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Proof that is not easily verified by hand
In the philosophy of mathematics, a non-surveyable proof is a mathematical proof that is considered infeasible for a human mathematician to verify and
Non-surveyable_proof
French computer scientist and mathematician
solutions for mathematical problems, especially for those that have a non-surveyable proof, such as the four color theorem. It has also been used in software
Thierry_Coquand
Cryptographic primitive
Non-interactive zero-knowledge proofs are cryptographic primitives, where information between a prover and a verifier can be authenticated by the prover
Non-interactive zero-knowledge proof
Non-interactive_zero-knowledge_proof
mathematical proofs. Such proofs often use computational proof methods and may be considered non-surveyable. As of 2011[update], the longest mathematical proof, measured
List of long mathematical proofs
List_of_long_mathematical_proofs
Mathematical proof at least partially generated by computer
verified for correctness. Since these proofs are generally human-surveyable (albeit with difficulty, as with the proof of the Robbins conjecture) they do
Computer-assisted_proof
Subfield of automated reasoning and mathematical logic
program's calculation (such proofs are called non-surveyable proofs). Another example of a program-assisted proof is the one that shows that the game of Connect
Automated_theorem_proving
Proving validity without revealing other data
In cryptography, a zero-knowledge proof (also known as a ZK proof or ZKP) is a protocol in which one party (the prover) can convince another party (the
Zero-knowledge_proof
Relation between sides of a right triangle
most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years
Pythagorean_theorem
Proof assistant
Cambridge, England and Benjamin Werner of INRIA used Rocq to create a surveyable proof of the four color theorem, which was completed in 2002. Their work
Rocq
Unique digital identifier that is recorded on a blockchain
Proponents claim that NFTs provide a public certificate of authenticity or proof of ownership, but the legal rights conveyed by an NFT can be uncertain.
Non-fungible_token
System that regulates the formation of blocks on a blockchain
Proof-of-stake (PoS) protocols are a class of consensus mechanisms for blockchains that work by selecting validators in proportion to their quantity of
Proof_of_stake
First article on transfinite set theory
constructive and non-constructive proofs have been presented as "Cantor's proof." The popularity of presenting a non-constructive proof has led to a misconception
Cantor's first set theory article
Cantor's_first_set_theory_article
Planar maps require at most four colors
this proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand. The proof has gained
Four_color_theorem
Decentralized digital cryptocurrency
based on proof of work, called mining, which is performed by purpose-built computers. Mining consumes large quantities of electricity, with surveyed miners
Bitcoin
Type of consensus algorithm
Proof of space (PoS) is a type of consensus algorithm achieved by demonstrating one's legitimate interest in a service (such as sending an email) by allocating
Proof_of_space
Canadian public opinion research companies
companies - Environics Research, the Environics Institute for Survey Research, and Proof - that provide public opinion research and strategic communications
Environics
Blockchain algorithm
(compared to proof-of-work and proof-of-stake). The most notable platforms using PoA are VeChain, Bitgert, Palm Network and Xodex. Proof-of-authority
Proof_of_authority
have made increasing progress in generating mathematical proofs at research level. Most such proofs have been conducted using models from OpenAI and Anthropic
List of mathematical discoveries by artificial intelligence
List_of_mathematical_discoveries_by_artificial_intelligence
Philosophical question
is insufficient proof that God exists. The following arguments deduce (some using science), mostly through self-contradiction, the non-existence of God
Existence_of_God
Theorem classifying finite simple groups
group of Lie type, in which case there would be 27 sporadic groups). The proof consists of tens of thousands of pages in several hundred journal articles
Classification of finite simple groups
Classification_of_finite_simple_groups
Mathematical logic concept
Gentzen's consistency proof is a result of proof theory in mathematical logic, published by Gerhard Gentzen in 1936. It shows that the Peano axioms of
Gentzen's_consistency_proof
Open-source blockchain computing platform
September 2022, Ethereum transitioned its consensus mechanism from proof-of-work (PoW) to proof-of-stake (PoS) in an update known as "The Merge", which cut the
Ethereum
Infinitely many prime numbers exist
Euclid in his work Elements. There are at least 200 proofs of the theorem. Euclid offered a proof in his work Elements (Book IX, Proposition 20), which
Euclid's_theorem
Relationship between programs and proofs
language theory and proof theory, the Curry–Howard correspondence is a direct relationship between computer programs and mathematical proofs. It is also known
Curry–Howard_correspondence
Kind of proof calculus
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to
Natural_deduction
Basic framework of mathematics
self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particular. This may also include the philosophical
Foundations_of_mathematics
Distributed data store for digital transactions
Papakyriakopoulos, Georgios; Patsakis, Constantinos (2019). "A Survey on Long-Range Attacks for Proof of Stake Protocols". IEEE Access. 7: 28712–28725. Bibcode:2019IEEEA
Blockchain
Digital asset using a distributed ledger
of coin ownership. The two most common consensus mechanisms are proof of work and proof of stake. Despite the name, which has come to describe many of
Cryptocurrency
Theorem in linear algebra
matrices. Given a non-negative A, assume there exists m, such that Am is positive, then Am+1, Am+2, Am+3,... are all positive. (Proof: Am+1 = AAm, so it
Perron–Frobenius_theorem
Three dimensional analogue of uniformization conjecture
Thurston announced a proof in the 1980s, and since then, several complete proofs have appeared in print. Grigori Perelman announced a proof of the full geometrization
Geometrization_conjecture
Classification theorem in group theory
of every non-trivial element is nilpotent. They showed that every CN group of odd order is solvable. Their proof is similar to Suzuki's proof. It was about
Feit–Thompson_theorem
System of dividing land in the United States
close to, the survey line. The purpose was to help retrace a surveyed line should that become necessary. It was also additional proof that the line had
Public_Land_Survey_System
Thesis on the nature of computability
Survey. New York: Wien Springer–Verlag. pp. 51ff. ISBN 978-3-211-82637-9. Gödel, Kurt (1936). "Über die Lāange von Beweisen" [On The Length of Proofs]
Church–Turing_thesis
Field of knowledge
shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas
Mathematics
Russian mathematician (born 1966)
22 December 2006, the scientific journal Science recognized Perelman's proof of the Poincaré conjecture as the scientific "Breakthrough of the Year"
Grigori_Perelman
Study of computable functions and Turing degrees
computability and definability. In these areas, computability theory overlaps with proof theory and effective descriptive set theory. Basic questions addressed by
Computability_theory
Multi-chain non-custodial cryptocurrency wallet
through third-party on-ramp partners such as MoonPay. Users can also stake proof-of-stake assets directly within the app to earn rewards, and store and transfer
Trust_Wallet
Computation model defining an abstract machine
+ ... +in completely determines the proof. The automatic machine carries out successively proof 1, proof 2, proof 3, ..." This is indeed the technique
Turing_machine
Existence of a line through two points
lines determined over every set of n {\displaystyle n} non-collinear points. Melchior's proof showed that t 2 ( n ) ≥ 3 {\displaystyle t_{2}(n)\geq 3}
Sylvester–Gallai_theorem
Curves of genus > 1 over the rationals have only finitely many rational points
1922 by Louis Mordell, and known as the Mordell conjecture until its 1983 proof by Gerd Faltings. The conjecture was later generalized by replacing Q {\displaystyle
Faltings'_theorem
Type of moisture control in building construction
Damp proofing in construction is a type of moisture control applied to building walls and floors to prevent moisture from passing into the interior spaces
Damp_proofing
2025 American film by Curry Barker
would "definitely recommend" it. Christian Zilko of IndieWire said it was "proof that the Cregger-ification of 2020s horror is in full effect, as its combination
Obsession_(2025_film)
Data structure for storing non-overlapping sets
disjoint-set forest data structure and formalized its correctness using the proof assistant Rocq (then: Coq). "Semi-persistent" means that previous versions
Disjoint-set_data_structure
Measure of algorithmic complexity
formal proofs in S by some procedure def nth_proof(n: int) which takes as input n and outputs some proof. This function enumerates all proofs. Some of
Kolmogorov_complexity
Subfield of mathematics
of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory)
Mathematical_logic
Generalization of Pythagorean theorem
through fragmentary quotations) commentary, Heron of Alexandria provided proofs of the converses of both II.12 and II.13. Using notation as in Fig. 2, Euclid's
Law_of_cosines
Percentage of people who answer a survey
Wiley. https://pprg.stanford.edu/wp-content/uploads/2007-TSMII-chapter-proof.pdf Seon Choung, Rok; Richard Locke, III; Schleck, Cathy D.; Ziegenfuss
Response_rate_(survey)
American computer scientist
accept new solutions, they couldn't start on the next problem. (see also: proof-of-work system). Although Szabo has repeatedly denied it, people have speculated
Nick_Szabo
Formal logic whose entailment relation is not monotonic
than attempting to remove it. Proof-theoretic formalization of a non-monotonic logic begins with adoption of certain non-monotonic rules of inference,
Non-monotonic_logic
Boolean satisfiability is NP-complete and therefore that NP-complete problems exist
machine and solvable in polynomial time by a non-deterministic Turing machine are equivalent, and the proof can be found in many textbooks, for example
Cook–Levin_theorem
Model of (first-order) Peano arithmetic that contains non-standard numbers
In mathematical logic, a non-standard model of arithmetic is a model of first-order Peano arithmetic that contains non-standard numbers. The term standard
Non-standard model of arithmetic
Non-standard_model_of_arithmetic
Blockchain and cryptocurrency
Algorand is a proof-of-stake blockchain and cryptocurrency. Algorand's native cryptocurrency is called ALGO. Algorand was founded in 2017 by Silvio Micali
Algorand
Document used to identify a person
them at all times as proof of identity, given the security situation in the country.[citation needed] NICs are not issued to non-citizens, who are still
Identity_document
Rules governing the functioning of cryptocurrency
technology, a public ledger that records all bitcoin transactions; mining and proof of work, the process to create new bitcoins and verify transactions; and
Bitcoin_protocol
Diving inside a protective cage to observe sharks in the wild
potentially dangerous shark species are present. In this application, the shark-proof cage may be used as a refuge, or as a diving stage during descent and ascent
Shark_cage_diving
Rejection of the Christian doctrine of the Trinity
Father were one entity either. While Trinitarians often use John 10:33 as proof for the divinity of Jesus, unitarian critics argue that the Pharisees accusing
Nontrinitarianism
Senate and signed into law by Trump, would have required voters to provide proof of citizenship in order to obtain or renew their voter registration. Voting
2026_United_States_elections
Surgical procedures to affirm gender identity
to qualify for insurance coverage, certain insurance plans may require proof of the following: a written initial assessment by a qualified licensed mental
Gender-affirming_surgery
Ability of a computing system to simulate Turing machines
July 2010. Boyer, Robert S.; Moore, J. Strother (May 1983). A Mechanical Proof of the Turing Completeness of Pure Lisp (PDF) (Technical report). Institute
Turing_completeness
Mathematical methods
logic, realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them. Formulas
Realizability
Resistance to a malicious network attack
Proof of personhood (PoP) is a means of resisting malicious attacks on peer-to-peer networks, particularly attacks that use multiple fake identities,
Proof_of_personhood
Indian national identification number
has several concerns. First, Aadhaar is not a proof of citizenship and such a linkage will not filter non-citizens. Another problem is the Unique Identification
Aadhaar
Cryptography secured against quantum computers
Post-quantum cryptography (PQC), sometimes referred to as quantum-proof, quantum-safe, or quantum-resistant, is the development of cryptographic algorithms
Post-quantum_cryptography
American professor, attorney, author, and political columnist
published the New York Times bestselling book Proof of Collusion (Simon & Schuster), which sought to establish "proof of collusion in the Trump-Russia case."
Seth_Abramson
Theorem in algebraic geometry
relative homology then gives the theorem. Neither Lefschetz's proof nor Andreotti and Frankel's proof directly imply the Lefschetz hyperplane theorem for homotopy
Lefschetz_hyperplane_theorem
Inequality in Riemannian geometry
uniform inequality for arbitrary 2-complexes with non-free fundamental groups is available, whose proof relies on the Grushko decomposition theorem. see
Gromov's systolic inequality for essential manifolds
Gromov's_systolic_inequality_for_essential_manifolds
Ethnic group
Indian diaspora (ISO: Bhāratīya Pravāsī), officially Non-Resident Indians (NRIs) and People of Indian Origin (PIOs), are people of Indian descent who
Indian_diaspora
Mathematical inequality relating inner products and norms
Bunyakovsky (1859) and Hermann Schwarz (1888). Schwarz gave the modern proof of the integral version. The Cauchy–Schwarz inequality states that for all
Cauchy–Schwarz_inequality
Virtual marketplace in the darknet
market, a vendor may have undergone an application process via referral, proof of reputation from another market or given a cash deposit to the market
Darknet_market
Concept in computer science
failures even when only requiring the non triviality property. This result is sometimes called the FLP impossibility proof named after the authors Michael J
Consensus_(computer_science)
Mathematics award
Yuri I. Manin, with the first-ever IMU silver plaque in recognition of his proof of Fermat's Last Theorem. Don Zagier referred to the plaque as a "quantized
Fields_Medal
Axiom of set theory
a set that is non-measurable even for any rotation-invariant finitely additive measure, as shown by the Banach–Tarski paradox. A proof requiring the axiom
Axiom_of_choice
Mathematical theory of data types
and its proof requires a term of that type. An example of a non-constructive proof is proof by contradiction. The first step is assuming that x {\displaystyle
Type_theory
Diagram that shows all possible logical relations between a collection of sets
entirely contained within the dairy-product zone—there is no zone for (non-existent) non-dairy cheese. This means that as the number of contours increases
Venn_diagram
Islamist terrorist attack in Indian-administered Kashmir
in the Parliament of Pakistan, "We don't consider TRF illegal. Show us proof they carried out the Pahalgam attack. Show ownership by TRF. We won't accept
2025_Pahalgam_attack
Blockchain protocol
Ouroboros is a family of proof-of-stake consensus protocols used in the Cardano and Polkadot blockchains. It can run both permissionless and permissioned
Ouroboros_(protocol)
Gastrointestinal disorder
pain, postprandial fullness, and early satiety. FD was formerly known as non-ulcer dyspepsia, as opposed to "organic dyspepsia" with underlying conditions
Functional_dyspepsia
In mathematics, a non-algebraic number
Lindemann published the first complete proof that π is transcendental. He first proved that ea is transcendental if a is a non-zero algebraic number. Then, since
Transcendental_number
Person in a country not having citizenship
The Rights of Non-Citizens. 8 U.S.C. § 1481 ("Loss of nationality by native-born or naturalized citizen; voluntary action; burden of proof; presumptions")
Alien_(law)
Conjecture on zeros of the zeta function
showed that all non-trivial zeros must lie in the interior of the critical strip 0 < Re(s) < 1. This was a key step in their first proofs of the prime number
Riemann_hypothesis
Apparent lack of pattern or predictability in events
Fast Quantum Random Bit Generator Chaitin: Randomness and Mathematical Proof A Pseudorandom Number Sequence Test Program (Public Domain) Dictionary of
Randomness
American multinational software company
Cloud". Wired. ISSN 1059-1028. Retrieved June 21, 2024. "Adobe's Piracy-Proof Software Already Pirated". NBC News. June 21, 2013. Archived from the original
Adobe_Inc.
American columnist, author and lecturer (born 1946)
proof by contradiction, and imaginary numbers. Especially contested was Savant's statement that Wiles' proof should be rejected for its use of non-Euclidean
Marilyn_vos_Savant
cryptocurrency tumbler Via Masternodes containing 1000 DASH held as collateral for "Proof of Service". Through an automated voting mechanism, one Masternode is selected
List_of_cryptocurrencies
Model theory concept
that I(T,ℵ0) cannot be 2. It is easy to find examples where it is any given non-negative integer other than 2. Morley proved that if I(T,ℵ0) is infinite
Spectrum_of_a_theory
23 mathematical problems stated in 1900
Irrationality and transcendence of certain numbers. Problems of prime numbers. Proof of the most general law of reciprocity in any number field. Determination
Hilbert's_problems
Australian and American mathematician (born 1975)
and proof assistants in mathematical research; Quanta Magazine has described him as an evangelist for machine-assisted mathematics. He surveyed the area
Terence_Tao
Mathematical model of the physical space
school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the Elements
Euclidean_geometry
Application layer cryptocurrency protocol
to ecosystem development. Pebblecoin (XPB) employed a memory-intensive proof-of-work algorithm, and Quazarcoin (QCN) originated as a Bitmonero relaunch
CryptoNote
1941 book
an additional chapter on recent progress in mathematics, including the proofs of the four-color map theorem and Fermat's last theorem, written by Ian
What_Is_Mathematics?
Method of deriving conclusions
inference play a central role in proofs as explicit procedures for deriving new lines of a proof from preceding lines. Proofs involve a series of inferential
Rule_of_inference
couples are entitled for recognition as co-owners of property if there is proof of actual contribution, in a case involving the disputed sale of a house
2026_in_the_Philippines
Bitcoin adoption in El Salvador
that year, 100 days after the Bitcoin Law came into force, according to a survey done by the Central American University: 34.8% of the population had no
Bitcoin_in_El_Salvador
Kitchen cooking appliance
some of the microwave radiation enters through the transparent floor. In a non-flatbed unit there is either a glass tray shaped to fit the chamber, keeping
Microwave_oven
Characterization of how many integers are prime
much easier to prove. D.J. Newman gives a quick proof of the prime number theorem (PNT). The proof is "non-elementary" by virtue of relying on complex analysis
Prime_number_theorem
Rape of a single victim by two or more violators
are almost always premeditated in their intent, target victim(s), social proof, and psychological causes.[clarification needed] Social factors such as
Gang_rape
Australian animated web series
in the middle of 2022, Glitch released character trailers that served as proofs of concept testing the series' animation style and visuals. A teaser trailer
The_Amazing_Digital_Circus
Mathematical theory by Shinichi Mochizuki
Mochizuki's arguments a proof of a quantitative result on abc, which could in principle give a refutation of the proof.[non-primary source needed] Mochizuki
Inter-universal Teichmüller theory
Inter-universal_Teichmüller_theory
Widespread moral panic alleging abuse
controversial, including its definition, the source of the allegations and proof thereof, testimonies of alleged victims, and court cases and criminal investigations
Satanic_panic
Phenomenon in materials science
identify Form 2 RHCl by reference to a 29-peak IR spectrum... proof of infringement requires proof that the drug alleged to infringe would exhibit all of those
Disappearing_polymorph
travel, tourism, insurance
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