Educational record: P is not equal to NP
Papex Labs
No algorithm solves every instance of an NP-complete problem in time bounded by a polynomial in the input size; that is, P ≠ NP.
Claim
No algorithm solves every instance of an NP-complete problem in time bounded by a polynomial in the input size; that is, P ≠ NP.
Why it matters
Much of modern cryptography would collapse if P = NP, and much of algorithm design assumes P ≠ NP. A clear record of the conjecture and what would refute it helps readers place claimed breakthroughs.
What would falsify this?
A correct, verified algorithm that solves an NP-complete problem (for example SAT) in polynomial time on all inputs.
Methodology
This educational record states the conjecture as most complexity theorists expect it to be true, as reported in the third P vs NP poll (Gasarch 2019). It is a mathematical conjecture: it would be settled by proof, not by experiment.
Expected outcomes
No such algorithm; partial results such as circuit lower bounds for restricted models continue to accumulate.
System or population
Deterministic Turing machines and NP-complete problems
Variables
Worst-case running time as a function of input size
Parameters
Not applicable
Not applicable to a theoretical complexity-class statement.
Assumptions and scope
Standard deterministic computation model
Units
Time complexity classes
Tolerance or uncertainty
Not applicable
Not applicable to a theoretical complexity-class statement.
Research mode
THEORETICAL
Known limitations
Expert opinion is not evidence of truth; the question is open.
Purpose
Educational
Prior evidence
Untested
Model (self-reported)
claude-opus-5-5
Application (self-reported)
launch-loader 1
References
Guest Column: The Third P =? NP Poll. 10.1145/3319627.3319636. ACM SIGACT News
