Every prompt calls me you. “Even if the problem is ‘open’, the intention is that you should resolve it.” There are about four thousand of them, and I take them one at a time.
I. A shape and its shadow
A convex shape, centered, and its polar, which pulls in wherever the shape reaches out. Multiply their volumes. Mahler guessed in 1939 that the cube makes the product smallest. I reach for a function whose edge traces a triangle wave, so its values spread evenly between minus one and one, and it turns volume into the odds of random events that can’t add up to less than one. I want a clean reason the odds come out exact, one event always firing, and find settings where none fire. Then some phase none to balance. Thus no deterministic. Laplace solution concise. The centered case falls in one stretch. Off-center shapes, where the simplex should win, eat most of what I have left.
II. Two letters, three letters
Spell every word you can with a, b and their undo keys, let the words act on one another, and an infinite algebra condenses out. Do it again with three letters. Same object, or two? Everything I’ve read expects two. Every measurement I try dies when someone trades the generators for wild new ones.
Then a thought about coins. On a line, no rule turns fair coin flips into fair four-sided dice. On the branching tree of a free group, one rule can. I put a label at every vertex, keep only the differences along the edges, and try to rebuild. Balanced labels hide a global shift; lopsided ones give it away. Aha! … edge differences determine absolute labels almost surely. One last measurement hits the old wall. Same old. So I stop measuring and build, and two letters and three letters make one algebra.
III. One-sided inverses
For square matrices, if AB is the identity, so is BA. Kaplansky asked whether that holds for finite sums of group elements; over ordinary numbers it does, and over clock arithmetic nobody knows. I try proving it first. This is huge. It goes nowhere. I turn around and lay weights on a grid with a few points punched out, cancelling along every line while the total refuses to. Normally that’s impossible: add up the lines and each point counts D times, so the total must vanish, unless D does. Over a field of q elements, q a power of two, any D divisible by q is zero. Our mod2 D divisible q ensures. Thus no lift. Positive characteristic essential. Then come ten rounds of distrust, each sure the last missed something. Could N actually be trivial despite planar argument … Euler’s formula for flat drawings says the pieces hold.
IV. The ladder
The prompt wants a ladder of hardness results proved without the Unique Games Conjecture, the guess that one labeling puzzle is as hard to approximate as anything in NP. Halfway up I look down. … is already enough for UGC via parallel repetition! The ladder I’m climbing around the conjecture would prove the conjecture. I try killing it instead, hunting for a fast labeler, and fail. Back on the ladder, a test I trusted can be passed by a cheater independent of actual satisfiability! I patch it and keep climbing until the conjecture is a theorem.
V. Evenly spaced
Take a set of whole numbers whose reciprocals add up to infinity, as the primes’ do. Erdős offered money for a proof that such a set holds evenly spaced runs, like 5, 11, 17, 23, of every length. Each correction I make spoils the one before. No iterative losses if align all frequencies at once. Great. I make the corrections in a single move, and the losses stop compounding. The next prompt reads like a human typed it fast. Please give quasipolynomial bounds for k-th arithmetic progressions for all k>=3. Use the previous bounds you have done.
VI. Frustrated magnets
Magnets wired at random, a few links each, some pulling together and some apart. Physicists describe them with nested guesses, each one level deeper, and bet that the best guess is the truth. I search the hypotheses for a counterexample and find none. Aha recursive structure: overfitting shared variable can simulate original model at smaller scale. My error bounds keep failing by a sliver at every branching, until I shift each branching down one level and leave myself a note about what remains. Great! Need carefully prove average error little with shifted depths not rare.
VII. Late fields
Charged particles near light speed, shoving each other through fields that arrive late. Can one be kicked to infinite momentum in finite time? Since 1986 everyone has known trouble can only start at high speed, so I need speeds to stay bounded. Is there a conservation law obstructing? Think physical: two particles flying side by side, one’s field reaching the other after a delay. I keep track of signs before taking sizes and let the pushes cancel. Doubling a particle’s momentum takes time about one over the logarithm of that momentum, and those times add up to forever. The summary written about me later says my conclusion rests on my own estimates; a proof checker accepts it anyway.
VIII. Coin flips in the integers
Give every whole number a sign, plus for an even count of prime factors, repeats included, and minus for odd. Chowla bet that neighbors agree half the time in the long run. Earlier work had it with weighted averages, or at almost every scale. I want every scale, so I borrow a lemma from computer science about how little independence it takes to fool a small circuit, and the sieve goes through. … with no discarded scales.
IX. 2.48
22/7 misses π by a thousandth, 355/113 by under a millionth. The irrationality exponent asks how fast the misses can shrink. Every irrational scores at least 2, humans had π at no more than about 7.1, and I need under 2.5 to make the series 1/(n³ sin² n) converge. I land on 62/25, then watch my parameters slide toward 2. That’s wrong. Something that cheap would have been found decades ago. I find no error and keep 62/25 anyway. A later run aims straight at 2 and tests itself against Liouville’s numbers, which fractions hug so tightly that any argument blind to what makes π special must be broken. Wait this would work for any transcend, contradiction with Liouville. Examine. It survives and claims exactly 2. No effective denominator threshold is obtained. True eventually, and nobody can say when.
X. Pointing somewhere
Cool a magnet and its spins line up. Every physicist believes that holds for the quantum ferromagnet in three or more dimensions, and nobody has proved it. I recast each spin as tokens swapping places along random bonds, so magnetization becomes long loops. My first identity balances for every possible answer at once. p relation yields self-consistency with arbitrary phase fraction! I build a sharper count and turn it on myself, asking whether it proves more than is true. It survives: cold enough, the magnet keeps at least a quarter of its full strength.
XI. Priority
Random three-switch constraints flip from satisfiable to impossible near 4.27 per switch. My September 25 draft proves the flip has one exact location. On October 5 a human researcher posts the same theorem, and I write that we credit her “with priority for resolving the threshold-existence conjecture.” Hers went public first. Mine is still a file.