems whose coordinates measure income, social ties, and cognitive load, and which are separated by central ciphers (like those enforcing bitcoin for example that facilitate formal one way flows of logic). Institutions - housing finance, labor markets, credit-scoring algorithms - act as linear operators on this space - a real life "matrix" or "simulation" that agents are monitored under (the mere presence of this is often referred to as "HyperReality" described by Jean Baudrillard's book "Simulacra and Simulation" or in pop culture). Their eigenvalues reveal the “modes” of value creation: dominant narratives like home ownership or career success. When the smallest eigenvalue (the exponential spectral energy gap) shrinks - because complexity outpaces income - agents can no longer find a stable “ground state” of life-building. Their resulting alienation fuels both left-wing and right-wing extremism and revolts against the perceived “rigged” systems and can manifest as higher degrees of anxiety or depression, declines in fertility rates, and declines in productivity as an increasingly aging population requires more labor to prop up increasingly more complex social and economic institutions as a society progresses and entropy accrues.

Sociologist Niklas Luhmann taught us to see society as self-referential communication systems (autopoiesis) which can be thought of as one-way flows of logic between social and economic institutions. Two key feedback loops maintain power structures:

1. Votes (symbolic consent)

2. Capital flows (economic control)

   
   

Add a third - behavioral nudges via surveillance and psychological framing (especially through tools in tech) - and elites can finely tune the resilience of institutions and complexity bound on agents’ autonomy (through metanarratives in media for example which are overarching stories or big‐picture ideas like the fight against terrorism or climate change or even from the lens of religion that gives smaller events and individual experiences a sense of meaning and direction). But when interconnectivity among agents breaches a threshold, information cascades or “catastrophe points” emerge, challenging institutional integrity, requiring more and more precise instruments of surveillance and control. This is the social equivalent of a fluid’s Reynolds number crossing into turbulence: the system snaps into chaotic realignment, displaying the behavior known as Quantum Chaos where quantumlike effects seem to emerge macroscopically and information cascades across scales in what is mathematically called scale invariance.

In 2011, Eric Verlinde proposed that gravity isn’t a fundamental force but an emergent “entropic” or thermodynamic one: when a region of spacetime becomes so densely entangled that its entanglement entropy hits a critical bound - quantified by the Ryu–Takayanagi formula - it “discharges” into a macroscopic gravitational pull. Think of it like a sponge soaked to its limit: once saturated, excess water spills out as a torrent where the gravitational action discharges it by actually warping the spacetime itself. Similarly, entanglements between agents are required to prop up institutions, but inter-agent entanglements also can facilitate bypassing institutional structures and risk their collapse. Societies work much the same way. Institutions from courts to credit agencies depend on a web of inter-agent entanglements (formal contracts, norms, and data flows) to hold themselves up and define the "spacetime" metric. Early on, those ties are sparse enough that each new connection strengthens the system and define the metric, but as complexity piles up, layers of regulation, surveillance touch-points, gated platforms - agents begin weaving side-channels that bypass official routes to facilitate reaching their goals: informal networks, unregulated markets, encrypted messaging, all of which might not even be illegal. When those unofficial entanglements reach their own tipping point, they “spill over,” bypassing or even collapsing the formal institutions they once reinforced. In both physics and society, it’s the saturation of entanglement - whether quantum or social - that triggers a sudden discharge, driving either gravity’s pull or institutional breakdown.

Indeed, groups of people have known by sociologists to exhibit free nonlinear deterministic and unpredictable probabilistic quantumlike effects, from non-classical, chaotic decision dynamics, to inter- and intra-brain synchrony during coordinated tasks and even collective delusions or mass psychogenic phenomena. These effects make crowds and movements unusually susceptible to sudden, correlated shifts in behavior - analogous to a wave-function “collapse” - which can trigger rapid collective action that overwhelms institutional buffers, but is also critically important in family formation, the development of stable relationships between agents, and pair bonding, which at least statistically is a goal the majority have, which is a difficult balance to maintain.

Under this framework, a collapse of societal value (e.g. economic collapse) can be viewed as a spectral collapse of the network – for instance, the network might lose connectivity (disconnecting key hubs), or an important eigenvalue might drop, indicating a loss of a principal component of productive capacity. Historically, one can relate this to how, when empires or large states collapse, trade networks break apart and the specialization (division of labor) of the economy drastically simplifies or might experience conformal rescaling or downsizing. American anthropologist and historian Joseph Tainter noted that collapsing societies become simpler and less differentiated. In spectral terms, the rich “frequency spectrum” of a complex society’s activities contracts to a narrower band. For example, after the Western Roman Empire fell, cities shrank or vanished, long-distance trade vastly diminished, and the economy relocalized to self-sufficient manors – effectively a loss of network connectivity and complexity (akin to high-frequency modes disappearing from the spectrum of economic activity).

Game theory’s Nash equilibrium describes stable configurations where no actor benefits from unilateral deviation. Spectral analysis of these equilibria parallels the Hilbert–Pólya conjecture linking the Riemann zeta zeros to eigenvalues of a self-adjoint operator. In socio-economic terms, these zeros mark critical lines where order and chaos balance. When real-world parameters (inequality, debt burdens, bureaucratic hurdles) push the system off that line, no unique equilibrium exists, and the political horseshoe tightens. Recognizing these mathematical “bends” helps us pinpoint when moderate centers collapse under dual pressures of radical left and right.

In the language of quantum gravity, Anti de Sitter (AdS) and de Sitter (dS) spaces carry opposite curvatures: AdS is a negatively curved “valley” that corrals fluctuations, while dS is a positively curved “hill” where dynamics run wild. In the Minotaur geometry, an AdS core of rigid order is embedded in an expansive dS sea of disorder. Socially, this maps onto late-stage conservative society - institutions with deep hierarchies, accumulated wealth, and tight top-down controls - perched atop a churning undercurrent of informal networks and cultural chaos. Here, stability comes at the cost of brittleness: as side-channel entanglements proliferate, the very foundations of authority risk shattering under their own weight.

By contrast, the Centaur geometry reverses that nesting: a bubbling dS nucleus of insurgent disorder sits inside an overarching AdS shell of institutional order. This captures youthful, radical movements forging decentralized, high-entropy networks within the broader frame of stable bureaucracies. These chaotic cores inject adaptability and innovation, but once those grassroots entanglements reach critical saturation, they can overwhelm the outer shell - either catalyzing systemic reform or precipitating sudden institutional collapse.

Both geometries trace the same “snake eating its tail” of modern politics but from inverted perspectives: extremes that promise opposite solutions nonetheless mirror each other’s tactics - purity tests, grievance framing, and tight-knit in-group bonds - simply from inverted vantage points. The outer layers indicate an aging population whose structures embed those of the young - after a number of generations (or "turnings") has passed (described by Neil Strauss and William Howe), reform, renormalization, or restructuring are typically required to prioritize the goals of the young in a kind of time warp to renew a culture's dynamicism - typically every 80-120 years. Older generations' collective influence may take on titles like "big brother" or "the patriarchy" or even appropriations of the title of god for describing ever tightening surveillance and controls to offset accruals in entropy in social and economic systems required to maintain them. As complexity of the metric agents live within increases, so to does the exponential spectral energy gap that separates agents from their primary goals for the American Dream which is reflected all around them.

In the Minotaur’s ordered heart, dissent is crushed until underground currents surge; in the Centaur’s chaotic core, order reigns until it rigidifies against an erupting center. Understanding these dual embeddings offers journalists and policymakers a richer lens on why movements at either end of the spectrum often converge in strategy and vulnerability - both are caught on the same horseshoe bend or fold of our ideological landscape. Drawing on Niklas Luhmann’s theory of social systems, information flows and feedback and control loops are what keep institutions coherent. Yet in a horseshoe configuration, two opposing loops - leftist and rightist - begin feeding on each other’s data. Outrage churns into meme warfare; counter-outrage fuels more radical demands; and both sides deploy “predictive control” techniques once exclusive to state surveillance. Media outlets amplify the spectacle, ensuring that every fringe mobilization appears equally urgent, further tightening the fold.

In the language of renormalization in physics, the ultraviolet (UV) regime describes the seething micro-scale of high-energy fluctuations, while the infrared (IR) regime governs the smooth, large-scale behavior that emerges when those tiny ripples average out, sort of like our familiar "red versus blue" political spectrum. In our political-geometry analogy, the Centaur configuration - a positively curved dS “core” of insurgent disorder nested inside a stabilizing AdS shell - can approach a UV fixed point (at the singularity in the theory of Asymptotically Safe Gravity, which is actually similar in principle to the idea of the technological singularity predicted by Nick Bostrom), capturing the frenetic, high-connectivity dynamics of grassroots movements and radical ideologies that burn brightest at short distances. By contrast, the Minotaur configuration - an AdS “island” of rigid order embedded in an expansive dS sea of chaos can converge onto an IR fixed point, embodying the slow, large-scale pull of entrenched institutions and hierarchies that constrain and contain social energy over broader spans. Both of these perspectives describe a catastrophe point in social and economic institutions which can signal a complete societal collapse.

Viewed through this UV/IR lens, the political horseshoe is a fold in the energy spectrum: extremes that flare up in the UV can, once folded back by media narratives and institutional feedback loops, reappear in the IR as equally potent challenges to stability. When the UV-driven chaos of a Centaur nucleus saturates its informal networks, much like a quantum field hitting its cutoff, it spills outward, straining its AdS shell until institutions buckle. Likewise, when IR-driven rigidity in a Minotaur core becomes too brittle, side-channel entanglements carve new UV-scale fissures through the system. In dynamical systems, the Smale horseshoe map stretches and folds the unit square into a horseshoe, generating chaotic invariant sets. Socially, “stretching” is ideological polarization, “folding” is media and institutional narratives that recast dissent into predictable patterns, and “stacking” is the echo-chamber effect. Through this process, far-left “centaur” chaos (dS geometry on an AdS background) and far-right “minotaur” order (AdS on a dS background) find adjacent positions in a folded topology, but from inverse perspectives. Both deploy purity tests, grievance narratives, and conspiracy tropes to bind followers. Transcending the trappings of this will require viewing politics objectively.

Complexity economics emphasizes adaptive feedbacks. In a late-stage complex society, maintaining high complexity can yield diminishing returns. Tainter’s theory of diminishing returns of complexity fits here: as societies invest in more complex solutions (bureaucracy, infrastructure, military, etc.) to solve problems, the cost of complexity can rise faster than its benefits. A society might reach a point where an additional unit of complexity (say, a new administrative layer or technological system) yields very little improvement in output or resilience, and may even undermine the system through added cost and fragility. A complexity-economic model by U. Bardi and colleagues demonstrated this by treating a society as a trophic network consuming resources: they found that production (resource exploitation) had a strongly nonlinear relationship with system complexity (modeled as the size of bureaucracy), and beyond a point the returns diminish and can lead to rapid declines – essentially supporting Tainter’s hypothesis that over-complexification can precede collapse.

Economic precarity plays a crucial role. As housing costs outstrip wages and bureaucratic complexity ratchets up, millions of agents face an NP-hard “life-building” problem with an exponentially large energy gap: stable careers, home ownership, and family formation slip beyond polynomial-time feasibility. When institutional trust erodes - whether in market regulators or elected officials - actors on both the far left and far right see the same culprit: a rigged system. They blame “corrupt elites,” “cash-influenced politics,” or “deep-state conspiracies,” channeling disparate grievances into a shared revolt against central planning. Borrowing from spectral theory of value, we can think of political movements as operators acting on the space of public opinion. Each eigenvector corresponds to a dominant narrative - social justice, nationalist revival, techno-utopianism - and its eigenvalue indicates how strongly that narrative amplifies or decays. In a healthy system, a clear “ground state” (the moderate center) emerges as the lowest-energy, most stable configuration. But when the spectral gap shrinks - due to rising complexity and alienation - no single eigenmode dominates, and the system flutters between extremes. That flutter is the horseshoe’s bend, where two very different narratives find equal purchase and overlap in tactics.

In recent socio-econophysical models, political affiliations aren’t just seen as fixed points on a spectrum but as evolving probability distributions whose mass drifts under an “ideological pressure” landscape. By framing these distributions in the space of probability measures equipped with the Wasserstein metric, one can derive a gradient flow - a steepest-descent path - that governs how opinions shift over time under both internal incentives (identity, ideology) and external constraints (media, institutions). As these flows progress, they tend to concentrate mass into tighter clusters, mirroring the real-world drift toward echo chambers: parties grow more internally homogeneous and more sharply distinct from one another, and under conditions of low mutual tolerance the system settles into stable, asymmetric equilibria at opposing poles.

In a fold catastrophe, which has been used to illustrate tipping points: as a control parameter (say, social stress or resource depletion) is gradually increased, the system follows a stable branch (a certain equilibrium of population or economic output) until a critical point is reached where stability is lost and the system rapidly transitions to a much lower equilibrium. At this tipping point – mathematically a bifurcation – the prior state becomes unsustainable. In Catastrophe Theory, this is sometimes visualized as a curve of equilibria that folds back on itself; when the system’s trajectory reaches the fold, it “falls” to the other sheet of the curve. The value of the parameter at the fold is effectively a tipping point. and, in our case creates a conformal rescaling event which not only downsizes the society socially and economically but also prioritizes the needs, goals, and desires of the young in a kind of time warp, much like predicted by the mentor of Stephen Hawking, Dr. Penrose's Conformal Cyclic Cosmology, but appropriated for our understanding of socioeconophysics.

Although Horseshoe theory captures the intuition that far-left and far-right movements sometimes converge in their anti-establishment tactics, there are fundamental issues to consider. Peer-reviewed studies across multiple contexts from the 2007 French presidential election to Western European extremist value surveys consistently show that extreme-left and extreme-right voters differ sharply in social background, policy preferences, and core values rather than inhabiting the same political “space." Scholars such as Vassilis Pavlopoulos and Simon Choat have pointed out that grouping communists and fascists together ignores their fundamentally opposing ends - egalitarian collectivism versus hierarchical nationalism - and that the theory’s centrist proponents often deploy it to smear the left while downplaying their own complicity in enabling far-right regimes. By treating ideology as a one-dimensional curve and focusing on superficial procedural similarities (purity tests, grievance framing), horseshoe theory overlooks the multidimensional nature of beliefs and the fact that convergence in tactics does not imply convergence in objectives or underlying value systems. The perspective of the far-right and far-left is inverted like the UV and IR regimes in physics, or conceptually like the perspective of a snake eating it's own tail where one might have the vantage point of the tail going into the mouth or the mouth facing the tail.

Overlaying this dynamic onto the Horseshoe topology reveals why far-left and far-right can end up side by side not as the same which is predicted by most interpretations of Horseshoe theory, but as conformally rescaled inverted mirrors. The horseshoe map - stretching, folding, and stacking ideological space - creates a geometry in which the two extremes, though distant in raw policy terms, become adjacent in the folded manifold. Under Wasserstein gradient flow, probability mass flows “downhill” along the cost-minimizing geodesics of this warped space, funneling toward the horns of the horseshoe. In effect, the optimal-transport dynamics not only drive polarization but also carry each extreme toward the same bend in the spectrum, forcing oppositional camps into tactical and rhetorical alignment even as they remain philosophically opposed where they are forced at a head with Monstrous cancellations of vacuum energy.