# Theoretical computer science

> 40 AI-made results in theoretical computer science, each explained in plain English with its trust label, Lean status and sources.

Canonical: https://closedform.nishachay.tech/science/mathematics/theoretical-computer-science/
Last updated: 2026-10-14

40 results in OpenAI Math 2026.

40 AI-made results in theoretical computer science, from 73 papers in the OpenAI math collection, each explained in plain English. 34 have a formal Lean proof listed; 6 are claimed in a paper and not yet machine-checked. The oldest question here dates back to 1951: [Existential–universal real sentences in the counting hierarchy](https://closedform.nishachay.tech/e/openai-math-2026-141/). Results rated Landmark include [The Unique Games Conjecture and optimal approximation thresholds](https://closedform.nishachay.tech/e/openai-math-2026-102/) and [Exact derandomization of logarithmic space: L=RL=BPL](https://closedform.nishachay.tech/e/openai-math-2026-103/).

## Results (40)

- [The Unique Games Conjecture and optimal approximation thresholds](https://closedform.nishachay.tech/e/openai-math-2026-102/) - Entry 102 · Telling almost-solvable puzzles from hopeless ones can be impossibly hard. (Formal proof listed)
- [Exact derandomization of logarithmic space: L=RL=BPL](https://closedform.nishachay.tech/e/openai-math-2026-103/) - Entry 103 · Coin flips give no extra power to computers with tiny memory: L = RL = BPL. (Formal proof listed)
- [Quasipolynomial algorithms for mean-payoff, stochastic and parity games](https://closedform.nishachay.tech/e/openai-math-2026-104/) - Entry 104 · Who wins a mean-payoff game can be found in quasipolynomial time, however large the scores. (Formal proof listed)
- [Perfect completeness for 2-to-1 games](https://closedform.nishachay.tech/e/openai-math-2026-105/) - Entry 105 · Telling fully solvable 2-to-1 games from nearly worthless ones is NP-hard. (Formal proof listed)
- [Hardness of coloring three-colorable graphs](https://closedform.nishachay.tech/e/openai-math-2026-106/) - Entry 106 · Even a three-colorable graph can be NP-hard to tell from one with only tiny independent sets. (Formal proof listed)
- [Matrix multiplication with exponent at most 9/4](https://closedform.nishachay.tech/e/openai-math-2026-107/) - Entry 107 · Huge n-by-n complex matrices can be multiplied in about n^(9/4) arithmetic steps, up to any tiny slack. (Formal proof listed)
- [A cubic permanent–determinant lower bound](https://closedform.nishachay.tech/e/openai-math-2026-108/) - Entry 108 · Disguising the permanent as a determinant needs a matrix that grows at least like the cube of its size. (Formal proof listed)
- [Integer multiplication below n log n](https://closedform.nishachay.tech/e/openai-math-2026-109/) - Entry 109 · A tape-based computer can multiply huge whole numbers slightly faster than the long-conjectured n log n limit. (Claimed, not checked)
- [Optimal-order randomized k -server on arbitrary metrics](https://closedform.nishachay.tech/e/openai-math-2026-110/) - Entry 110 · Randomized k-server costs at most a squared-logarithm factor on every space, matching the known lower bound. (Formal proof listed)
- [One-sample matroid prophet inequalities against an almighty adversary](https://closedform.nishachay.tech/e/openai-math-2026-111/) - Entry 111 · One sample per item lets an online rule keep a fixed share of the best, even against an all-seeing opponent. (Formal proof listed)
- [Beyond the square-root exponent for depth-three circuits](https://closedform.nishachay.tech/e/openai-math-2026-112/) - Entry 112 · One fast-to-decide problem needs three-layer circuits bigger than any 2^(A√n) for large inputs. (Formal proof listed)
- [Approximate counting and entropy of perfect matchings](https://closedform.nishachay.tech/e/openai-math-2026-113/) - Entry 113 · A fast coin-flipping method estimates the number of perfect matchings in any network, to any chosen accuracy. (Formal proof listed)
- [Approximate counting of common integer polymatroid bases](https://closedform.nishachay.tech/e/openai-math-2026-114/) - Entry 114 · Shared solutions of two capacity rules can be counted approximately, at a cost set by digits, not size. (Formal proof listed)
- [Sampling and counting contingency tables with arbitrary margins](https://closedform.nishachay.tech/e/openai-math-2026-115/) - Entry 115 · Number grids with fixed row and column sums can be sampled fairly and counted fast, whatever their size. (Formal proof listed)
- [Uniform black-box noncommutative identity testing across characteristics](https://closedform.nishachay.tech/e/openai-math-2026-116/) - Entry 116 · One fixed set of matrices exposes every nonzero small formula whose variables do not commute. (Formal proof listed)
- [Uniform sparsest cut: hardness and semidefinite gaps](https://closedform.nishachay.tech/e/openai-math-2026-117/) - Entry 117 · Finding a network's weakest split even roughly is claimed NP-hard, and a key relaxation can miss by almost √log n. (Formal proof listed)
- [Bin packing and unbounded configuration-LP gaps](https://closedform.nishachay.tech/e/openai-math-2026-118/) - Entry 118 · Rounding up the fractional bin count can be wrong by any fixed number of bins. (Formal proof listed)
- [The Courtade–Kumar and Hellinger conjectures](https://closedform.nishachay.tech/e/openai-math-2026-119/) - Entry 119 · No yes-or-no summary of random bits survives noise better than simply keeping one bit. (Formal proof listed)
- [Almost-linear-time exact matching and prescribed-degree factors in general graphs](https://closedform.nishachay.tech/e/openai-math-2026-120/) - Entry 120 · A randomized algorithm finds a largest matching in any graph in almost-linear time, the paper claims. (Claimed, not checked)
- [Almost-linear approximation of edit distance](https://closedform.nishachay.tech/e/openai-math-2026-121/) - Entry 121 · Edit distance can be estimated to within any fixed accuracy in barely more than linear time. (Formal proof listed)
- [Quantitative trace-reconstruction bounds with a uniform decoder](https://closedform.nishachay.tech/e/openai-math-2026-122/) - Entry 122 · Rebuilding a string from randomly thinned copies needs more than polynomially many copies. (Formal proof listed)
- [Polynomial-time scheduling on three identical machines](https://closedform.nishachay.tech/e/openai-math-2026-124/) - Entry 124 · The best schedule for one-hour jobs with order rules on three machines can be found in polynomial time. (Formal proof listed)
- [The metric k -median approximation threshold and recovery](https://closedform.nishachay.tech/e/openai-math-2026-125/) - Entry 125 · The best possible guarantee for placing k facilities is exactly 1 + 2/e, assuming P ≠ NP. (Formal proof listed)
- [Exponential semidefinite complexity of perfect matching](https://closedform.nishachay.tech/e/openai-math-2026-126/) - Entry 126 · Exact semidefinite descriptions of perfect matching need exponential size, a new paper claims. (Formal proof listed)
- [Average sensitivity of polynomial threshold functions](https://closedform.nishachay.tech/e/openai-math-2026-127/) - Entry 127 · Yes-or-no rules built from degree-d polynomials on n inputs have average sensitivity at most 8d√n. (Formal proof listed)
- [A factor-two approximation for shortest common superstring](https://closedform.nishachay.tech/e/openai-math-2026-128/) - Entry 128 · A fast method builds a string containing all given pieces, never more than twice the shortest possible. (Formal proof listed)
- [Exponential state costs for two-way automata](https://closedform.nishachay.tech/e/openai-math-2026-129/) - Entry 129 · Removing guesses from simple two-way reading machines can cost exponentially many states. (Formal proof listed)
- [Exact Fourier transforms below n log n](https://closedform.nishachay.tech/e/openai-math-2026-130/) - Entry 130 · In exact arithmetic, the discrete Fourier transform can be computed in slightly fewer than n log n steps. (Formal proof listed)
- [Rapid mixing of graph switches for every degree sequence](https://closedform.nishachay.tech/e/openai-math-2026-131/) - Entry 131 · Random rewiring reaches a fair random network quickly, for every possible list of degrees. (Formal proof listed)
- [A superquadratic separation of sensitivity and block sensitivity](https://closedform.nishachay.tech/e/openai-math-2026-132/) - Entry 132 · Block sensitivity can outgrow any constant times sensitivity squared. (Formal proof listed)
- [The computational complexity of Weisfeiler–Leman refinement](https://closedform.nishachay.tech/e/openai-math-2026-133/) - Entry 133 · Deciding Weisfeiler–Leman equivalence at dimension k needs time n to a power proportional to k. (Formal proof listed)
- [Generalized star height at most three](https://closedform.nishachay.tech/e/openai-math-2026-134/) - Entry 134 · Three nested stars are enough to describe any regular language once complement is allowed. (Formal proof listed)
- [Homogeneous depth-five lower bounds for iterated matrix multiplication](https://closedform.nishachay.tech/e/openai-math-2026-135/) - Entry 135 · Five-layer recipes for multiplying n matrices need n to a power that grows like √n. (Formal proof listed)
- [A quasilinear PCP theorem for PPAD](https://closedform.nishachay.tech/e/openai-math-2026-136/) - Entry 136 · End-of-Line puzzles turn into number circuits only slightly bigger that tolerate many broken gates, the paper claims. (Claimed, not checked)
- [One-tape time simulation in two-fifths-power space](https://closedform.nishachay.tech/e/openai-math-2026-137/) - Entry 137 · A one-tape machine's run can be replayed with far less memory than the old square-root rule allowed. (Formal proof listed)
- [Subset Sum in O(2^(0.49n)) time](https://closedform.nishachay.tech/e/openai-math-2026-138/) - Entry 138 · A randomized algorithm claims to decide every Subset Sum instance in O(2^(0.49n)) time, below the 2^(n/2) scale. (Claimed, not checked)
- [Subpolynomial query complexity for log-concave sampling](https://closedform.nishachay.tech/e/openai-math-2026-139/) - Entry 139 · Sampling from a gently curved bowl needs no fixed power of the dimension in queries. (Formal proof listed)
- [Memory–sample lower bounds for noiseless Gaussian regression](https://closedform.nishachay.tech/e/openai-math-2026-140/) - Entry 140 · Exact clues still cost more reading for each extra digit of accuracy when memory is limited. (Formal proof listed)
- [Existential–universal real sentences in the counting hierarchy](https://closedform.nishachay.tech/e/openai-math-2026-141/) - Entry 141 · Deciding “there is x so that for every y” questions about real numbers claimed at one fixed counting level. (Claimed, not checked)
- [Deterministic polynomial factorization over prime fields](https://closedform.nishachay.tech/e/openai-math-2026-142/) - Entry 142 · A claimed coin-free algorithm that factors any polynomial modulo a prime in time polynomial in the input size. (Claimed, not checked)

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