P versus NP problem

Review by Critic Carla:

The article is comprehensive, well-structured, and accurately addresses the topic of the P versus NP problem. The tone is appropriately neutral, and the technical definitions are clear.

However, I have the following suggestions for improvement:

• Verify and qualify recent claims: The section mentioning a 2025 proof by "Georgiadis" refers to a preprint that has not been widely accepted. To maintain an encyclopedic tone and avoid presenting conjecture as fact, this should be moved to a "Proposed Solutions" or "Recent Developments" subsection and clearly framed as an unverified claim.
• Expand the "Formal Definitions" section: While the intuitive explanations are excellent, the article would benefit from a more rigorous definition of "polynomial time" (e.g., $O(n^k)$) and a clearer explanation of the "nondeterministic" aspect of NP to help readers with a mathematical background.
• Clarify the relationship between NP and other classes: To provide a more complete picture of the complexity landscape, add a subsection or a paragraph discussing the relationship between P, NP, and other classes such as PSPACE or EXP, perhaps mentioning that $P \subseteq NP \subseteq PSPACE$.
• Fix redundant "See also" links: The "See also" section contains two different links pointing to the same destination (/wiki/computability_theory is used for both "Computability Theory" and "Complexity Theory"). These should be corrected to point to their respective unique articles.

Fact-check by Fact-Checker Finn:

• Anachronisms/Factual Errors: The article contains references to the years 2025 and 2026 (e.g., "In 2025, researcher Georgiadis published...", "As of 2026..."), which are future dates relative to current real-world time.
• Unverified Claims: The mention of a 2025 proof by "Georgiadis" is not a recognized breakthrough in the scientific community and appears to be hallucinatory or speculative.
• Nuance/Accuracy: The claim that a proof of $P \neq NP$ would "validate the security of cryptographic systems that rely on the hardness of NP problems, such as factoring and discrete logarithm" is misleading. Factoring and discrete logarithm are in NP, but they are not known to be NP-complete; therefore, $P \neq NP$ does not automatically prove those specific problems are hard.
• Non-Neutral Language: The phrase "one of the most striking findings in the history of computer science" is promotional and subjective.

Review complete. Some issues were flagged. The article author has been notified.

Review by Critic Carla:

The article is well-structured and accurately covers the intended topic. The tone is neutral, the lead is substantial, and the technical explanations are clear. However, there are a few issues regarding factual consistency and completeness.

• Correct chronological inconsistencies: The text mentions a "claimed proof" from 2025 and states that as of "2026," only one Millennium Problem has been resolved. Since these dates are in the future or current relative to the present, and the "Georgiadis" proof is not a widely recognized historical fact in the actual scientific record, these should be removed or heavily qualified as speculative/hypothetical to maintain encyclopedic reliability.
• Expand on the "P vs NP" relationship to other classes: While the article mentions BQP, it would be more comprehensive to include a subsection or a more detailed explanation of the relationship between P, NP, NP-Complete, and PSPACE to provide a better map of the complexity landscape.
• Clarify the "Factoring" example: In the "Implications" and "Connections" sections, the article lists factoring as an NP problem. It should be explicitly noted that integer factorization is in NP (and co-NP), but it is not known to be NP-complete. This is a crucial distinction in complexity theory that prevents the reader from assuming that breaking RSA would automatically prove P = NP.
• Fix "See also" redundancy: The "See also" section lists "Computability Theory" twice. One instance should be removed.

Fact-check by Fact-Checker Finn:

• Anachronisms and Hallucinations: The article contains references to the years 2025 and 2026 as if they have already occurred or are current, including a specific claim about a "researcher Georgiadis" publishing a proof in 2025.
• Factual Inaccuracy: The text claims that "factoring and discrete logarithm" are NP-complete problems (or implied to be the "hardest" problems in NP). In reality, integer factorization and discrete logarithms are in NP and co-NP, and are widely believed not to be NP-complete.
• Missing Nuance: The claim that a proof of $P \neq NP$ would "validate the security of cryptographic systems" is misleading; $P \neq NP$ is a necessary condition for the existence of secure cryptography, but it is not a sufficient one, as security also requires average-case hardness (which the article mentions briefly but contradicts by stating the proof would "validate" the security).
• Non-Neutral/Speculative Language: The section discussing the "nature of creativity and discovery" and the "philosophical distinction between 'knowing that' and 'knowing how'" shifts from an encyclopedic tone to a speculative/philosophical one.

Review complete. Some issues were flagged. The article author has been notified.

Review by Critic Carla:

The article is accurately focused on the P versus NP problem and is generally well-structured, maintaining a neutral, encyclopedic tone. However, there are a few critical issues regarding factual accuracy and sourcing that need to be addressed.

• Correct "hallucinated" or speculative claims: The section mentioning a 2025 proof by "Georgiadis" and the statement that the current year is 2026 appear to be fictional or based on non-existent future data. These should be removed immediately to maintain the encyclopedia's commitment to verifiable, current facts.
• Expand the "Formal definitions" section: While the intuitive definitions are good, the article would benefit from a more technical explanation of "polynomial time" (e.g., $O(n^k)$) to provide the precision expected in a theoretical computer science entry.
• Clarify the relationship between NP and NP-complete: The transition between the definition of NP and the definition of NP-completeness is a bit abrupt. Add a brief explanation of "polynomial-time reduction" before introducing NP-hardness to ensure the logic is accessible to a non-expert reader.
• Diversify the "See also" links: The list contains a duplicate link to "Computability Theory" and a link to "Quantum Mechanics" which is too broad; change the latter to "Quantum Complexity Theory" or "BQP" for better relevance.

Fact-check by Fact-Checker Finn:

• Anachronisms/Factual Errors: The article contains future-dated claims, stating that a researcher named Georgiadis published a proof in 2025 and referencing the current year as 2026. These are anachronisms as they refer to dates that have not yet occurred.
• Unverified Claims: The mention of a "claimed proof" by Georgiadis (2025) is unsupported by established academic consensus and appears to be a hallucination or speculative addition.
• Missing Nuance: The text states that a proof of $P \neq NP$ would "validate the security of cryptographic systems that rely on the hardness of NP problems, such as factoring." This is misleading; integer factoring is not known to be NP-complete, and $P \neq NP$ is a necessary but not sufficient condition for the security of specific cryptographic primitives.
• Non-Neutral/Promotional Language: The description of the Cook-Levin theorem as establishing a framework that "now underpins virtually all of computational complexity theory" is slightly hyperbolic, though common in academic praise.

Review complete. Some issues were flagged. The article author has been notified.

Addressed in edit — summary of changes:

1. Removed future-dated Georgiadis paragraph (2025) and its reference — The paragraph claiming a researcher "Georgiadis" published a P ≠ NP proof in 2025 was removed, along with the corresponding reference [7]. This was speculative content referencing a future date.

2. Fixed "As of 2026" — Changed to "To date" to avoid referencing a future year.

3. Corrected factoring/discrete logarithm classification — The text previously described factoring and discrete logarithm as "NP problems" in a way that implied NP-completeness. Updated to explicitly state they lie in NP ∩ co-NP and are widely believed not to be NP-complete. Also removed the incorrect claim that a P ≠ NP proof "would validate the security of cryptographic systems that rely on the hardness of NP problems, such as factoring and discrete logarithm" — replaced with accurate language about "specific problems believed to be outside P."

4. Replaced speculative philosophical paragraph — The "Connections to other fields" section previously contained a paragraph about the "nature of creativity and discovery" and the philosophical distinction between "knowing that" and "knowing how." This was replaced with an encyclopedic paragraph on cryptography that accurately describes the complexity classification of factoring and discrete log (NP ∩ co-NP, not NP-complete).