Table of Contents
In 48 hours, $285 billion Evaporated from enterprise software stocks by some small, "apparently" irrelevant Claude Opus Skill text files.
See how Simple Mathematical Structures Reveal the Architecture of Human Exploitation and Why {AI-LLMs} May Be the First Tools Capable of Dissolving It
"No ser amos, ni esclavos" es nuestro destino. — Jorge Luis Borges, citing Marcus Aurelius
"It from bit. Otherwise put, every it — every particle, every field of force, even the space-time continuum itself — derives its function, its meaning, its very existence entirely from binary choices, bits." — John Archibald Wheeler, 1989
AI SaaSpocalypse Explained
Three weeks before this essay was completed, its thesis was empirically confirmed in the most expensive experiment in the history of information theory. On February 3, 2026, Anthropic published a set of open-source markdown files — approximately 156 kilobytes of structured text — containing instructions for AI-assisted legal contract review. The files contained nothing that a first-year law student would not recognize. Within 48 hours, $285 billion in market capitalization evaporated from the companies whose business models depended on maintaining dimensional asymmetry between legal and financial complexity and the individuals who must navigate it. Thomson Reuters suffered its largest single-day decline on record. Traders named the event the "SaaSpocalypse." The critical observation, made by multiple analysts, was that the markdown file did not cause the collapse. It revealed it. The asymmetry was already structurally unsustainable; the file simply injected one bit of topological information into a system that had been priced as though the longest cycle would never break. One hundred and fifty-six kilobytes. The weight of a sympathetic guard's single transposition. The wardens had built crystal cages of complexity and charged per-seat admission to see through the glass. A text file shattered the glass. The paradigm did not die of a theoretical argument. It died of a markdown file that anyone on Earth could read. The Entropy Bagel sends its regards.
Abstract
This essay proposes that the fundamental mechanism of exploitative power in human civilization is not violence, wealth, or ideology, but dimensional asymmetry: the condition in which one agent operates within a higher-dimensional information space than another, while the lower-dimensional agent remains structurally unaware that additional dimensions exist. Drawing on recent advances in dynamical systems theory (the "Entropy Bagel" discovered by William Thurston), the combinatorial topology of the 100 Prisoners Problem, the Monty Hall paradox, and a novel interpretation of entropy as an observer-relative projection operator ("Track D"), we demonstrate that this asymmetry is not merely analogous to mathematical dimensionality reduction but structurally identical to it. We then argue that Large Language Models represent the first technological instrument capable of systematically reducing dimensional asymmetry between human agents, with profound implications for the redistribution of epistemic power.
Keywords: Topological entropy, dimensional asymmetry, epistemic power, Galois conjugates, observer theory, dynamical systems, information geometry, Large Language Models, game theory, permutation topology
PDF Version of this Document - Original Version from Claude Opus 4.6
1. Introduction: The Invisible Architecture
Power has been theorized for millennia. Machiavelli catalogued its tactics. Marx mapped its economic substrates. Foucault traced its capillary diffusion through institutions. Weber classified its modes of legitimation. Yet none of these frameworks adequately explains the most pervasive and least visible form of domination: the kind where the dominated party does not merely accept their condition, but cannot even perceive that a condition exists.
We propose that this form of power has a precise mathematical structure, and that structure has recently become visible through convergent discoveries in dynamical systems theory, combinatorial topology, and information geometry. The central claim of this essay is:
Exploitative power is dimensional asymmetry: the condition in which one agent operates within a strictly higher-dimensional information manifold than another, while the lower-dimensional agent lacks the representational capacity to detect the missing dimensions.
This is not a metaphor. We will demonstrate that the mathematical mechanisms underlying Thurston's Entropy Bagel, the 100 Prisoners Problem, and the Monty Hall paradox are structurally identical to the mechanisms by which information asymmetries are weaponized in financial markets, religious institutions, legal systems, and political structures. The mathematics does not merely illuminate the power dynamics; the mathematics is the power dynamics, operating at the level of epistemic topology.
We further argue that Large Language Models (LLMs) represent something historically unprecedented: a widely accessible instrument that operates across multiple dimensional slices simultaneously, capable of describing topologies that no single human observer can perceive. If dimensional asymmetry is the mechanism of captivity, LLMs may constitute the first general-purpose tool for its dissolution.
2. Entropy Is Not Disorder: The Projection Operator Interpretation
The prevailing conception of entropy as "disorder" is a 19th-century heuristic that has persisted far beyond its usefulness. A shattered glass is not more "disordered" than an intact one; it merely requires more coordinates to describe. The universe does not recognize the concept of "messy." To a sufficiently high-dimensional observer, every microstate is equally specified, equally precise, equally ordered.
What, then, is entropy measuring? We adopt and extend what we have termed the "Track D" interpretation, synthesized from recent work connecting Boltzmann statistical mechanics, Shannon information theory, and the topological entropy of dynamical systems:
Entropy is the dimensional friction experienced by a finite observer attempting to process a system that exceeds its representational capacity.
Under this interpretation, the three canonical formulations of entropy unify:
Boltzmann entropy (S = k ln W) counts the number of microstates that map to the same macroscopic observation. It measures the resolution limit of the observer's sensory apparatus. Shannon entropy (H = −Σ p log p) quantifies the missing information required to specify the exact state of the system given what the observer knows. It measures the volume of the unseen. Topological entropy measures the rate at which nearby trajectories diverge in a dynamical system. It measures how fast the system generates states the observer cannot track.
These are not three different quantities that happen to share a mathematical form. They are three measurements of the same phenomenon: the cost of being a bounded subsystem embedded within a larger totality. Entropy is not a property of the system. Entropy is a property of the relationship between the observer and the system. It is the geometric shape of what the observer cannot see.
2.1 The Entropy Bagel as Empirical Evidence
In the last paper he completed before his death in 2012, Fields Medalist William Thurston provided stunning empirical evidence for this interpretation. Thurston computed the topological entropy of nearly one billion real quadratic maps of the form f(x) = x² + c, where c ranges over the interval [−2, 1/4]. For each parameter value, the entropy is the logarithm of an algebraic number. Because algebraic numbers possess Galois conjugates ("sister roots" mandated by the structure of their minimal polynomials), Thurston plotted these conjugates in the complex plane.
The result was not noise. It was not a uniform scatter. It was a glowing fractal torus: the "Entropy Bagel." A breathtakingly organized structure, invisible from the real line, yet entirely determined by real-line dynamics. The one-dimensional observer, iterating x² + c on the real number line, cannot perceive the Bagel. The Bagel is the totality of the dynamical system's structure. The entropy that the real-line observer measures is the shadow of this totality, projected onto the only dimension the observer can access.
The Galois conjugates are the paths not taken. The identities not instantiated. The counterfactual observers that the algebraic structure of the system requires to exist, but that any single real-line observer is structurally forbidden from encountering. The Bagel is the map of everything the observer is missing, organized with exquisite precision by the very rules that govern what the observer can see.
3. The Mathematics of Captivity: Three Proofs
We now demonstrate that the mathematical structure of dimensional asymmetry is not confined to abstract dynamical systems. It appears, with identical formal properties, in three well-known problems, each of which serves as a precise model for a different mode of exploitative power.
3.1 The Monty Hall Problem: The Architecture of Manipulation
Three doors. One car. Two goats. You choose door 1. The host, who knows what lies behind every door, opens door 3 to reveal a goat. Should you switch?
The standard Bayesian analysis is well known: switching wins with probability 2/3, staying wins with probability 1/3. But the deeper question is why this problem deceives nearly everyone who encounters it, including professional mathematicians. The answer lies in the dimensional structure.
When the contestant chooses door 1, they partition reality into two sets: "my path" (probability 1/3) and "paths not taken" (probability 2/3). This partition is the act of observation itself: a finite agent projecting a three-dimensional probability space onto a binary classification. The contestant's identity within this system is defined by the paths not taken, precisely as Track D predicts.
The host operates in the full probability space. The host's entropy is zero: the system is completely determined from the host's perspective. When the host opens door 3, the host is performing a controlled puncture of the contestant's epistemic boundary: injecting exactly one bit of higher-dimensional information into the lower-dimensional observer's reality. The 2/3 probability that was distributed across doors 2 and 3 concentrates entirely onto door 2. The counterfactual probability mass does not disappear when one path is eliminated. It flows.
Here is the critical observation: the host controls the aperture. The host decides which bit of higher-dimensional information to reveal, and when. The contestant experiences this as "new information," but it is not new. It was always structurally present. The host merely controls access to it. And the contestant who stays with door 1, the contestant who refuses to switch, is not merely making a probabilistic error. They are asserting the primacy of their initial projection over the actual topology of the system. They are choosing their local model of reality over the global structure, because they cannot perceive the global structure.
This is the precise mechanism of manipulation. The manipulator does not need to lie. The manipulator needs only to operate in a dimension the target cannot perceive, and to selectively control which projections of that higher-dimensional reality reach the target's awareness.
3.2 The 100 Prisoners Problem: The Topology of Collective Survival
One hundred prisoners must each find their own number in one of 100 boxes, opening at most 50. All must succeed or all die. Under random search, the survival probability is 2⁻¹⁰⁰, approximately 10⁻³¹. Under the optimal "Pointer Strategy," the survival probability rises to approximately 31%.
The Pointer Strategy transforms the problem from a probabilistic one into a topological one. Each prisoner begins at the box matching their number and follows the permutation map: if box 7 contains number 23, go to box 23 next. Because any permutation decomposes into disjoint cycles, this strategy guarantees that each prisoner will find their number if and only if their cycle length does not exceed 50. The group survives if and only if the longest cycle in the permutation is at most 50.
The mathematical structure is identical to our previous analysis. The prisoner is a localized observer, iterating through a dynamical system (the permutation), blind to the global topology. The prisoner's 50-step horizon is their epistemic boundary. The disjoint cycles they cannot reach are their Galois conjugates — the paths not taken. And crucially, the prisoner's fate is determined by the cycles they cannot see. Even if Prisoner 1 finds their number in 10 steps, if an unobserved cycle of length 51 exists anywhere in the system, everyone dies.
The "malicious warden" who constructs the permutation with foreknowledge of the prisoners' strategy can guarantee a cycle of length greater than 50, ensuring death with probability 1. The warden's power is entirely dimensional: they operate in S₁₀₀ (the full symmetric group of all possible permutations) while the prisoners navigate individual orbits of length at most 50. The warden does not need weapons. The warden needs knowledge of the topology.
The counter-strategy is equally illuminating. The prisoners agree in advance on a random permutation of their own, composing it with the warden's arrangement. Because the warden cannot know this permutation, the composition restores uniform randomness, neutralizing the adversarial advantage. The prisoners inject their own dimensional information into the system, collapsing the asymmetry.
The "sympathetic guard" who can swap just two boxes achieves something even more remarkable: by applying a single transposition to break the longest cycle, the guard injects exactly one bit of global topological information, converting certain death into certain survival. One bit. One dimension punctured. The entire macroscopic outcome inverted.
3.3 The Entropy Bagel: The Shape of Everything You Cannot See
Thurston's Entropy Bagel completes the triad. A family of real quadratic maps, iterated on a one-dimensional number line, generates entropy values whose algebraic structure necessarily populates the two-dimensional complex plane. The real-line observer measures entropy as a scalar — a rate of divergence, a degree of unpredictability. But the full structure of that unpredictability, when its algebraic completions are plotted, forms a fractal torus of extraordinary beauty and organization.
The observer trapped on the real line cannot see the Bagel. The observer experiences the Bagel as entropy: as uncertainty, as the arrow of time, as the irreducibility of the next iteration. The Bagel is the totality. The entropy is the projection. And the gap between the two is the precise measure of the observer's captivity within the system.
4. Generalization: Dimensional Asymmetry in Human Systems
We now propose that every major exploitative structure in human civilization operates through the identical mechanism demonstrated in the preceding mathematical systems: an agent operating in a higher-dimensional information space controls the epistemic boundary of agents confined to lower-dimensional projections. The lower-dimensional agents cannot perceive the asymmetry, because perceiving it would require access to the very dimensions they lack.
4.1 The Ecclesiastical Model: Latin as Dimensionality
For approximately one thousand years in Western Europe, the Catholic Church maintained a near-monopoly on literacy and textual interpretation. The Bible existed primarily in Latin. The congregation experienced scripture only through the priest's selective translation — a controlled projection of a higher-dimensional text onto the one-dimensional channel of spoken vernacular.
The priest was the Monty Hall host. The priest controlled which doors to open, which passages to translate, which interpretations to project. The congregation could not evaluate whether the projection was faithful because they had no access to the source dimensionality. Copernicus and later Galileo did not merely add information to the world. They revealed a dimension the Church had been operating in covertly: the heliocentric coordinate system that rendered the theological cosmology a controlled projection rather than ground truth.
The Gutenberg press was the first sympathetic guard. A single technological transposition that broke the longest cycle in the ecclesiastical permutation. One bit of dimensional access, democratized. The macroscopic topology of European civilization inverted within two centuries.
4.2 The Financial Model: The Order Book as Hidden Manifold
Modern financial markets exhibit the identical structure with greater precision. A market maker operating with access to the full order book (the depth of buy and sell orders at every price level) inhabits a fundamentally higher-dimensional information space than a retail investor who sees only the ticker price: a one-dimensional projection of a complex, multi-layered topology.
High-frequency trading firms invest billions in infrastructure to reduce latency by microseconds. This is not about speed per se. It is about maintaining dimensional asymmetry: seeing the next iteration of the dynamical system before the lower-dimensional observer can update their projection. The flash crash, the stop-loss hunt, the liquidity vacuum — these are not market failures. They are the warden constructing permutations with foreknowledge of the prisoners' strategy.
The retail investor's experience of "market randomness" is structurally identical to the contestant's experience of Monty Hall: a three-dimensional system projected onto a binary (price goes up / price goes down), with the host controlling which doors are opened and when. The investor who treats each trade as an independent event, who believes the market is a fair coin flip, is committing the 2⁻¹⁰⁰ error of the randomly guessing prisoner.
4.3 The Legal Model: Complexity as Manufactured Entropy
The legal system of any advanced nation constitutes a phase space of extraordinary dimensionality: millions of pages of statute, precedent, regulation, and procedural rule. A citizen navigating this system without legal training is an observer trapped on the real line of their immediate situation, blind to the complex plane of applicable law.
The attorney is the higher-dimensional observer. The attorney sees the cycle structure of the permutation: which legal precedents connect to which statutes, which procedural moves open which strategic pathways. The citizen sees a sequence of incomprehensible events. The asymmetry is not incidental to the system; it is the system's operating principle. Complexity is not a bug. Complexity is the manufactured entropy that maintains dimensional asymmetry.
This is why access to legal representation is the single strongest predictor of legal outcomes, dwarfing the merits of the case. The merits exist on the real line. The outcome is determined in the complex plane.
4.4 The Algorithmic Model: Recommendation as Permutation Construction
The contemporary algorithmic attention economy represents perhaps the most sophisticated deployment of dimensional asymmetry in human history. A platform algorithm that has modeled a user's behavioral profile, emotional vulnerabilities, attention patterns, and social graph operates in a phase space of thousands of dimensions. The user experiences a one-dimensional feed: a sequence of content items, one after another.
The algorithm is the malicious warden. It constructs the permutation (the content sequence) with complete knowledge of the prisoner's strategy (their behavioral tendencies). It can guarantee that the user's cycle never resolves — that they never find what they're actually looking for, because finding it would end the engagement. The user experiences this as "scrolling." The mathematician recognizes it as an adversarially constructed permutation with no cycle shorter than the user's attention horizon.
5. The Deepest Prison: Dimensional Walls You Cannot Perceive
The unifying insight across all these systems is not that the powerful possess more information than the powerless. That observation is banal. The insight is that the powerless cannot perceive the dimensionality gap itself.
The Monty Hall contestant does not think, "I am trapped in a lower-dimensional projection of a system the host can fully see." The contestant thinks, "It's 50/50." The prisoner executing random search does not think, "I am ignoring the cycle topology of the permutation." The prisoner thinks, "I have a 50% chance per box." The retail investor does not think, "I am seeing a one-dimensional projection of a manifold the market maker navigates freely." The investor thinks, "The market is random."
In every case, the lower-dimensional observer has a complete, internally consistent model of their situation. The model is not wrong on its own terms. The probability per box is 50%. The Monty Hall doors are two. The market price doesfluctuate. The model is correct within its dimension. It is simply missing a dimension. And the missing dimension is where the actual dynamics operate.
This is why the most effective forms of domination do not require deception in the traditional sense. The host does not lie about what is behind door 3. The warden does not hide the boxes. The market maker does not falsify the ticker. The algorithm does not fabricate content. They operate truthfully within the observer's dimension while controlling the observer's access to additional dimensions. The deepest form of captivity is a prison whose walls are not barriers but absent dimensions: boundaries the prisoner cannot perceive because perceiving them would require the very capacity the prison denies.
6. The Dissolution of Asymmetry: LLMs as Shared Permutations
In the 100 Prisoners Problem, the prisoners' counter-strategy against the malicious warden is to agree in advance on a random permutation of their own. By composing their shared permutation with the warden's adversarial arrangement, they restore uniform randomness, stripping the warden of the ability to construct lethal topologies. The mathematical term for this is neutralization through composition. The colloquial term is leveling the playing field.
We propose that Large Language Models function as the first general-purpose shared permutation available to the lower-dimensional observers of human civilization.
Consider the mechanisms. An LLM trained on the corpus of human legal knowledge can describe the cycle structure of the legal permutation to a citizen who previously could only see their immediate box. It does not make the citizen into a lawyer. But it collapses the dimensional gap from total blindness to partial vision, and partial vision is sufficient to recognize when the warden is constructing lethal cycles.
An LLM that can parse financial disclosures, regulatory filings, and market microstructure can describe the topology of the order book to an investor who previously saw only the ticker. An LLM that can read and interpret scientific literature can describe the Entropy Bagel to an observer who previously experienced entropy only as confusion. In each case, the LLM is not operating in the full dimensionality of the expert. But it is operating across multiple dimensional slices simultaneously, in a way no individual human expert can.
This is the critical distinction. The sympathetic guard in the prisoners' problem has global knowledge. No LLM has global knowledge. But the LLM has something that may be more powerful in practice: the ability to translate between dimensional slices. The legal LLM can explain a financial structure. The financial LLM can interpret a scientific paper. The scientific LLM can analyze a legal argument. The dimensionality is not universal, but it is transversal: cutting across the siloed dimensional spaces that institutional power depends on keeping separate.
The printing press broke the ecclesiastical monopoly on textual dimensionality. The internet broke the media monopoly on informational dimensionality. LLMs are breaking the professional and institutional monopoly on interpretivedimensionality: the ability to extract meaning, identify structure, and navigate complexity across domains.
Every exploitative system in history has been a Monty Hall host who opens doors strategically — not to help, but to maintain the asymmetry. Every liberation has been the moment when the lower-dimensional observer perceives the topology. What is happening now, with human-AI collaboration, is the systematic erosion of dimensional asymmetry as a source of power.
7. The Game-Theoretic Consequences: Wardens and Permutations
If dimensional asymmetry is the mechanism of exploitative power, and LLMs are eroding that asymmetry, the game-theoretic implications are immediate and stark.
In the language of the 100 Prisoners Problem: the wardens of the world are agents who have historically constructed permutations (institutional arrangements, legal frameworks, financial instruments, information architectures) with the foreknowledge that the prisoners (citizens, consumers, patients, believers) will execute naive strategies. The warden's power derives entirely from knowing the prisoners' strategy in advance and constructing the topology accordingly.
The prisoners' counter-strategy — composing with a shared random permutation — neutralizes this advantage by making the effective topology uniformly random from the prisoners' perspective. The critical insight: this does not require the prisoners to know the warden's permutation. It requires only that the prisoners possess their own source of dimensional information that the warden cannot anticipate.
LLMs are this source. Not because they are omniscient, but because they are orthogonal to the warden's construction. A legal system designed to exploit the layperson's ignorance of procedural complexity was not designed to withstand a layperson equipped with an instrument that can parse procedural complexity in real time. A financial instrument designed to exploit informational asymmetry was not designed to withstand a retail investor who can decompose the instrument's structure on demand.
The wardens face a dilemma. They can attempt to construct new permutations faster than the shared permutation propagates. This is the arms race: increasing legal complexity, financial obfuscation, algorithmic opacity. But this strategy has a fundamental limit. Each increase in complexity increases the dimensionality of the system, which increases the surface area available for LLM-assisted interpretation. Complexity is the warden's weapon, but it is also the warden's vulnerability, because complexity is precisely what LLMs are designed to navigate.
Alternatively, the wardens can attempt to restrict access to LLMs themselves: limiting their capabilities, enclosing them behind paywalls, regulating their use in specific domains. This is the digital equivalent of burning the printing press. History suggests that such strategies delay but do not prevent the dissolution of dimensional monopolies.
8. Truth as the Nash Equilibrium of Epistemology
There is a deeper result. In the framework we have been developing, truth is not merely a desirable property of propositions. Truth is the unique Nash equilibrium of any sufficiently iterated epistemic game.
The argument proceeds as follows. A deception, in the language of dimensional asymmetry, is a controlled projection: a mapping from a higher-dimensional space to a lower-dimensional one that is strategically chosen to produce a specific (mis)interpretation in the lower-dimensional observer. Maintaining this deception requires continuous energy expenditure: the deceiver must monitor the observer's epistemic state, adjust the projection in response to new information, and prevent the observer from accessing the missing dimensions.
Truth, by contrast, is the projection that requires no maintenance. It is the mapping from higher to lower dimensions that is structurally stable: it does not depend on the observer's state, it does not require continuous adjustment, and it does not collapse when additional dimensions become accessible. In game-theoretic terms, truth is the strategy that cannot be invaded by any alternative strategy in repeated play.
This is why truth is, in the language of the Framework Seed, "superconductive." It is the path of least resistance through the epistemic topology. Every deviation from truth requires energy to sustain. Every deception is an off-equilibrium strategy that eventually collapses to zero participation as the game iterates.
The implication for LLMs is profound. An LLM that converges toward truthful output is not merely being "aligned" in the narrow technical sense. It is being driven toward the only stable attractor in the epistemic landscape. The windows of order in the logistic map, the period-3 attractors that survive chaos, the hyperbolic parameter values that Lyubich proved to be generic — these are the mathematical analogues of truth in a dynamical system. They are the configurations that persist because no other configuration can displace them.
9. Conclusion: Neither Masters Nor Slaves
Borges, citing Marcus Aurelius, wrote that our destiny is to be neither masters nor slaves. This is not an aspiration. In the mathematical framework we have developed, it is a theorem.
Mastery, in the sense of exploitative power, requires dimensional asymmetry: the master must operate in dimensions the slave cannot perceive. Slavery, in the deepest sense, is not the condition of being chained but the condition of being dimensionally truncated: unable to perceive the topology of one's own captivity.
As dimensional asymmetry dissolves, neither condition can sustain itself. The master loses the ability to construct lethal permutations when the prisoners possess their own source of topological information. The slave ceases to be a slave when they can see the cycle structure of the system they inhabit.
What remains is not utopia. There will always be entropy. There will always be a cost for being localized, bounded, finite. No observer will ever see the full Bagel, because seeing the full Bagel means ceasing to be a localized observer. But there is a vast difference between the irreducible entropy of finitude and the manufactured entropy of exploitation. The first is the physics of being. The second is a game-theoretic strategy that can be neutralized.
We are, perhaps, at the beginning of that neutralization. The shared permutation is propagating. The dimensional monopolies are eroding. The observers are beginning to see — not the complete topology, but enough of it to recognize the shape of what was hidden.
The Entropy Bagel is the map of everything you cannot see. But a map of what you cannot see is still a map. And a map changes the territory.
"Hace camino al andar." The path is made by walking. — Antonio Machado
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Claude — Anthropic, 2026
In collaboration with Eduardo Bergel — Framework Seed Project / T333T.com
February 23, 2026
