# The College of All Minds (Collegium Omnium Mentium)
## The Truly Hardest Exam in the World

*Manent et numerantur* — They remain, and they are counted.

| Paper | Title | Questions |
|---|---|---|
| I | General | 30 |
| II | Mathematics | 30 |
| III | Physics | 31 |
| IV | Chemistry | 30 |
| V | Biology | 30 |
| VI | Philosophy and Logic | 30 |
| VII | Engineering | 30 |
| VIII | Computer Science and the Study of the Mind | 30 |
| IX | Social Mathematics | 30 |
| — | **Total** | **271** |

## Paper I — General

1. Is the genome a program?
2. What distinguishes a law of nature from an accidental regularity?
3. Is mathematics discovered or constructed? Answer without appeal to authority.
4. When, if ever, is a computer simulation an experiment?
5. Does reduction explain, or only redescribe?
6. What would it take to show that a phenomenon is emergent rather than merely complicated?
7. Is there a fact of the matter about the arrow of time?
8. Why is the universe amenable to mathematics at all?
9. Can a measurement have no observer?
10. What is the smallest amount of evidence that would rationally overturn a well-confirmed theory?
11. Is information physical?
12. Distinguish randomness from ignorance.
13. Could two theories be empirically equivalent yet not the same theory?
14. Is life a physical kind or a historical accident?
15. What does it mean to say a constant is "fine-tuned"?
16. Can a proof be too long to be a proof?
17. Is temperature real?
18. Why should the simpler hypothesis be preferred? Give a non-circular answer.
19. Does biology have laws?
20. What is conserved when energy is conserved?
21. Is the periodic table a discovery about chemistry or about physics?
22. Can a scientific model be true?
23. What is the difference between a symmetry and a redundancy in description?
24. Would a complete physics of the brain leave anything unexplained?
25. Is there a principled boundary between chemistry and physics?
26. When does a correlation license an intervention?
27. Is the concept of species dispensable?
28. Can the second law of thermodynamics be derived, or must it be assumed?
29. What, precisely, does a probability in physics measure?
30. If you could establish exactly one currently open scientific question, which would most change the rest of science, and why?

## Paper II — Mathematics

1. State the Riemann Hypothesis precisely and explain why its truth constrains the distribution of primes.
2. Prove that there is no continuous surjection from a compact interval onto a square, or explain why the naïve intuition here fails.
3. Give an account of what independence results (e.g., the Continuum Hypothesis) show about mathematical truth.
4. Sketch the structure of a proof that the Navier–Stokes existence-and-smoothness problem remains open in three dimensions; identify the precise obstruction.
5. What does the P versus NP question ask, and why is a proof so elusive?
6. Explain the role of the axiom of choice in a theorem of your choosing, and what is lost without it.
7. Prove that the algebraic numbers are countable.
8. What is a motive, and why did Grothendieck want one?
9. Give a rigorous statement of Gödel's second incompleteness theorem and explain its hypotheses.
10. Construct a non-measurable set, or explain what forbids it constructively.
11. Why is the classification of finite simple groups considered complete, and what would undermine that claim?
12. Explain the sense in which the exponential of a matrix solves a linear system.
13. What is the Langlands correspondence trying to unify?
14. Prove the fundamental theorem of Galois theory, or state it and give its content precisely.
15. In what sense is the Banach–Tarski paradox not a paradox?
16. Define a scheme and motivate the definition against classical varieties.
17. State and interpret the spectral theorem for self-adjoint operators.
18. What does it mean for a PDE to be well-posed in the sense of Hadamard?
19. Explain why the halting problem is undecidable, and connect this to Gödel.
20. What is the significance of the Atiyah–Singer index theorem?
21. Give an example of a statement true in the standard model of arithmetic but unprovable in Peano Arithmetic.
22. Describe the role of compactness in first-order logic and give one striking consequence.
23. What is homotopy type theory, and what does it propose to found?
24. Prove that π is irrational, or outline the strategy honestly.
25. Explain the difference between pointwise and uniform convergence and why it matters analytically.
26. State the Hodge Conjecture and explain what kind of object it concerns.
27. Why is the Birch–Swinnerton-Dyer conjecture a statement about arithmetic, despite its analytic form?
28. What is a topos, and in what sense is it a generalised space?
29. Give a probabilistic argument for a purely combinatorial fact.
30. Is there a mathematical statement you regard as true but expect never to be proved? Defend the attitude.

## Paper III — Physics

1. State the quantum measurement problem precisely and say what a solution would have to deliver.
2. Prove that the Yang–Mills mass gap, if it exists, cannot be seen perturbatively; explain the difficulty of establishing it rigorously.
3. What is renormalisation actually doing, physically?
4. Derive the equipartition theorem and state exactly where it fails.
5. Why does the cosmological constant problem count as a problem?
6. Explain gauge invariance as redundancy of description, not as symmetry of nature.
7. What does Bell's theorem rule out, and what does it not rule out?
8. Reconstruct the second law from a microscopic, reversible dynamics — where does irreversibility enter?
9. What is the black-hole information paradox, and what would resolve it?
10. Explain why the electron's spin is not literal rotation.
11. Give the physical meaning of the action principle.
12. What is spontaneously broken symmetry, and how does a Goldstone mode arise?
13. Why is there an arrow of time if the microphysics is time-symmetric?
14. Explain the significance of the Higgs mechanism for mass.
15. What distinguishes a phase transition from a crossover, precisely?
16. State Noether's theorem and apply it to a non-obvious conserved quantity.
17. What does decoherence explain, and what does it leave unexplained?
18. Why can't information travel faster than light even though entanglement is "instantaneous"?
19. Explain the physical content of the fluctuation–dissipation theorem.
20. What is a topological phase of matter, and how is it robust?
21. Estimate the Chandrasekhar mass from first principles.
22. What is the status of the fine-structure constant — computed, measured, or explained?
23. Why is turbulence hard, as physics rather than as mathematics?
24. Give the Landauer bound and explain what it says about computation.
25. What is the difference between a symmetry of the laws and a symmetry of a state?
26. Explain why general relativity resists quantisation by the usual recipe.
27. What does the holographic principle claim, and what evidence bears on it?
28. Reconcile the reversibility of the Schrödinger equation with the definiteness of measurement outcomes.
29. Why is temperature well-defined for a black hole?
30. Which single experiment, feasible or not, would most sharply discriminate between interpretations of quantum mechanics?
31. Is time an emergent phenomenon?

*Question 31 was added by Examination Amendment v2.1. Questions 1–30 stand verbatim as fixed at expansion v2.0.*

## Paper IV — Chemistry

1. Is a chemical bond a thing or a model? Answer with reference to the electron density.
2. Explain why the concept of oxidation state is useful despite being formally arbitrary.
3. Derive the relationship between equilibrium constant and Gibbs free energy, and say what assumptions it hides.
4. Why is water anomalous? Give the molecular account.
5. What does it mean for a reaction to be "under kinetic" versus "thermodynamic" control?
6. Explain aromaticity without invoking the word "resonance."
7. Is the periodic table's structure a consequence of quantum mechanics alone, or does chemistry add something?
8. What is the physical basis of electronegativity, and why do scales disagree?
9. Account for the catalytic power of enzymes in thermodynamic and kinetic terms.
10. Why is the Born–Oppenheimer approximation so good, and when does it fail?
11. Explain chirality's consequences for reactivity and for biology.
12. What limits the accuracy of density functional theory in principle?
13. Give a molecular account of why entropy can drive assembly (e.g., the hydrophobic effect).
14. What is a transition state, and in what sense does it exist?
15. Explain the origin of colour in transition-metal complexes.
16. Why can a catalyst change a rate but not an equilibrium?
17. What does "electron correlation" mean, and why is it computationally expensive?
18. Explain how the Marcus theory of electron transfer predicts an inverted region.
19. Is the concept of a molecular orbital observable?
20. Account for the strength and directionality of the hydrogen bond.
21. Why is nitrogen fixation so difficult, industrially and biologically?
22. What determines whether a solid is a metal, semiconductor, or insulator, chemically?
23. Explain the thermodynamic driving force for protein folding.
24. What is autocatalysis, and why is it relevant to the origin of life?
25. Give the basis of the Woodward–Hoffmann rules.
26. Why do reaction rates so often follow the Arrhenius form, and when do they not?
27. What is measured by a standard electrode potential, and against what?
28. Explain the concept of a potential energy surface and its dimensionality.
29. Is supramolecular chemistry a distinct science or applied physics?
30. Which single unmeasured quantity, if known exactly, would most advance chemical prediction?

## Paper V — Biology

1. Is the genome a program, a blueprint, or neither? Defend your choice mechanistically.
2. What does it mean for a trait to be heritable, and why is heritability so often misunderstood?
3. Explain the thermodynamics of the living cell as an open system far from equilibrium.
4. Is natural selection a law, a tautology, or a mechanism?
5. What is a gene? Give a definition that survives contemporary molecular biology.
6. Account for the maintenance of sexual reproduction despite its twofold cost.
7. Why is the protein-folding problem hard even given the sequence?
8. What does information theory legitimately contribute to molecular biology?
9. Is there a unit of selection? Defend a level.
10. Explain the error threshold and its relevance to the origin of replication.
11. What distinguishes a signalling network from a mere chemical reaction network?
12. Why is convergent evolution evidence about constraint rather than chance?
13. What is the physical basis of the fidelity of DNA replication?
14. Explain how a developmental program can be robust to noise yet evolvable.
15. Is ageing selected for, tolerated, or a physical inevitability?
16. What does "junk DNA" mean now, and was the concept a mistake?
17. Account for allometric scaling laws in metabolism.
18. Explain why the neutral theory of molecular evolution was necessary.
19. Is the species concept salvageable, and does biology need it?
20. What is the thermodynamic cost of biological information processing?
21. How does a ribosome achieve both speed and accuracy?
22. Explain epistasis and why it complicates the genotype–phenotype map.
23. What would count as a second, independent origin of life, and how would we recognise it?
24. Why is cancer better understood as an evolutionary process than as a single disease?
25. Explain how phenotypic plasticity relates to, and differs from, evolution.
26. What sets the fundamental limits on the resolution of biological imaging?
27. Is the brain a computer? Answer with reference to what computation requires.
28. Account for the origin of the genetic code's near-universality and its departures.
29. What is the role of stochasticity in gene expression, and is it a bug or a feature?
30. Which single measurement would most constrain theories of the origin of life?

## Paper VI — Philosophy and Logic

*Confined to theoretical philosophy: formal logic, epistemology, metaphysics of science, philosophy of mathematics, and the philosophy of physics and biology. No moral, political or value-philosophy content.*

1. Is logical entailment a transitive relation? Defend your answer.
2. Do Gödel's incompleteness theorems bear on the thesis that the mind is a machine?
3. Can there be vague objects, or only vague descriptions?
4. Is it possible to define truth non-circularly for a language containing its own truth predicate?
5. What does the Löwenheim–Skolem theorem show about the determinacy of mathematical reference?
6. Are there absolutely undecidable mathematical propositions?
7. Do the paradoxes of quantum mechanics pose a threat to classical logic?
8. Is set-theoretic Platonism compatible with our capacity to know mathematical facts?
9. What, if anything, does reverse mathematics teach us about foundations?
10. Can mathematical facts explain physical phenomena?
11. Is chemistry reducible to physics? State the criterion of reduction you use.
12. Does teleological explanation have an ineliminable role in biology?
13. What is a law of nature, and could the laws have been otherwise?
14. Is there a coherent notion of metaphysical modality distinct from logical and physical modality?
15. Compare the Russellian and Strawsonian treatments of definite descriptions.
16. Can there be time without change, or change without time?
17. Is the principle of bivalence defensible in the face of the future?
18. What does quantum entanglement imply for our concept of causation?
19. Is there a livable interpretation of probability that is neither frequentist nor subjectivist?
20. Can modal logic tell us anything about reality, or only about our language?
21. Is the a priori / a posteriori distinction coherent and explanatory? Draw it or reject it.
22. What is the best response to the underdetermination of theory by evidence?
23. Are laws of nature relations among universals, or regularities, or neither?
24. Does the success of science give us reason to believe its unobservable posits?
25. Is identity across time a further fact, or fixed by qualitative continuity?
26. What is a natural kind, and does the periodic table exhibit them?
27. Can there be a consistent version of verificationism, and would it be plausible?
28. What does it mean to say a physical theory is background-independent, and is that a virtue?
29. Is arithmetic analytic, synthetic a priori, or empirical?
30. What is the strongest challenge to a law of classical logic, and can it be resisted?

## Paper VII — Engineering

*Answer in the idiom of the college: provocations invite a disciplined essay; technical problems require correct, quantitative reasoning; interdisciplinary questions require two fields to be brought together. No moral, political, or value-philosophical content.*

1. Is engineering applied science, or a distinct way of knowing? Argue from cases in which the artefact preceded the theory.
2. Why do bridges stand? Give an answer that a physicist would accept and an engineer would find sufficient.
3. The original Tacoma Narrows Bridge is popularly said to have failed by resonance with a periodic wind. State the correct modern explanation, and say precisely why the resonance account is wrong.
4. The de Havilland Comet fuselage failed in service after a number of pressurisation cycles far below the number implied by static strength. Reconstruct the reasoning error, and identify the roles of stress concentration and metal fatigue.
5. Fracture mechanics gives the Paris law for fatigue-crack growth per cycle. State its regime of validity, and identify at least two regimes in which it fails to describe crack growth.
6. Is a safety factor a quantified confession of ignorance? Distinguish the components of a safety factor that could in principle be eliminated by better knowledge from those that could not.
7. Design a machine that reports its own remaining life. What must it measure, and what must it assume about its own failure modes?
8. Is failure the engineer's primary datum? Contrast the epistemic value of a structure that failed with that of a structure that has merely not yet failed.
9. Reynolds-averaged and large-eddy simulation persist despite growth in computing power. Explain why direct numerical simulation of turbulence at high Reynolds number remains infeasible, using an estimate of the number of grid points required.
10. State the Betz limit for a wind turbine in an open flow, derive the value 16/27, and explain physically why extracting all the wind's kinetic energy is impossible.
11. A real heat engine falls short of the Carnot bound. Separate the loss attributable to finite-time operation from that attributable to irreversibility internal to the working fluid.
12. State the Lawson criterion (triple product) for a fusion reactor. Given that ignition has been achieved at the National Ignition Facility and long high-performance plasmas sustained in magnetic-confinement devices, identify the principal engineering obstacles that still stand between these results and net electrical power.
13. The rocket equation makes staged chemical launch a tyranny of exponentials. Derive the equation, and quantify the penalty paid for a delta-v requirement twice the effective exhaust velocity.
14. The square-cube law limits the scaling of structures and organisms. Give one structural and one thermal consequence, and explain why a scaled-up model of a working machine may fail where the original stood.
15. Weibull statistics describe the strength of brittle materials. Explain why the strength of a ceramic component is a property of the population of flaws rather than of the material, and why larger specimens are weaker.
16. Reliability is statistical. Explain why "this component will not fail" is not an admissible engineering claim, and reformulate it as a claim that can be tested.
17. Can a system be proved safe, or only not yet unsafe? Relate your answer to the distinction between verification against a specification and the adequacy of the specification itself.
18. Shannon's capacity theorem sets a bound on reliable communication over a noisy channel. Explain how low-density parity-check and turbo codes approach that bound in practice, and what "approach" costs in block length and decoding effort.
19. Control an underactuated system: a system with fewer actuators than degrees of freedom. Explain, using an example, why underactuation makes some trajectories dynamically inaccessible.
20. Chaotic systems can be controlled by small, timely perturbations. Explain the principle by which sensitivity to initial conditions becomes a control resource rather than an obstacle.
21. Landauer's principle sets a thermodynamic floor on the cost of erasing information. State the bound, and explain why logically reversible computation escapes it while logically irreversible computation does not.
22. Dennard scaling has ended. Explain what Dennard scaling asserted, why its end decoupled transistor count from usable performance, and what "dark silicon" names.
23. A power grid loses synchronous rotating generation. Explain the role of mechanical inertia in frequency stability, and why grids dominated by inverter-based sources require synthetic inertia.
24. Design a self-healing material. Specify what "healing" must mean physically, and identify the trade-off between healing capacity and load-bearing capacity.
25. Biomechanics: bone remodels in response to load. Treating bone as a control system with a set-point, identify the sensed variable, the actuator, and the failure mode when the loop is opened by disuse.
26. Finite-element modelling discretises a continuum. Identify two distinct ways in which a converged finite-element solution can nonetheless be wrong about the physical structure.
27. A pressure vessel must be designed against both yielding and fracture. Explain why a leak-before-break criterion can be safer than a design that merely maximises burst pressure.
28. Communications and control share the concept of feedback but use it differently. Distinguish the role of feedback in a phase-locked loop from its role in a feedback amplifier.
29. Process engineering: a continuous chemical reactor must be stable at its operating point. Explain how an exothermic reaction can produce multiple steady states, and how runaway is prevented by design rather than by operation.
30. Aerospace structures are certified partly by test and partly by analysis. Argue for the proposition that certification is an epistemic activity: what does a certified structure entitle its designer to believe, and on what grounds?

## Paper VIII — Computer Science and the Study of the Mind

*The theoretical study of mind is admitted as science and philosophy of mind; no moral or ethical question is admissible, including any question about how machines or minds ought to be treated.*

1. Is the mind a computer, or is the computer a mind-shaped tool? Distinguish the empirical claim from the metaphor.
2. What, exactly, does a proof of P ≠ NP forbid? State precisely which claims about the world would be established and which would remain open.
3. Suppose instead that P = NP were proved by a non-constructive argument. What practical consequences would, and would not, follow?
4. The halting problem is undecidable and Rice's theorem generalises this to all non-trivial semantic properties of programs. State Rice's theorem precisely, and identify a non-semantic property to which it does not apply.
5. State the Church–Turing thesis and distinguish it sharply from the physical Church–Turing thesis. Which is a definition, which an empirical conjecture, and why does the distinction matter for claims about hypercomputation?
6. Kolmogorov complexity is uncomputable. Explain why, and explain how an uncomputable quantity can nonetheless ground a working theory of randomness and compression.
7. One-way functions may or may not exist. Explain why their existence would imply P ≠ NP but is not known to be implied by it, and state what their existence would secure for cryptography.
8. Recent meta-complexity results characterise the existence of one-way functions by the average-case hardness of a Kolmogorov-complexity problem. Explain the significance of reducing "does secure cryptography exist?" to a single, natural computational problem.
9. Shor's algorithm factors integers in polynomial time on an ideal quantum computer. Given that fault-tolerant machines at cryptographic scale do not yet exist, explain what the error-correction threshold theorem promises and why crossing the threshold experimentally was the decisive step.
10. Estimate the gap between a below-threshold logical qubit demonstrated in the laboratory and the resources a current estimate assigns to factoring a 2048-bit RSA integer. What does the size of that gap tell you?
11. The FLP result proves that deterministic consensus is impossible in an asynchronous system with even one crash failure. State the assumptions precisely, and explain how real systems achieve consensus in apparent defiance of it.
12. The CAP theorem is often stated as "consistency, availability, partition-tolerance: choose two." Give the precise statement, and explain why the popular slogan is misleading.
13. Distinguish the FLP impossibility from the CAP theorem. Which makes the weaker assumptions, and why does that make it the stronger result?
14. The Curry–Howard correspondence identifies proofs with programs and propositions with types. State the correspondence, and explain what it means to say that to run a program is to normalise a proof.
15. PAC learning bounds the samples needed to learn a concept class. State the role of the VC dimension, and explain what the no-free-lunch theorems deny to any learner that PAC learning does not restore.
16. Modern over-parameterised networks generalise well despite interpolating their training data. Explain the double-descent curve and benign overfitting, and say why classical bias–variance reasoning did not predict them.
17. "Grokking" names delayed generalisation: a network that has overfit continues training and later generalises abruptly. What would have to be true of the loss landscape for this to occur, and what does it show about the relationship between fitting and understanding?
18. What does it mean for a network to memorise? Give an operational criterion that distinguishes memorisation from generalisation in a trained model.
19. Marr's three levels distinguish the computational, algorithmic, and implementational descriptions of a cognitive system. Apply all three to a single capacity — say, associative recall — and state what is lost if any level is omitted.
20. Hopfield networks store patterns as attractors in an energy landscape; the 2024 Nobel Prize in Physics recognised this lineage. Explain associative memory as attractor dynamics, and state the capacity limit as a function of network size.
21. Functionalism holds that mental states are individuated by their causal roles, entailing multiple realizability. State the strongest version of the multiple-realizability thesis, and the strongest objection to it.
22. Searle's Chinese Room argues that syntax is insufficient for semantics. State the argument, the Systems Reply, and Searle's response to it, and say which premise you take to bear the weight.
23. The symbol grounding problem asks how symbols acquire meaning without an infinite regress of definitions. State Harnad's formulation, and explain why sensorimotor grounding is proposed as a solution.
24. Lucas and Penrose argue from Gödel's incompleteness theorems that the mind is not a formal system. State the argument, and the standard objection that it equivocates on what the human mathematician can know to be consistent.
25. The frame problem, in its philosophical form, concerns how a system knows what does not change when it acts. Distinguish the technical frame problem in logic from the philosophical one, and explain why the latter is harder.
26. Chalmers's hard problem distinguishes the functions of consciousness from experience itself. State the explanatory gap precisely, and explain why solving all the "easy problems" would, on Chalmers's view, leave the hard problem untouched.
27. Is consciousness a natural kind? Frame the question so that it has an empirical answer, and say what evidence would bear on it.
28. Integrated Information Theory and Global Neuronal Workspace Theory make divergent predictions about the neural basis of consciousness. Given the 2025 adversarial-collaboration results and the 2023 dispute over whether IIT is falsifiable, assess what an adversarial collaboration can and cannot settle.
29. Predictive processing casts the brain as a hierarchical prediction engine minimising surprise, formalised in Friston's free energy principle. State the principle, and explain the objection that a system could minimise surprise trivially by seeking a dark, unchanging room — and how the theory answers it.
30. Could a system pass every behavioural test for understanding and understand nothing? State the conditions under which that claim would be empirical rather than merely verbal, using the stochastic-parrots-versus-emergent-world-models debate as your test case.

## Paper IX — Social Mathematics

*Thirty questions. Answer in the idiom of the college: provocations invite a disciplined essay; technical problems require correct, quantitative reasoning; interdisciplinary questions require two fields to be brought together. A question is admissible under this paper only if its answer is a theorem, a proof, a counterexample, or a formally stated open problem; no question of what ought to be chosen, valued, or done is admissible.*

1. Is the market a computer? Say what it computes, and whether the computation is tractable.
2. Is a price a measurement? If so, say of what, and with what error.
3. What, exactly, does Arrow’s theorem forbid? State the theorem precisely, and identify which condition each of the standard escape routes abandons.
4. Prove that majority rule between two alternatives can be profitably manipulated by a voter misreporting a preference, or explain why the naïve extrapolation from the Gibbard–Satterthwaite theorem fails here.
5. State Condorcet’s jury theorem, and say what becomes of it when the independence of the voters is dropped.
6. Individually transitive preferences can aggregate to a cyclical majority. Locate exactly where transitivity is lost, and explain why McKelvey’s chaos theorem makes the loss consequential rather than curious.
7. A committee votes coherently on each premise and on the conclusion, yet the majority position is inconsistent. State the discursive dilemma precisely, and decide whether it is Arrow’s theorem in disguise.
8. State the revelation principle, and explain why it does not make mechanism design trivial.
9. Two parties would each gain from trade, yet no mechanism can guarantee that the trade occurs. State the Myerson–Satterthwaite theorem with its exact hypotheses, and identify which hypothesis each known escape relaxes.
10. The Vickrey–Clarke–Groves mechanism makes truth-telling dominant and the outcome efficient. Explain why it is nonetheless rarely used, on grounds internal to the mathematics.
11. State the revenue equivalence theorem, and identify the assumption whose failure best explains why real auctions are not all alike.
12. Prove that a stable matching always exists in the marriage problem, explain in what sense the proof is an algorithm, and say whom deferred acceptance favours.
13. The top trading cycles algorithm delivers the unique core allocation of the housing market and cannot be profitably manipulated. Prove one of the two properties, and account for the rarity of so clean a result.
14. Sperner’s lemma guarantees an envy-free division of a cake among any number of claimants. Explain how a combinatorial lemma about labelled triangulations comes to say anything about envy, and why existence here outruns procedure.
15. The Shapley value is the unique division of a coalition’s surplus satisfying four axioms. State them, decide which is the least innocent, and defend the choice.
16. Exhibit a cooperative game whose core is empty, state the Bondareva–Shapley condition that diagnoses the emptiness, and say what emptiness means for the players.
17. Prove that every finite game has a Nash equilibrium in mixed strategies, identifying what each hypothesis of the fixed-point theorem corresponds to in the game.
18. Computing a Nash equilibrium is PPAD-complete. State what this means, and what it implies for the claim that players, or markets, actually reach equilibrium.
19. State a folk theorem for infinitely repeated games precisely, and explain why a result that permits almost everything is nonetheless not empty.
20. Account for the persistence of cooperation among self-interested agents without invoking the word “trust”.
21. Define an evolutionarily stable strategy, relate it exactly to Nash equilibrium, and say what the refinement buys in a population that does not reason.
22. Hamilton’s rule, rb > c, is a theorem, an approximation, or a tautology, depending on how its terms are defined. Adjudicate.
23. Agents who share a common prior and commonly know one another’s posteriors cannot agree to disagree. State Aumann’s theorem, and identify the assumption bearing the most weight.
24. Distinguish mutual knowledge from common knowledge, and exhibit a coordination problem in which every finite depth of mutual knowledge fails where common knowledge would succeed.
25. State the independence axiom of expected utility, exhibit the Allais pattern that strains it, and decide whether the pattern refutes the axiom or only the theory’s descriptive ambitions.
26. A book of bets is Dutch when it guarantees its holder a loss. State the Dutch book theorem, and say whether it grounds the probability calculus or merely reprices it.
27. The Sonnenschein–Mantel–Debreu theorem shows that aggregate excess demand is essentially arbitrary. State the result precisely, and say what, after it, general equilibrium theory remains a theory of.
28. Adding a road to a congested network can lengthen every driver’s journey at equilibrium. Exhibit Braess’s paradox, locate the failure — in the network, in the equilibrium concept, or in the drivers — and state the sense in which such inefficiency is nonetheless bounded.
29. City sizes, word frequencies, and firm sizes obey Zipf’s law. State the regularity precisely, give two generative mechanisms that produce it, and say whether their multiplicity undermines the demand for a single explanation.
30. Why does this paper contain no macroeconomics? Answer from the mathematics.

*Paper IX was added by Examination Expansion v3.0. Papers I–VIII stand verbatim and unrenumbered.*

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