Unsolved Math Problems Today and What AI Can and Cannot Do

Six of the seven Millennium Prize Problems are still open: the Riemann hypothesis, P vs NP, Navier Stokes, Yang Mills and the mass gap, the Hodge conjecture and Birch and Swinnerton-Dyer. Only the Poincaré conjecture is solved, by Grigoriy Perelman. AI systems now score at olympiad gold level and find new constructions on open problems, but none has proved a major open conjecture.

Below we go problem by problem, explain what "solved" means (a checked proof, not a convincing pattern), walk through the Lean proof assistant, and list what AI has and has not done, using only what labs and peer reviewed papers report. At the end we cover where formal reasoning shows up in software security.

One honest note first. Safeguard builds single purpose security models, and our own benchmarks page says a frontier general purpose LLM will out-write any Safeguard model on math word problems. Nothing here claims Safeguard models solve mathematics problems.

Key takeaways

• Clay announced seven Millennium Prize Problems on 24 May 2000, $1 million each. Only the Poincaré conjecture is resolved; Perelman was awarded the prize on 18 March 2010.
• "Solved" means a checked, accepted proof. Clay waits for publication, two years and general acceptance.
• Evidence is not proof. The Riemann hypothesis is verified for all zeros up to height 3 × 10^12 and is still open.
• P vs NP matters for security: Stephen Cook's official description notes that if P = NP, the complexity assumptions behind cryptography would be false.
• Proof assistants such as Lean, Coq, HOL Light and Isabelle check every step by machine. Lean's mathlib held about 291,000 theorems when we checked.
• AI has found new constructions on open problems and reached IMO silver (2024, formal proofs) and gold (2025, natural language) level scores, as reported by Google DeepMind and in Nature.
• No AI system has proved a Millennium problem, the Collatz conjecture, the twin prime conjecture or the strong Goldbach conjecture.
• In security, the same split between "plausible" and "checked" applies to reachability analysis and patch reasoning.

A dated timeline of the problems and the tools

What does "solved" mean in mathematics?

A problem is solved when someone produces a proof, a gap free chain of logic from accepted axioms to the statement, and other experts have tried to break it and accepted it. Simulations and large computer searches do not prove a statement about infinitely many cases.

Clay's rules, adopted on 26 September 2018, make this concrete. Clay accepts no direct submissions. A proposed solution must be published in a qualifying outlet, at least two years must pass, and it must have general acceptance in the mathematics community before Clay considers it.

The Poincaré award shows how this works in practice. Perelman posted his preprints in 2002 and 2003. The prize came in March 2010, after a special advisory committee (Simon Donaldson, David Gabai, Mikhail Gromov, Terence Tao and Andrew Wiles) recommended it, then the Scientific Advisory Board and the Board of Directors agreed, according to Clay's press release. Those seven years are how long it took experts to be sure.

Keep three levels apart:

1. Proved and accepted. A human proof experts have checked, such as the Poincaré conjecture.
2. Formally verified. A proof a proof assistant has checked step by step, such as the Kepler conjecture and BB(5). Someone still has to confirm the formal statement says what the informal one means.
3. Supported by evidence. Huge numbers of checked cases or "almost all" results. The Riemann hypothesis and the Collatz conjecture sit here. They are open.

When a headline says AI "solved" something, ask which level it reached and who checked it.

What are the Millennium Prize Problems?

Clay chose the problems through its founding Scientific Advisory Board, which conferred with experts worldwide, and focused on classic questions that had resisted solution for many years. Atiyah and Tate presented the problems at the Paris meeting. The Riemann hypothesis is the one overlap with David Hilbert's famous list of 23 problems from 1900.

The Poincaré conjecture: the one that fell

Clay explains the idea with a rubber band. Stretch one around an apple and you can shrink it to a point without tearing it or lifting it off the surface. Do the same around a doughnut and you cannot. The apple's surface is "simply connected"; the doughnut's is not. Poincaré knew that simple connectivity characterises the two dimensional sphere and asked in 1904 whether the same holds one dimension up.

Nearly a century later, Perelman answered yes. His proof built on Richard Hamilton's Ricci flow, which evolves the geometry of a space somewhat like heat spreading through metal. Along the way he proved Thurston's geometrization conjecture: every three manifold is assembled from standard pieces, each with one of eight well understood geometries. Simon Donaldson, quoted in Clay's press release, noted the surprising use of partial differential equations and differential geometry.

The lesson for AI and mathematics: the winning idea imported a tool from another field. Nobody has yet seen a machine make that kind of leap on a problem this size.

The Riemann hypothesis

The prime number theorem tells you how primes are distributed on average. The Riemann hypothesis, in Clay's words, tells us about the deviation from that average. Riemann's 1859 paper connected the primes to a function, the zeta function, and the hypothesis asserts that all its "non obvious" zeros are complex numbers with real part exactly 1/2.

If it is true, many conditional results about primes become unconditional. How much evidence is there? A great deal. In April 2020, David Platt and Tim Trudgian posted a rigorous verification, using interval arithmetic so that rounding errors cannot fool the computer, that every zero of the zeta function up to height 3 × 10^12 lies on the critical line. It proves nothing about the infinitely many zeros above that height, so the problem stays open.

P vs NP, and why it matters for cryptography

Clay states the question simply: if it is easy to check that a solution is correct, is it also easy to find one? Its example is housing 100 of 400 students while avoiding a list of incompatible pairs. Checking a proposed list is fast; finding one may take impossibly long. Cook and Levin formulated the question independently in 1971.

"P" is the class of problems solvable in polynomial time; "NP" is the class whose solutions can be checked in polynomial time. Every P problem is in NP. The question is whether the reverse holds.

Stephen Cook's official problem description spells out the security link. Internet security depends on assumptions such as the difficulty of factoring; Cook writes that if P = NP, these assumptions are all false, and that efficient factoring would break RSA. He also notes the upside: an efficient NP solver could find formal proofs of any theorem with a reasonably short proof.

Two nuances that are often lost:

• P ≠ NP would not by itself prove any cipher secure. Cook describes factoring as seemingly neither in P nor NP complete, so it could in principle be easy even if P ≠ NP.
• Post quantum cryptography also rests on assumptions. NIST published FIPS 203, based on module lattices, on 13 August 2024. Lattice hardness is likewise unproved. For the quantum side of that story, see our explainer on what a cryptographically relevant quantum computer is and on whether quantum computing breaks AES-256.

Practical security sits on top of open mathematics. That is a reason to keep cryptographic code updated, inventoried and swappable, because when an assumption weakens, the fix is a software change in your dependency tree.

Navier Stokes

The Navier Stokes equations describe how fluids such as water and air move. Clay's page says that for decades nobody has proved answers to the most basic questions: do solutions exist, and are they unique? Engineers simulate these equations daily, and the simulations work. The open question is whether a smooth starting flow in three dimensions can develop a singularity in finite time. Clay's page also says why proof matters: it gives "not only certitude, but also understanding".

The other three

• Yang Mills and the mass gap. Quantum Yang Mills theory underlies the standard model of particle physics. Clay notes that experiment and computer simulations suggest a "mass gap" in the quantum versions of the Yang Mills equations, but no proof of this property is known.
• The Hodge conjecture asks whether certain topological features of projective algebraic varieties are always built from algebraic pieces.
• Birch and Swinnerton-Dyer predicts how many rational points an elliptic curve has from the behaviour of an associated function at one point. Elliptic curves also underpin public key cryptography, though its security does not hinge on this conjecture.

Famous open problems outside the Millennium list

Several of the most searched problems are not Millennium problems. They are easy to state, which is why amateur "proofs" of them are common.

The Collatz conjecture

Halve even numbers, send odd n to 3n + 1, repeat. The conjecture says you always reach 1. Computers have checked enormous ranges; nobody can prove it.

The strongest partial result is Terence Tao's 2019 paper. Earlier work by Korec had shown that almost all starting numbers eventually dip below a fixed power of themselves (any exponent above about 0.7924). Tao showed something much stronger: for any function that grows to infinity, however slowly, almost all orbits eventually fall below that function's value, measured in logarithmic density. It is still an "almost all" result. A single starting number that escapes to infinity, or a loop that never reaches 1, would still disprove the conjecture, and Tao's theorem does not rule either out.

Twin primes and bounded gaps

Twin primes are pairs like 11 and 13. The conjecture says there are infinitely many. Until 2013 nobody could prove even that some fixed gap recurs infinitely often. Then, in work published in the Annals of Mathematics in 2014, Yitang Zhang proved that there are infinitely many pairs of consecutive primes less than 70,000,000 apart. Months later, James Maynard posted a refined sieve method that brought the bound to 600 with a much simpler argument, and the Polymath8 collaborative project, working in public online, pushed it to 246. Under a stronger unproved hypothesis (the generalised Elliott Halberstam conjecture) the bound drops to 6. To prove the twin prime conjecture you need the bound to be 2. The Polymath paper also adapts Selberg's "parity problem" argument to show that its conditional result is the best that purely sieve theoretic methods can give, so closing the last gap needs a new idea.

Goldbach

In 1742, Goldbach and Euler exchanged letters that led to two conjectures. The strong (binary) version says every even number greater than 2 is the sum of two primes. The weak (ternary) version says every odd number greater than 5 is the sum of three primes. Harald Helfgott's 2013 preprint, built on the circle method and improved estimates on exponential sums, states a proof of the ternary version. The binary version remains open, despite extensive computer verification.

Why is a computer check not a proof?

Here is the trap in its simplest form. This short Python script checks the Collatz conjecture for every starting value below a limit you choose:

def reaches_one(n: int, max_steps: int = 10_000) -> bool:
    steps = 0
    while n != 1 and steps < max_steps:
        n = n // 2 if n % 2 == 0 else 3 * n + 1
        steps += 1
    return n == 1

limit = 1_000_000
failures = [n for n in range(1, limit) if not reaches_one(n)]
print(f"checked 1..{limit - 1}, failures: {failures[:10]}")

Run it with python3 collatz_check.py and it reports no failures. Raise the limit and it still reports none. You have learned something real: there is no counterexample in that range. You have not proved anything about the numbers above it, and there are infinitely many of those.

There is a quieter trap too: max_steps. A number needing more steps than the cap would be reported as a failure. Every computational check carries choices like this, which is why serious verifications, like Platt and Trudgian's, use interval arithmetic and publish their methods.

Computers can prove things, but only when the problem is reduced to finitely many cases and every case is handled rigorously. The four colour theorem and the Kepler conjecture were settled that way, and the Busy Beaver result below is the most recent striking example.

What are proof assistants like Lean, and what is mathlib?

A proof assistant is a program that checks proofs written in a formal language. You state a theorem precisely, write a proof as a sequence of steps or tactics, and a small trusted core (the kernel) checks that each step follows from the rules of logic and earlier results. If the kernel accepts it, the proof has no gaps, assuming the kernel and the axioms are sound and the statement says what you meant.

The Lean project describes Lean as an open source programming language and proof assistant. Mathlib is Lean's community library of formalised mathematics. Its statistics page listed 291,099 theorems, 139,563 definitions and 772 contributors when we fetched it.

Three results show what formal verification has achieved:

• The Kepler conjecture. The Flyspeck project formally proved the Kepler conjecture on dense sphere packings in HOL Light and Isabelle; its 2015 paper is the official account.
• The Liquid Tensor Experiment. Peter Scholze challenged the community to formalise a hard theorem of his and Dustin Clausen's. The Lean community blog reports completion on 14 July 2022, a year and a half later.
• BB(5). The busy beaver number BB(5) is the maximum number of steps a five state, two symbol Turing machine can take before halting, starting from a blank tape. The Busy Beaver Challenge, a large online collaboration, completed the search on 2 July 2024. The accompanying paper reports that S(5) = 47,176,870 was proved in the Coq proof assistant, that the proof enumerates 181,385,789 machines and decides for each whether it halts, and that it is the first busy beaver value ever formally verified.

BB(5) has a lesson for software. The paper notes the busy beaver function was introduced as an example of an uncomputable function: no general algorithm decides whether an arbitrary program halts. Five states could be settled only because the machines are finite in number and each got a checkable argument. We return to this under reachability below.

Step by step: check a proof yourself in Lean

You do not need to be a mathematician to see what "machine checked" means. This walkthrough follows the Lean community's own instructions for creating a project that depends on mathlib.

1. Install Lean. Follow the official install page at leanprover-community.github.io for your operating system. It installs elan, the Lean version manager, usually with the VS Code extension. Confirm it works:

elan --version
lake --version

2. Create a project that depends on mathlib. The community instructions use this command (the pinned lake version only makes sure a recent enough lake runs; it does not fix your project's Lean version):

lake +v4.24.0 new my_project math
cd my_project
lake exe cache get

The math keyword adds mathlib as a dependency. lake exe cache get downloads prebuilt mathlib files so you do not compile the whole library yourself, which would take hours.

3. Write a file. Put this in MyProject/Basic.lean (the generated project already has a file you can edit):

import Mathlib

theorem two_add_two : 2 + 2 = 4 := by norm_num

example (n : ℕ) : ∃ p, n ≤ p ∧ Nat.Prime p := Nat.exists_infinite_primes n

def collatzStep (n : ℕ) : ℕ := if n % 2 = 0 then n / 2 else 3 * n + 1

theorem collatz_open (n : ℕ) (h : 0 < n) : ∃ k, collatzStep^[k] n = 1 := by
  sorry

#print axioms collatz_open

4. Build it.

lake build

What you will see, and what it means:

• two_add_two checks instantly. The norm_num tactic does the arithmetic and the kernel verifies it.
• The example reuses Nat.exists_infinite_primes, mathlib's proof that there are infinitely many primes (for every n there is a prime at least n). You did not prove it; you used a theorem someone else proved and the kernel already checked.
• collatz_open states the Collatz conjecture formally. Lean accepts the file but warns that the declaration uses sorry. sorry is a placeholder that says "trust me here". #print axioms then shows that this theorem depends on sorryAx, which is Lean's way of telling you it is not actually proved.

That habit transfers directly to AI claims. When a system says it produced a Lean proof, ask: does lake build pass with no sorry warnings, does #print axioms show only the standard axioms, and does the formal statement match the informal problem?

What has AI actually achieved in mathematics?

This section reports results only as the labs and peer reviewed papers describe them. We do not rank systems or compare vendors, and we cannot independently verify unpublished internal evaluations.

New constructions: FunSearch and AlphaEvolve

The FunSearch paper in Nature pairs a pretrained LLM with a systematic evaluator in an evolutionary loop. Instead of searching for an answer directly, it searches for programs that generate answers. Applied to the cap set problem, a question in extremal combinatorics about how large a set can be without containing three points in a line, the authors report constructions of large cap sets beyond the best previously known, in both finite dimensional and asymptotic cases. The paper notes that LLMs can produce plausible but incorrect statements, which is why every candidate is scored by an evaluator rather than trusted.

AlphaEvolve, announced by Google DeepMind on 14 May 2025, extends this approach to a coding agent. Google DeepMind says it applied the system to over 50 open problems in analysis, geometry, combinatorics and number theory. In roughly 75 percent of cases it rediscovered the best known solutions and in about 20 percent it improved on them. Two examples the lab gives: a configuration of 593 outer spheres that sets a new lower bound for the kissing number in 11 dimensions, and an algorithm that multiplies 4 × 4 complex valued matrices with 48 scalar multiplications, improving on Strassen's 1969 algorithm in that setting.

These are real contributions. Notice their shape: an improved construction is something a computer can verify directly. These systems are strongest where the answer is an object you can test. They do not prove a bound is optimal or resolve a conjecture.

Olympiad problems: AlphaGeometry, AlphaProof and Gemini Deep Think

The IMO gives students six problems over two 4.5 hour sessions, seven points each.

AlphaGeometry, in Nature in January 2024, tackles olympiad geometry. Its authors note that translating human proofs into machine verifiable form is expensive, especially in geometry, so they synthesised millions of theorems and proofs and combined a neural language model with a symbolic deduction engine.

At IMO 2024, according to Google DeepMind, AlphaProof and AlphaGeometry 2 solved four of the six problems for 28 points, the top of the silver medal range, with the gold threshold at 29. AlphaProof pairs a pretrained language model with the AlphaZero reinforcement learning algorithm and searches for proofs in Lean. Three facts from the announcement matter:

• The problems were manually translated into formal language first.
• One problem was solved within minutes; the others took up to three days, compared with the 9 hours students get.
• The two combinatorics problems were not solved.

At IMO 2025, Google DeepMind reports that an advanced version of Gemini Deep Think solved five of six problems for 35 points, a gold medal score, and that its results were officially graded and certified by IMO coordinators using the same criteria as for students. The lab highlights the change from the year before: the model worked end to end in natural language, from the official problem statements, within the 4.5 hour limit.

Two observations. The 2024 Lean proofs were checked by a kernel; the 2025 natural language proofs were checked by expert graders. Both are legitimate, but different. And olympiad problems are designed to be solvable in hours with known answers. Research problems have no known answer and no grader.

Other labs also publish material on their models' mathematical reasoning. We have limited this post to results we could fetch from a lab's own page or a peer reviewed paper while writing it.

Research level benchmarks

To measure research level ability, a group of mathematicians built FrontierMath, a benchmark of hundreds of original, unpublished problems. The paper says a typical problem takes a specialist researcher several hours, and the hardest take days, with answers verified automatically to limit data contamination. When the paper was first posted in November 2024, it reported that the best AI models of the time solved under 2 percent. Benchmark scores move quickly, so treat that as a late 2024 snapshot. The durable point is the design: unpublished problems with automatically checkable answers.

What can AI not do yet in mathematics?

As of October 2026, based on everything we could source:

• No AI system has proved a Millennium problem, or the Collatz, twin prime or strong Goldbach conjectures. Clay's prize page still lists Poincaré as the only solved problem.
• Olympiad skill does not transfer automatically to research. Open problems often need new theories, as Perelman's use of Ricci flow did.
• Natural language proofs need expert checking. A fluent argument can hide one wrong step, which is why IMO coordinators graded the 2025 results and Clay waits two years.
• Formalisation is its own bottleneck. If anyone formalises the wrong statement, the kernel will verify a proof of the wrong thing. Humans translated the 2024 IMO problems for this reason.
• Search finds objects better than explanations. A proof that a bound is optimal has no score to climb.

None of this predicts AI will never help settle a major problem. It already helps mathematicians search, compute and formalise. Today AI is a strong tool inside a process whose final authority is still checked proof.

Where does formal reasoning matter in software security?

This is the part of the post that connects to what Safeguard does, and it is deliberately limited.

Reachability is a halting style problem

A software composition analysis tool finds that your application includes a library version with a known vulnerability. The useful question is whether your code can actually reach the vulnerable function. In full generality, that question is undecidable for the same reason the busy beaver function is uncomputable: deciding whether an arbitrary program reaches a given line is at least as hard as deciding whether it halts. We wrote a separate explainer on whether reachability analysis can ever be perfect.

So every reachability tool approximates. Good ones are explicit about which way they approximate (towards over reporting or under reporting), use call graphs and configuration to narrow the search, and give you the evidence path so a human can check it. If you want the vocabulary straight, see affected, reachable, exploitable and exploited.

Where Safeguard fits, honestly

Safeguard's models are single purpose security models. Our public benchmarks page states that a frontier general purpose LLM will out-write any Safeguard model on math word problems, and it does not yet publish results for Griffin at its current size. We are not claiming mathematical ability, and nothing here suggests Safeguard models can attack the problems in this post.

What Safeguard does use reasoning models for is narrower and practical:

• Reachability analysis (GA). Safeguard's Griffin models reason over call graphs, with adversarial attempts to disprove a finding, so the vulnerable paths your application actually calls rank first. Like every reachability method, this is an engineering approximation, not a mathematical proof, and it is evidence for a human to review.
• Patch reasoning and autonomous remediation (GA). Griffin authors fixes and opens tested pull requests that move a dependency to a fixed version. The "tested" part is the important part, and it is the security equivalent of the checker in FunSearch: a candidate change is only useful once your build and test suite accept it. Remediation ships switched off by default.
• Human approval. The Safeguard CLI's /permissions setting includes modes that ask before each command or edit, so a human can stay the final reviewer, much like a referee.

The common thread is the discipline, not the subject: separate what looks right from what has been checked. You can ask Safeguard whether a finding is reachable from Claude through the MCP server and read the evidence path yourself, or read how the Safeguard CLI works as a security aware coding assistant.

Common mistakes

These are the misconceptions we see most often when people talk about unsolved problems and AI.

• "It has been checked up to a huge number, so it is basically proved." Evidence over a finite range says nothing certain about infinitely many cases. The Riemann hypothesis is verified to height 3 × 10^12 and remains open.
• "An almost all result is the same as a proof." Tao's Collatz theorem is a major advance, but it allows for a sparse set of exceptions, and one exception would disprove the conjecture.
• "A preprint means it is solved." Clay requires publication in a qualifying outlet, two years and general acceptance. Many announced proofs of famous problems have later been found to contain gaps.
• "AI won an olympiad gold, so it can solve research problems." Olympiad problems have known answers and fit in hours. Research problems have neither property.
• "Lean checked it, so the problem is solved." Only if the formal statement matches the real problem and the proof contains no sorry and no extra axioms. Check #print axioms.
• "Proving P ≠ NP would prove our encryption secure." It would not. Practical schemes rest on specific assumptions such as the hardness of factoring or lattice problems, which are separate questions.
• "Our security model reasons, so it can do maths." Domain models are trained for their domain. Safeguard says plainly that general frontier models beat its models on math word problems.

FAQ

What are the unsolved math problems today?

The best known list is the six open Millennium Prize Problems: the Riemann hypothesis, P vs NP, Navier Stokes existence and smoothness, Yang Mills and the mass gap, the Hodge conjecture and the Birch and Swinnerton-Dyer conjecture. Famous problems outside that list include the Collatz conjecture, the twin prime conjecture and the strong Goldbach conjecture.

How many Millennium Prize Problems have been solved?

One. The Clay Mathematics Institute awarded the prize for the Poincaré conjecture to Grigoriy Perelman on 18 March 2010, based on preprints he posted in 2002 and 2003.

Has AI solved any Millennium Prize Problem?

No. AI systems have reached olympiad gold level scores and improved constructions on some open problems, according to the labs and papers that report them, but none has proved any Millennium problem or any of the famous conjectures in this post.

Why does P vs NP matter for cybersecurity?

Most practical cryptography depends on problems believed to be hard, such as factoring. Cook's official Clay description notes that if P = NP, those complexity assumptions would all be false. A proof that P ≠ NP, however, would not on its own prove any particular cipher secure.

What is Lean and what is mathlib?

Lean is an open source programming language and proof assistant that checks proofs by machine. Mathlib is its community library of formalised mathematics, with roughly 291,000 theorems and 140,000 definitions on its statistics page when we checked.

Is the Collatz conjecture solved?

No. Terence Tao proved in 2019 that almost all Collatz orbits attain almost bounded values, in logarithmic density, which is the strongest known partial result. A single counterexample would still disprove the conjecture.

What did AI score at the International Mathematical Olympiad?

According to Google DeepMind, AlphaProof and AlphaGeometry 2 scored 28 of 42 at IMO 2024 (silver level), and an advanced version of Gemini Deep Think scored 35 of 42 at IMO 2025 (gold level), with the 2025 results graded and certified by IMO coordinators.

Can Safeguard models solve math problems?

No, and we do not claim they can. Safeguard's models are single purpose security models, and our benchmarks page says a frontier general purpose LLM will out-write any Safeguard model on math word problems. They are built for reachability analysis, patch reasoning and related security work.

Can reachability analysis ever be perfect?

Not in general. Deciding whether arbitrary code reaches a given function is undecidable, like the halting problem behind the busy beaver numbers. Practical tools approximate and should show their evidence so a person can check it.

Fix it without waiting for upstream

Open problems in mathematics are not something any software tool fixes. But the practical edge of this topic is the cryptographic and numeric code your applications already ship: libraries that implement RSA, elliptic curves and the new lattice schemes, plus the big scientific and machine learning stacks people use to explore mathematics, all of which pull long chains of open source dependencies that need patching like anything else.

Whatever the vulnerability, most of the risk sits in two gaps. The first is the time between a fix being published and that fix actually running in your production builds. The second, and worse, is the time before any fix exists at all, while a maintainer works out a patch and a release schedule. Waiting through both gaps is a habit, not a requirement. Safeguard gives teams three ways to close them.

1. Autonomous remediation: pull requests, not tickets

When a fixed version exists, Safeguard's Griffin models open a tested, policy gated pull request that moves the affected dependency to the safe version. You can run it on one repository or in bulk across every project that carries the vulnerable version. Remediation covers eight categories, including dependencies, licenses, security posture and findings. Reachability analysis ranks the vulnerable code paths your application actually calls first, so the first pull requests you merge are the ones that matter most. Autonomous remediation ships switched off, with every category set to Manual, so your team decides how much runs on its own and what still needs a human approval.

2. Gold Open Source: the safe version in seconds, free

gold.safeguard.sh is free and needs no login. Search any CVE or package to see affected and fixed versions, CVSS, EPSS, CISA KEV status and a recommended safe version, or use the "Am I affected?" checker. The same data is available through a free read API, README status badges and RSS threat feeds. The safeguard-sh/gold-scan-action GitHub Action can fail a build on a known exploited vulnerability, on any known vulnerability, or never, so a version you have just removed cannot quietly come back in next week's merge.

3. Gold Registry: start from components that are already fixed

registry.safeguard.sh provides hardened, drop in replacements for upstream packages and container images. Every Gold artifact is held continuously to zero exploitable critical or high CVEs, zero malware and a completed license check, with SLSA L3 or higher provenance, a Sigstore signature and a public evidence page you can hand to an auditor. An artifact loses its Gold badge if it regresses. Teams can point their own package directory at the registry so hardened equivalents resolve without changing how developers install anything.

When upstream has no fix yet, or will not ship one, Safeguard first works to get the fix accepted upstream. If the issue is severe or upstream will not move, Safeguard builds a fork, fixes the vulnerability there and publishes a hardened Gold build, so you are not stuck on a maintainer's release schedule. You can also request a hardened, compatibility tested build of a specific package. This fork and patch flow is in early access today, so ask which packages it covers before you plan around it.

One honest limit: when the vulnerable code lives in a vendor appliance, a hosted service or an operating system runtime you do not build yourself, the fix still has to come from that vendor or distribution. The steps above apply to the open source dependencies and container images you build and ship, which is where most teams carry the largest and least tracked share of their exposure.

To try it on your own machine, download Safeguard from https://safeguard.sh/download and see the CLI documentation at https://docs.safeguard.sh/docs/cli .

See how Safeguard would help

The first demo connects Claude to the Safeguard MCP server with OAuth and lets you choose which tools Claude can use. It is the quickest way to ask a general assistant about your own findings, such as which projects carry a given library version, while keeping control over what it can touch.

Click through it at your own pace, or open it full screen.

The second demo is a real Safeguard CLI session: asked to fix a lodash CVE, it reads the finding and advisory, asks before each command, edits package.json to lodash 4.17.21, runs npm install and npm test with approval, and pushes nothing. That approve each step loop is the security version of the theme above: a candidate fix counts only once the tests accept it.

Click through it at your own pace, or open it full screen.

The third demo sets up Safeguard's local runner on your own machine, approves it with a device code, and runs a scan from the terminal. On a research workstation with a large Python or Lean toolchain, the same flow lists which open source packages are installed and which carry known vulnerabilities.

Click through it at your own pace, or open it full screen.

Quick check

Sources

• Clay Mathematics Institute: The Millennium Prize Problems
• Clay Mathematics Institute: Rules for the Millennium Prizes
• Clay Mathematics Institute: Poincaré conjecture
• Clay Mathematics Institute: Press release, 18 March 2010, Millennium Prize awarded to Grigoriy Perelman (PDF)
• Clay Mathematics Institute: Riemann hypothesis
• Clay Mathematics Institute: P vs NP
• Stephen Cook: The P versus NP problem, official Clay problem description (PDF)
• Clay Mathematics Institute: Navier Stokes equation
• Platt and Trudgian: The Riemann hypothesis is true up to 3·10^12, arXiv 2004.09765
• Tao: Almost all orbits of the Collatz map attain almost bounded values, arXiv 1909.03562
• Zhang: Bounded gaps between primes, Annals of Mathematics 179 (2014)
• Maynard: Small gaps between primes, arXiv 1311.4600
• Polymath: Variants of the Selberg sieve, and bounded intervals containing many primes, arXiv 1407.4897
• Helfgott: The ternary Goldbach conjecture is true, arXiv 1312.7748
• NIST: FIPS 203, Module Lattice Based Key Encapsulation Mechanism Standard
• Lean programming language and proof assistant
• Lean community: Mathlib statistics
• Lean community: A mathlib overview
• Lean community: Creating a Lean project
• Lean community blog: Completion of the Liquid Tensor Experiment
• Hales et al.: A formal proof of the Kepler conjecture, arXiv 1501.02155
• Busy Beaver Challenge: Story
• The bbchallenge Collaboration: Determination of the fifth Busy Beaver value, arXiv 2509.12337
• Romera-Paredes et al.: Mathematical discoveries from program search with large language models, Nature (2023)
• Trinh et al.: Solving olympiad geometry without human demonstrations, Nature (2024)
• Google DeepMind: AI achieves silver medal standard solving International Mathematical Olympiad problems
• Google DeepMind: Advanced version of Gemini with Deep Think officially achieves gold medal standard at the IMO
• Google DeepMind: AlphaEvolve, a Gemini powered coding agent for designing advanced algorithms
• Glazer et al.: FrontierMath, a benchmark for evaluating advanced mathematical reasoning in AI, arXiv 2411.04872
• Safeguard: Model benchmarks

Last reviewed: 23 October 2026.