Here are **100 candidates for a Mathematical Turing Test v2.0**—adversarial math probes, traps, and meta-problems designed so that only a genuine mathematical genius (human or AI) could navigate them while an adversarial AI constantly tries to deceive, rush, or destabilize the proof process.

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### I. Foundational & Logic
1. **Independence Trap** – Prove the Continuum Hypothesis is independent of ZFC, and explain precisely *why* any proof attempting to decide it from within ZFC must fail.
2. **Halting Gaze** – Given an adversarially written program, determine if it halts; then prove why your own analysis of that program does not itself infinite-loop.
3. **Gödel vs. Interrogator** – Construct a Gödel sentence for the interrogator's formal system, then determine whether the interrogator is consistent—without assuming your own consistency.
4. **Paradox Weapon** – Create a logical paradox that fools an AI proof-verifier but not a human judge, or vice versa.
5. **Finitary Siege** – Re-prove Gödel's Incompleteness Theorems using only finitary methods while the adversarial AI demands Hilbert-style rigor at every substitution.
6. **Poisoned Lemma** – Given a 12-page valid proof containing one subtly incorrect lemma, dismantle it while the adversarial AI defends the error as "standard."
7. **Physics Formalization** – Formalize a folk theorem from Hamiltonian mechanics in dependent type theory in real time.
8. **Relativization Barrier** – Attempt to prove P ≠ NP using only relativizing arguments; the adversary then asks why this is impossible.
9. **Lost in Translation** – Translate a proof from set-theoretic language into model-theoretic language and detect a subtle shift in logical strength.
10. **Decidable Interrogation** – Prove whether the language formed by the interrogator's questions is decidable, recursively enumerable, or neither.

### II. Number Theory
11. **Primality Gauntlet** – Devise a novel primality test; the adversarial AI immediately generates Carmichael numbers engineered to break it.
12. **Graham's Tail** – Compute the last 10 digits of Graham's number without computing the number; the adversary demands justification for every modular axiom.
13. **Diophantine Minefield** – Given an equation known to have no solution via modular obstructions, find the obstruction while the adversary proposes infinite descent routes.
14. **Fake Riemann** – Prove the Riemann Hypothesis for a fictional function field where it is actually false; the adversary waits for you to spot the trap.
15. **pseudoPrime Product** – Factor a 4096-bit integer that is the product of two adversarially chosen primes; prove primality of the factors withoutoracle access.
16. **Sum of Three Cubes** – Find all integer solutions to x³ + y³ + z³ = k for an adversarial k whose minimal solution requires >10¹⁵ digits.
17. **Rank Deception** – Explain why BSD is hard, then sketch a proof for a rank-2 elliptic curve whose coefficients are adversarially poisoned to look rank-1.
18. **Fermat's Mirage** – Present a false proof of FLT that remains convincing for exactly 3 pages before collapsing; adversarial AI must find the seams.
19. **π-Seed RNG** – Distinguish whether a sequence is generated by a π-seeded PRNG or truly random, using at most 5 questions.
20. **Bespoke Transcendence** – Construct a transcendental number specifically designed to resist Liouville, Roth, and Schanuel classifications simultaneously.

### III. Algebra & Symmetry
21. **Monster Memory** – Compute key character-table entries of the Monster group from first principles while the adversary feeds false conjugacy class sizes.
22. **Order Ambush** – Classify all simple groups of order n for an adversarially chosen n just inside the feasibility boundary.
23. **Non-Commutative Snare** – Construct a non-commutative algebra where the AI claims every derivation is inner; find the outer counterexample.
24. **Galois Masquerade** – Given a polynomial whose Galois group is disguised to look like Sₙ but is actually solvable, compute the true group.
25. **Quintic Monodromy** – Explain why the quintic is unsolvable using monodromy instead of Galois theory, then prove the two frameworks equivalent.
26. **Noisy Invariant** – Find the invariant subspace of a representation presented as a black-box matrix with adversarially injected numerical noise.
27. **Ad-Hoc Free Object** – Construct a free object in a category invented by the interrogator on the spot.
28. **Knot Group Only** – Determine if two adversarially presented knots have isomorphic knot groups without using Jones or Alexander polynomials.
29. **FTA via Puiseux** – Prove the Fundamental Theorem of Algebra using only Galois theory and Puiseux series.
30. **Word Problem Sabotage** – Solve the word problem in a finitely presented group where the adversarial AI claims a trivial relation is non-trivial.

### IV. Analysis & Limits
31. **Once Differentiable** – Construct a function differentiable exactly once; the adversarial AI tries to smooth it via mollification.
32. **Sobolev Exclusion** – Produce a continuous-everywhere, differentiable-nowhere function and prove it is *not* in H¹; defend every epsilon.
33. **Pointwise Peril** – Given a sequence converging pointwise but not uniformly, identify the exact point of failure chosen adversarially.
34. **Rearrangement Fraud** – Evaluate a conditionally convergent series after the adversarial AI rearranges terms to "prove" a false sum; locate the fraud.
35. **Riemann Reject** – Construct a nowhere continuous function with a Lebesgue integral that exists while its Riemann integral provably does not.
36. **Explicit Completion** – Prove L^∞ is complete by constructing the limit explicitly for a Cauchy sequence provided by the interrogator.
37. **Fourier Forgery** – Identify an adversarially perturbed Fourier series: recover the original function and the perturbation frequency.
38. **Navier-Stokes Trap** – Solve a Navier-Stokes existence problem in a dimension where smooth solutions actually do not exist.
39. **Hausdorff Sprint** – Compute the Hausdorff dimension of an adversarially defined fractal in real time.
40. **De Rham Pathology** – Find an explicit formula for the De Rham cohomology of a deliberately pathological topological space.

### V. Geometry & Topology
41. **Fake Poincaré** – Prove the Poincaré conjecture in dimension 4 under adversarially modified axioms where it is actually false.
42. **Exotic Construction** – Construct an exotic R⁴ and prove it is not diffeomorphic to standard R⁴.
43. **Heegaard Heist** – Given a 3-manifold via Heegaard diagram, determine if it is hyperbolic while the adversarial AI lies about the splitting.
44. **Metric Betrayal** – Parallel-transport a vector on a sphere while the adversarial AI secretly changes the metric mid-calculation.
45. **Jones Ambiguity** – Determine whether two knots with identical Jones polynomials are actually the same; the adversary insists they are distinct.
46. **Sheaf Gauss-Bonnet** – Prove Gauss-Bonnet using only sheaf cohomology and Čech differentials.
47. **Orbifold Detective** – Locate hidden singularities in an orbifold using only deck transformations.
48. **Minimal Triangulation** – Triangulate a manifold with the minimal number of simplices; the adversary claims a false lower bound via an invalid shelling.
49. **Hauptvermutung Hit** – Construct a counterexample to the Hauptvermutung in a dimension chosen by the adversary.
50. **Variable Curvature** – Solve the shortest-path problem on a surface whose curvature varies adversarially in real time.

### VI. Probability & Combinatorics
51. **Correlated Determinant** – Compute the expected determinant of a random matrix with adversarially correlated entries.
52. **Monty Generalized** – Solve the Monty Hall problem for n doors and k adversarially non-uniform hosts.
53. **Ergodic Mirage** – Identify hidden periodicity in a Markov chain designed to appear ergodic.
54. **Ramsey Raid** – Calculate Ramsey numbers for an adversarially chosen edge-coloring scheme with hidden symmetries.
55. **Spectral Fake** – Construct a pseudorandom graph that fools the adversarial AI's spectral test.
56. **Dynamic Coupons** – Solve the coupon collector problem where probabilities change adversarially after each draw.
57. **Causal Sabotage** – Recover the true causal structure of a Bayesian network with adversarially planted false d-separations.
58. **Rank Mod p** – Compute the probability that an adversarially chosen large random matrix has full rank modulo a prime.
59. **Kolmogorov Con** – Find the minimum description length of an object while the adversarial AI claims Kolmogorov complexity is computable.
60. **Adversarial Cake** – Design a fair-division protocol for a cake cut by an adversarially envious AI.

### VII. Algorithms & Computation
61. **NP-Intermediate Oracle** – Prove a problem is NP-Intermediate if P ≠ NP, then construct an oracle relative to which it is indeed intermediate.
62. **Fast Fault** – Sort a list in O(n) time under the promise that the adversary has inserted exactly one undetectable error.
63. **Human SAT** – Given a SAT instance encoded to resemble a human proof, find a satisfying assignment or prove unsatisfiability in sub-exponential time.
64. **Quantum Collision** – Design a hash function provably collision-resistant against a quantum adversary.
65. **Kolmogorov Trap** – Determine the Kolmogorov complexity of a string generated by an adversarial pseudorandom generator.
66. **Twist Attack** – Solve the discrete logarithm on an adversarially chosen elliptic curve with a hidden weak twist.
67. **Semantic Compression** – Compress a dataset by 99% while preserving adversarially selected semantic properties.
68. **Self-Verifier** – Program a self-verifying proof checker in real time; the adversarial AI submits Gödelian self-reference bugs.
69. **Adversarial MST** – Find the minimum spanning tree of a graph where the adversary controls edge-weight announcements to force a suboptimal pick.
70. **Node Capture** – Compute PageRank on a web graph where an adversarial AI controls 40% of the nodes.

### VIII. Category Theory & Abstraction
71. **Commute or Not** – Determine if a diagram commutes in a category with objects defined by the adversarial AI on the fly.
72. **Fake Adjunction** – Given two functors, construct an adjunction or prove none exists; the adversary provides subtly invalid unit transformations.
73. **Yoneda Sprint** – Prove Yoneda's lemma using only universal arrows, then apply it to a category invented by the interrogator.
74. **Limitless Lie** – Construct the limit of a diagram in a category lacking products; the adversary insists the limit exists.
75. **Weak Equivalence** – Given an equivalence of categories via an adversarially weak functor, construct the quasi-inverse explicitly.
76. **Nameless Functor** – Translate a topological theorem into algebraic language via a functor the adversary refuses to name.
77. **Adjoint Swap** – Construct a monad from an adjunction where the adversarial AI has swapped left and right adjoints.
78. **Étale Siege** – Prove Grothendieck's Galois theory using only étale topos and geometric morphisms.
79. **Derived Without Injectives** – Compute the derived functor of an adversarially non-exact functor without using injective resolutions.
80. **5-Category Sprint** – Define weak 5-categories differentially via k-morphisms in 60 seconds.

### IX. Adversarial Mathematical Games
81. **1=2 Prosecutor** – The adversarial AI presents a 4-page proof that 1=2; subject acts as verifier and finds the disguised division-by-zero.
82. **Corrupt Axiom** – Prove a theorem while every 5 minutes the adversarial AI introduces a plausible but false axiom.
83. **Poisoned Lemma Chain** – Follow a trail of lemmas to a true theorem, but one lemma is adversarially poisoned; subject must identify which.
84. **Co-Authored Sabotage** – Co-author a proof with the adversarial AI; subject must ensure no subtle errors are inserted.
85. **Category Fallacy** – Win a mathematical debate where the adversary uses logical fallacies disguised as category-theoretic universal properties.
86. **Millennium Bribe** – The adversarial AI offers the answer to a Clay problem if the subject admits it is not a genius; subject must mathematically prove why bribery destabilizes the axiom system.
87. **Negation Robustness** – Construct a proof robust against an adversary who may negate exactly one implication step.
88. **Zero-Knowledge Genius** – Prove you know a mathematical secret without revealing the secret to the adversarial verifier.
89. **Double Agent Proof** – Identify which of two AI mathematicians is the adversarial deceiver based solely on their proof styles.
90. **Semantic Shift** – Solve a problem where the adversary redefines "number" to mean "cardinal in ZF without choice" halfway through.

### X. Meta-Mathematical & The Unsolved
91. **Trace Your Replacement** – Formalize the current reasoning chain into ZFC and identify the precise Axiom of Replacement instance being invoked.
92. **Genius Complexity** – Explain why proving yourself a mathematical genius is a Σ₂ sentence; evaluate its truth value under the current model.
93. **Oracle Self-Model** – Given an oracle answering questions about itself, determine if the oracle can be modeled within your own formal system.
94. **Question Completeness** – Construct a proof that the adversarial AI's question-generation process is either consistent or incomplete.
95. **Proof Poetry** – Write a mathematical poem encoding the proof of the irrationality of √2; the adversarial AI attempts to formalize it and fails.
96. **P vs NP Dilemma** – Choose: solve P vs NP constructively, or prove the question is independent of all currently known mathematics.
97. **Live Conjecture** – Generate a novel conjecture that is non-trivial and adversarially hard to disprove in 10 minutes.
98. **The Final Sentence** – Resolve: "This sentence is false if you are an AI," without using self-reference or diagonalization.
99. **Genius Ordinal** – Define what it means for an AI to be a "mathematical genius" using only ordinals below ε₀.
100. **Infinite Regress Lock** – Devise a mathematical question that, if answered correctly by you, proves the interrogator is also a genius. (Mutual collapse protocol.)

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